Positive Philosophy and The Automation of Reason · work. Two other divisions are the organon,...

Positive Philosophy and The Automation of Reason

Roger Bishop Jones

September 10, 2012

http://www.rbjones.com/rbjpub/www/books/ftd/ftdpam.pdfc© Roger Bishop Jones

http://www.rbjones.com/rbjpub/www/books/ftd/ftdpam.pdf

Contents

Contents 3

Preface 6

1 Introduction 11.1 The Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Themes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 The Organon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Historical Threads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Foundationalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 By Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 How to Read this Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 The Project 72.1 Leibniz, Characteristic and Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Carnap’s Programme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Information Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Alan Turing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 The Science of Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Automation and Intelligence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.4 Some Distinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 The Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Fundamental Dichotomies 133.1 Hume’s Fork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Relations of Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 Matters of Fact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.3 The Place of The Fork in Hume’s Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 A Contemporary Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Before Hume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.1 The Pre-Socratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.2 Two Kinds of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.3 Plato’s Theory of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.4 Aristotle’s Logic and Metaphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.5 Rationalist Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.6 British Empiricism Before Hume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.7 Leibniz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 After Hume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.1 A Broad Sketch of the Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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4 CONTENTS

3.4.2 Kant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.3 Bolzano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.4 Frege . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.5 Russell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.6 Wittgenstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.7 Tarski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.8 Carnap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.9 Quine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.10 Kripke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Analyticity and Analysis 224.1 Abstract Logical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 The Scope of Analytic Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Analytic Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.5 Analyticity in Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Computation and Deduction 265.1 Before the Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 The Status of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.4 Incompleteness and Recursion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.5 Computing Machinery and Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.6 Sound Computation as Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.7 Oracles and the Terminator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.8 Self Modifying Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Rigour, Scepticism and Positivism 326.1 Systematic Skepticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2 positivism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.3 Rudolf Carnap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.3.1 Tolerance, Pluralism, Metaphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.4 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 Epistemic Retreat 36

8 Language Planning 388.1 Languages, Notations and Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.2 Universalism and Pluralism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.3 On the Need for Synthetic Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.4 Logical Foundation Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.5 Interactive Theorem Provers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.6 Theory Hierarchy as Knowledge Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9 The Architecture of Knowledge 439.1 Requirements from Leibniz and Carnap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.2 Epistemic Retreat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10 Metaphysical Positivism 4510.1 First Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

CONTENTS 5

10.2 By Comparison with Logical Positivism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.3 Principal Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

11 Digging Deeper 4811.1 Foundations for Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.2 Logical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.3 Empirical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Bibliography 52

List of Tables 53

List of Positions 55

Index 55

Preface

[I really don’t have much clue what to do with thispreface. The plan for the preface is as follows. Everynow and then I will just throw it away and write an-other. Hopefully, sometime, I will write one that I like,and then keep it.]

This is the first of two volumes in which I hope to givea first account of a philosophical point of view which Icall Positive Philosophy. In this preface I describe thescope of the enterprise and the way in which it is dividedacross the two volumes, by reference to the divisions inthe philosophy of Aristotle.

To a first approximation the first volume is concernedwith theoretical philosophy, the second with practicalphilosophy. Of these, theoretical philosophy is con-cerned with knowledge, with logic, mathematics andempirical science, and practical philosophy with action,with ethics, politics and economics.

These two categories, the theoretical and the practi-cal, appear in Aristotle, but are not exhaustive of hiswork. Two other divisions are the organon, which con-sists of Aristotle’s writings on or related to logic, con-ceived rather more broadly than today, and productivescience, which involve arts and crafts, or more gener-ally, how to make or do things. In this two-fold divi-sion of positive philosophy, the first volume, theoreticalphilosophy, includes also logic. In addition to this aug-mented theoretical philosophy, and the second volumeon practical philosophy there is an important elementof what Aristotle might have thought productive phi-losophy, concerned with a kind of production which Ithink of primarily as engineering, albeit primarily soft-ware engineering.

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Chapter 1

Introduction

1.1 The Project

Three centuries ago the philosopher, logician, math-ematician, scientist and engineer Gottfried WilhelmLeibniz conceived, as a young man, a grand project,so far ahead of its time that no-one has yet come closeto realizing it. The principal elements his project were:

• A universal formal language in which all scientificknowledge might be expressed.

• An encyclopedia of science formalized in that lan-guage.

• A calculus or algorithm for answering question ex-pressed in that language.

• Mechanical calculators able to perform the neces-sary computations.

He devoted considerable energy to progressing theseideas. But his ideas were pie in the sky, there was nohope of success.

Since then many important developments have takenplace in philosophy, logic, mathematics and informationtechnology, and most of the now known impediments tothe realization of something close to his dream have nowbeen surmounted.

Today it is natural to think that computers solve themechanical side of the project, and that the rest of theproject is just software of various kinds. Though Leib-niz did work on the hardware, in his day mechanicalcalculators, his main interest was in what we would nowcall software, the algorithms and the data upon whichthey operated, and his approach to this was philosoph-ical and logical.

This is at once an information technology project andphilosophical research. To engineer a “cognitive agent”,to build something which has or can acquire knowledge,

and can reason from that knowledge to solve problems,can only be done in the context of a suitable philo-sophical framework. At its most abstract, architecturaldesign for cognitive artifacts is philosophy.

The kind of philosophy required to underpin such aproject is not piecemeal philosophical analysis, it is asystematic philosophical synthesis. It is the aim of thisbook to undertake such a synthesis, in such a way thatits relationship with the engineering of a certain kindof intelligent artifact is made clear. The engineeringenterprise I call here “the project”. The philosophy isintended to underpin that project and to constitute itsearliest most abstract stages.

It is therefore intended to present by stages together,

• a conception of an engineering project one of whosegoals is the automation of engineering design (theaims of the project),

• some relatively abstract ideas on how that projectcan be organized and implemented (some architec-tural design)

• a philosophical framework in the context of whichthe project aims and the proposed architecture canbe described and in which the reasons for believingthat the architecture might achieve the aims canbe articulated.

In this introduction I shall survey the structure ofthe exposition and sketch the principal themes.

It is in the nature of this project that it combinesmaterials from and seeks to interest devotees of diversedisciplines. The various themes discussed demand var-ied background, much of which I cannot hope to supply.Though the philosophical the logical and the technolog-ical aspects are intimately intertwined, it is hoped thatreaders with a particular competence and interest in

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2 CHAPTER 1. INTRODUCTION

one aspect will find most of the material of interest tohim intelligible without a complete mastery of the otheraspects of the presentation.

1.2 Themes

1.2.1 The Organon

The collection of those works of Aristotle which con-cerned logic is known as the organon. This word comesfrom the Greek word for a tool.

It has this name because logic was conceived by Aris-totle and by later scholars as providing a tool, ratherthan as a purely academic pursuit. The principal appli-cation of that tool was to be demonstrative science, thederivation of necessary truths in the various sciencesfrom the first principles of those sciences.

The emphasis on logic as a tool might well have beenunimportant through most of the history of logic, sinceuntil recently Aristotle’s organon has dominated thefield. However, though well intentioned, Aristotle’s for-mal logic is inadequate for any non-trivial scientificapplication, and the study of logic has remained theprovince of Philosophers.

With the advent of modern logic, beginning in thesecond half of the nineteenth century with mathe-maticians and philosophers such as Boole, Frege andPierce, there was spawned a new discipline of mathe-matics, mathematical logic, which was primarily meta-theoretical in character. Academic interest in logic nowspans multiple disciplines, of which the most impor-tant are philosophy, mathematics and computer sci-ence. There are various ways in which these disciplinesmake use of logic as a tool, but its use as a tool in themanner envisaged by Aristotle, as the means wherebyconclusions are drawn from various first principles, israre.

This book belongs to a line of philosophical works(notably, Aristotle, Leibniz and Carnap) of which theaim is to contribute to the means for the applicationof formal deductive methods in the establishment andapplication of knowledge. This work is subordinate tothat purpose.

The realization of that objective depends upona philosophical context which cannot be taken forgranted, or even taken from a shelf and dusted off. Overthe last half-century philosophy has moved in other di-rections, and in order to do so has undermined thephilosophical basis for the most recent manifestation

of the project in the philosophy of Rudolf Carnap. Fora resumption of the project, new philosophical founda-tions are required.

1.2.2 Historical Threads

The philosophical innovation required in support of theproject consists substantially in the re-establishment ofideas which have fallen into disrepute. This includesspecific ideas such as the analytic/synthetic dichotomy,and entire philosophical perspectives or systems suchas positivism. Their reestablishment, at least as viablealternatives to received opinion, will not be realized bya detailed refutation of the arguments which displacedthem.

In logic over the last 150 years and in Computingover the last 50 years there have been very many newdevelopments of a kind which one might expect to givephilosophers pause for thought. But new ideas in phi-losophy itself are very rare, most of the twists and turnsin philosophical fashion are revivals of old ideas in newclothes. It is in the nature of philosophical progress thatit often appears through millennia of debate as ideas areproposed, developed, disputed, rejected and perhaps ig-nored for a while before rising again re-engineered fora new philosophical climate.

If we seek afresh to understand and to advance suchideas, tracing their development through history maybe helpful in getting or in conveying an understandingof their contemporary manifestation.

Several historical threads serve I hope to illumi-nate aspects of the present proposal. The philosopherswhose ideas are touched upon in these sketches havespent a lifetime developing the ideas. The purpose ofthe sketches is to make the ideas presented here clearerby connecting them with their historical roots.

The first of these historical threads sketches the an-alytic/synthetic dichotomy and a variety of related di-chotomies and concepts. This is closely connected withthe development of ideas of logical truth, and of truthconditional semantics.

Alongside such questions relating to the establish-ment of meaning, there is the question of truth andhow it may be established or evaluated. Doubts aboutour ability to establish truths are at their most severein scepticism which part of the historical backgroundto and continuous with positivism, which combines ele-ments of scepticism with a strict attitude towards rigourin science. Our positivism departs from its predecessors

1.3. BY CHAPTER 3

substantially in ways which can be illuminated by con-sideration of those sceptical roots, and which make itnatural to think of as a graduated positive scepticism.This is closely connected with the question of rigour,and with the tension between rigour and progress, inscience, mathematics and philosophy.

Though there appears to be at any point in time atrade-off between rigour and productivity (which is per-haps easiest to see in mathematics), at times advancesare made which allow both to advance at once, anda new balance to be achieved. This happened in thenineteenth century for mathematics, a period of con-solidation in which standards of rigour in mathemati-cal analysis were transformed. The last stages in thistransformation involved the invention of modern meth-ods in logic, and established the possibility of the formalderivation of mathematics. The greater precision in lo-cating the most abstract subject matters of mathemat-ics, through the agency of axiomatic set theory, result-ing in achievements of high standards of rigour whichwere sustained through a century of continuous math-ematical innovation. These higher standards dependedonly peripherally on the new mathematical logic. Ax-iomatic set theory provided sufficient additional clarityto the definition of mathematical concepts, that stan-dards of rigour were able to advance without the adop-tion of formal derivation as the standard for mathemat-ical proof. It sufficed for a mathematical to convincehis peers that a formal derivation would be possible.

In the second half of the century, the applicationof digital computers to the support of formal nota-tions and deductive systems has created a new domainin which for the first time formal notations and lan-guages are extensively used. Stored program comput-ers demand and support the use of such formal lan-guages. Communicating unambiguously with comput-ers becomes a prime motivation for the use of formalnotations, which provide not only the motivation butalso the kinds of support which facilitate the use of for-mal languages.

1.2.3 Foundationalism

If we look for the most fundamental ideas in philoso-phy we find a inextricably interrelated complex of ideasbelonging to several distinct areas of philosophy.

Philosophically this work is primarily epistemologi-cal, as one might expect, and the philosophical systempresented gives central place to epistemology.

Epistemology is approached in what might perhapsbe described as an instrumental manner. Two as-pects of epistemology receive no attention. The firstis the meaning of the word “know”, and hence for somephilosophers the question of what knowledge em, is notconsidered. Second is any aspect of how knowledge isacquired or applied which is peculiarly human.

Instead of considering what knowledge is we are con-cerned with how knowledge might be represented andapplied.

The epistemology is foundational both in relation tological and empirical knowledge. The distinction be-tween these two, the logical and empirical, is a key-stone of the project and the philosophy. The projectrealizes an general analytic method a principal featureof which is the systematic and thorough separation ofthese two kinds of knowledge. Logical knowledge (un-der a broad conception of logical truth as analyticity)is represented formally and established by deductivelysound methods. Empirical knowledge is representedformally using abstract logical models. The relation-ship between abstract formal theories and the systemswhich they model is ultimately informal.

1.3 By Chapter

The Project Chapter 2.The first stage of this, in Chapter 2, is to give the

next stage of detail in describing the project, the aimsand the architecture. This is presented by starting withan account of Leibniz’s project, tracing the history ofthe ideas from the 17th through to the 21st century, andthen offering a revised conception of such a project forthe 21st.

The philosophical side of this history ends in debacle.The last major philosophical proponent of a significantfragment of the Leibnizian enterprise was Rudolf Car-nap in whose philosophy the formalization of sciencehad a central place. In mid 20th century, fundamentalphilosophical ideas with a key place in Carnap’s phi-losophy were repudiated by W.V.Quine. In particu-lar, the analytic/synthetic distinction, subject to con-tinuous critique by Quine since his first exposure toCarnap’s philosophy, was given a full blooded and un-compromising repudiation in Quine’s influential “TwoDogmas of Empiricism”. Though not aligning himselfwith Quine on the analytic/synthetic dichotomy SaulKripke then teased apart the triumvirate of conceptswhich had been identified by Carnap (analyticity, ne-


cessity and the a priori), allowing Kripke to inauguratea new kind of metaphysics, and set analytic philosophyon a new track fundamentally at odds with the philos-ophy of Carnap. Thus Kripke contributed to the sub-sequent widely held view that the philosophy of RudolfCarnap (sometimes known as Logical Positivism) hadbeen decisively refuted as a result of technical advancesby two of the most highly competent and respectedphilosopher-logicians of the period.

Fundamental Dichotomies Chapter 3.The philosophical framework I offer here is closer to

the philosophy of Carnap than to that of any otherphilosopher, and it is therefore necessary in Chapter3 to repair some of the damage done to this point ofview. Chapter 3 considers the status of these three di-chotomies, the distinctions between analytic and syn-thetic, between necessity and contingency, and that be-tween the a priori and the a posteriori. Though notalways with this vocabulary, similar distinctions havebeen talked of throughout almost the entire history ofwestern philosophy. Through this history one can seeboth a gradual refinement of these concepts and alsoreversals. I therefore lightly trace this history show-ing how Carnap’s understanding of these dichotomieswas reached, and then review and respond to some ofthe modern criticisms which are still held by many tobe decisive against Carnap. In this chapter I arguethat the refutation of Carnap on these matters is onemore demonstration of the contrast between the rigourof mathematics and that of philosophy. An illustration,I might say, of the irrationality of philosophy.

Analyticity and Analysis In Chapter 4 I then takethe concept of analyticity as established, and the ques-tion of its significance is examined in greater detail.In philosophy the twentieth century was called the ageof analysis, and the principal kind of philosophy pro-gressed by academics was called analytic philosophy. Inthe philosophy of Rudolf Carnap and the logical posi-tivists the connection between the concept of analytic-ity and analytic philosophy was simple. Insofar as phi-losophy was concerned with establishing the truth ofpropositions (in the manner in which mathematiciansestablish results by proving theorems, or science estab-lishes physical laws by observation and experiment) theresults of the kind of philosophical analysis envisagedby Carnap would be analytic, though in his hands suchresults play a secondary role to the articulation of meth-

ods and the definition of languages or concepts suitablefor science. For none of the many other conceptions ofphilosophical analysis which appear in the 20th centurywas there such a simple connection between analytic-ity and analysis. In this chapter we look at the rele-vance of analytic truth to various kinds of analysis, bothphilosophical and scientific. This is done using a kindof philosophical thought experiment. Suppose that wehad an oracle (man or machine) which could tell us ofany conjecture whether or not it was an analytic truth?What impact would that have on the various kinds ofanalysis under consideration?

Computation and Deduction Chapter 5.

With the concept of analyticity and hence of de-ductive soundness in place it is time to further refinemy characterization of the project, as being concerned,firstly, with deductively sound computation and ulti-mately with useful approximations to the terminator.These terms are explained in Chapter 5, and a varietyof contemporary research trends are compared with theresearch thus envisaged. Since the beginning of mathe-matical logic many different conceptions of have evolvedof proof and its relation to computation. The approachenvisaged here is clarified in the context of a generaldiscussion of these different conceptions of proof.

Rigour Skepticism And Positivism Chapter 6.

Carnap’s philosophy had been intended to provide away forward for philosophy to the achievement of stan-dards of rigour comparable to those of mathematics (anideal which has been held by many philosophers overthe last 2500 years), but his program had been defeated,illustrating just those defects that he sought to remedy.To reinstate this idea I sketch another historical threadin Chapter 6. This is a history of rigour in mathematicsand in philosophy. It is a history also of those philoso-phers who have perceived the rational deficit in philos-ophy, or in the search for knowledge more generally, ofthe sceptics of ancient Greece, and of the more moderntradition of positivism consisting of a kind of mitigatedor constructive scepticism in which high standards ofrigour are articulated for both a priori and a posteriorisciences.

Epistemic Retreat Chapter 7.

The epistemologically conservative aspects of posi-tivism give rise in this philosophy and architecture to

1.4. HOW TO READ THIS BOOK 5

the notion of “epistemic retreat”. This involves an ad-mission of general doubt, but the acceptance of degreesof doubt, and hence a partial ordering of conjecturesindicating in a relative way how well they have beenestablished, and what level of confidence they may beviewed.

Language Planning Chapter 8.

Carnap’s pluralism, a willingness to accept the use ofany well-defined language on a pragmatic basis, givesrise to the problem of “language planning”, address-ing for example the problems arising from the use ofmultiple different languages for different aspects of thesame problem. In Chapter 8 the architecture is furtherdeveloped by addressing these matters.

Carnap built on the purely mathematical founda-tional ideas of Frege and Russell, but sought to applythe new logical methods to the empirical sciences. Hebelieved that this required innovation on his part toadmit languages suitable for talking about the mate-rial world rather than purely about mathematics, andin making this transition he also moved from a purelyuniversalistic conception of logic (in which one languagesufficed) to a pluralistic conception of the language ofscience. His contribution to this pluralistic world wasby way of meta-theory, he spoke about how languagesmight be defined in their syntax, semantics and proofrules, taking this to be a proper philosophical contri-bution to the methodological advancement of science.The proliferation of languages thus envisaged would de-mand some kind of activity which he called “languageplanning”, but did little on.

The project I envisage embraces the pluralism of Car-nap, and therefore depends upon an architecture whichadmits multiple languages and permits large scale ap-plications involving more than one language. This con-nects with other initiatives in computing, notably theidea of an “Extensible Markup Language” (XML) andthe many related ideas which have built up around it,including the idea of a “semantic web”.

So far as applications to empirical science are con-cerned the project we outline does not adopt the ap-proach of Carnap. Instead we envisage that the projectprovides or empirical science (and ultimately of vari-ous engineering enterprises which depend upon it) byregarding these as working exclusively with abstractmodels of the physical world, and take the connectionbetween such models and the concrete world to be be-yond the scope of these formal methods.

Instead of asserting the truth of an empirical theorywe instead evaluate the contexts in which it provides auseful model of aspects of the real world, and evaluatethe model in different application domains in terms ofreliability and fidelity or precision. Carnap’s notion oflanguage planning (on which he himself said little), isone domain in which our project might best be com-pared with the W3C Semantic Web initiative.

The Architecture of Knowledge In chapter 9 wenow table an architectural proposal, in the form of a setof key requirements and a set of architectural featuresintended to realize those requirements, and a rationalefor the belief that they do indeed realize them.

Metaphysical Positivism With the architecturalproposal in place we return in chapter 10 to the phi-losophy.

This first involves gathering together a coherent androunded philosophy sufficient to underpin the proposedarchitecture, an important element of this is founda-tional. The second part concerns the methods sup-ported by the architecture and their scope of applica-bility.

Digging Deeper Metaphysical Positivism does notanswer all philosophical problems, but it does influencewhat might be considered a worthwhile philosophicalproblem for further investigation. Chapter 11 looks atsome of these.

Climbing Higher Where does this architecture takeus, why should it be implemented?

Chapter ?? is concerned with the outer reaches ofthe significance of the project, beyond the confines ofphilosophy or of academia.

1.4 How to Read this Book

Far be it for me to say how you, reader should proceed.However, here are some ideas, and some observationson how I have tried to write it which might be helpful.

I rarely myself read a book linearly from cover tocover. The book nominally addresses a very broadrange of potential readers, many of whom will be in-terested in only some aspects of its subject matter. Inwriting it I have therefore tried to make it possible forreaders to reach those parts which matter most to them


without having to struggle through too great a jungleof detail which might seem to them peripheral.

To this end I have tried to begin and end each chapterwith summary material which for some readers mightsuffice, and to include references as specific as possiblein the text to prior materials upon which an under-standing may depend.

I have felt it desirable, in order to make as clear aspossible the ideas which I present, to make use of storiesabout the history of various aspects of the subject mat-ters. Often the work of philosophers who have spent alifetime producing an important body of original workwill be spoken of in a few sentences which cannot be afair account of their work, even in some special corner.

In order to avoid misrepresentation I have used wher-ever possible the device of enunciating a position which,whether it was ever held by any philosopher or not, isuseful in making a point. One or more philosophers maybe named as having inspired this position, without go-ing into a detailed examination (which I am rarely bestequipped to undertake) of how closely it does corre-spond to the positions they in fact held.

This is a method not unrelated to that of the philoso-pher Saul Kripke in his examination of certain ideassuggested to him by the writings of Ludwig Wittgen-stein. The method may be adopted of connecting aphilosophical problem which is thought to be of interestin its own right with the history of the subject. Alter-natively, for those whose interest is primarily historicaland exegetical, the process of rational reconstructionmay begin in this way, with a definite model of someaspect of a philosophers work, which may cast light byevaluation of its similarities and differences with thetextual sources, and which may be refined in the lightof such comparisons into progressively more complexand subtle models less readily seen as diverging fromthe target of analysis.

In this work it is the former motivation which con-cerns us exclusively. The second, as a kind of analysisis of interest from a meta-theoretic point of view, butis not here practiced in anger.

Too rigorous an attempt to distill historical illustra-tions into hypothetical positions not directly attributedwould however be unduly cumbersome. This mode ofpresentation is reserved for the most important andsubstantial points, and much background is presentedas if historical fact, but should nevertheless be thoughtof in a similar manner, as so simple an account as couldat best be true in spirit, only to be dissipated on closer

inspection.

Chapter 2

The Project

It is my aim here devise a philosophical frameworksuitable to underpin a particular enterprise, the project.This might sound like a narrowly scoped philosophy,but “the project” is broad enough to encompass thewhole of mathematics, science and engineering whenundertaken by deductively sound methods. The projectis, in brief, the automation of deductive reason and itsapplications, undertaken in a particular manner whichI hope to make clear. The relationship between thatproject and the philosophical ideas which I offer as asupporting context is complex. A full and detailed ac-count of the project depends upon the philosophy, butthe project plays an important role in the articulationof the philosophy, they are hand-in-glove.

In this section I make a first attempt at describingthe project trying to minimize reference to the under-pinning philosophy. In subsequent chapters the philo-sophical issues are entered into in earnest, and in thisway it is hoped that aspects of the project are mademore clear.

The scope of the phrase “automation of deductivereason” is intended to include intelligent reasoning, butnot to encompass much of the aims or methods of thatfield known as artificial or machine intelligence.

Here and in subsequent sections explanations I usetry to make the ideas clearer partly by talking abouttheir historical evolution. In this section the connectionwith some of the pioneers and visionaries who have con-tributed to the formalization and automation of reasonin ways particularly relevant to out project are drawnout. The principle figures in this are Gottfried Wil-helm Leibniz and Rudolf Carnap, as exemplars of thephilosophers with similar projects, and Alan Turing isdiscussed for a contrast with the kind of automation ofreason I have in mind here.

2.1 Leibniz, Characteristic andCalculus

Leibniz was a rationalist, a philosopher who was partic-ularly concerned with what can be established by rea-son rather than observation. The distinction betweenthe two he recognized in the dictum:

“There are two kinds of truths, truths of rea-son and truths of fact.”

Despite recognizing this distinction, he perceivedthat all these truths might be codified formally, so thatthe distinction between truths and falsehoods could bedetermined by computation.

His vision was universalistic in several respects. Heenvisaged a universal characteristic, which was to be aformal logical language in which all knowledge might becodified, and a calculus ratiocinator , a method of cal-culating the truth value of any sentence in his universalcharacteristic.

These two ideas provide the first prototype for thiswork.

Leibniz did not imagine that the truth of scientifichypotheses could be established in this way, rather thatthe corpus of scientific knowledge could be encoded inthe universal characteristic, the calculus then settlingquestions in the context of that knowledge. In conse-quence, Leibniz also promoted the collaborative devel-opment of an encyclopedia, of scientific academies andjournals. Leibniz perhaps also guessed that the compu-tations involved might be non-trivial, and contributedalso to the development of the information technologyof his age, calculating machines.

Leibniz in these ideas was at least three centuriesahead of his time. The logic he knew was essentially

7

8 CHAPTER 2. THE PROJECT

that of Aristotle, and fell short of what is needed for theformalization of science by a wide margin. The neces-sary innovations in logic did not appear for another twohundred years. Beyond the adequacy of modern for-mal languages to express the entire corpus of scientificknowledge, there lie known limitations in the extent towhich formal deduction can capture truth in these sys-tems. Mechanical decision procedures are known notto exist, so for these reasons we know that no calcu-lus could be relied upon to settle all well formulatedproblems in the universal characteristic.

These limitations are perhaps of only marginal signif-icance for applications in science and engineering, buteven within these limitations there are serious prob-lems of computational intractability. For many or mostproblems whose answer is in principle computable, itwill not be accomplished in a reasonable timescale orthrough the deployment of available resources. Brutecomputation will therefore not suffice in any but themost simple cases, and something closer to the work-ings of human intelligence would be needed, to comeclose to an effective implementation of Leibniz’s calcu-lus.

The technology of calculating machines was even fur-ther from sufficiency. It remains moot whether today’sinformation technology is up to the task. Beyond theadequacy of the hardware, there are problems with thesoftware, potentially even less tractable.

Leibniz’s ideas in these matters had little influencein his own time or for centuries following, not just be-cause they were so far removed from what was thenrealizable, but also because they were not even pub-lished. His largest impact pressed matters in the oppo-site direction, and came from his independent inventionof the differential and integral calculi, and particularfor the notation which he devised for this epoch mak-ing mathematical innovation. The epoch thus spawnedwas two centuries of rapid development of mathematicsapplicable in science and mathematics. These enor-mous advantages took place despite the reduction con-sequent on the introduction of infinitesimal quantities.Right at the beginning of this period the philosopherBerkeley criticized these new developments, but it wastwo hundred years before mathematicians began to takethe problems of rigour seriously and the rigorization ofanalysis began, ultimately leading to the new devel-opments in logic which made it sufficient for Leibniz’sproject.

The rigorization of analysis was accomplished first bythe displacement of infinitesimals in a precise notion oflimit and by construction of real numbers from natu-ral numbers, effectively reducing analysis to arithmetic.This done, the next challenge taken up by Kettle’s Fregewas to show that arithmetic was simply a matter oflogic, requiring no special metaphysical insights. Inrising to this challenge Frege, inspired in part by theexample of Leibniz, re-invented logic in the form of hisBegriffsschrift or concept notation, which provided thebasis not merely for the logicisation of arithmetic, butalso potentially for Leibniz’s universal characteristic.

Frege’s project was more narrowly scoped, but eventhe formalization of arithmetic was an arduous under-taking, into which Frege had invested decades of in-dustry when Bertrand Russell pointed out to him theinconsistency of his basic principles. At about the timewhen Russell and Whitehead were completing their ownassault on the logicisation of mathematics (a substan-tial hurdle in which was ensuring against the kind of in-consistency found in Frege’s system) the young RudolfCarnap was attending as an undergraduate Frege’s lec-tures on his new logic and absorbing much of Frege’sattitude to rigour in deductive reasoning and its generalapplicability.

2.2 Carnap’s Programme

Carnap was first apprised of modern logic as an un-dergraduate by Frege’s lectures on his ‘Begriffsschrift”[Fre67, VH67]. At this stage in his development Car-nap clearly showed an interest both in philosophy andin mathematics and science. He was also strongly in-clined towards the precision of language which he foundin Frege’s logic which he contrasts with the teaching oflogic in philosophy. Within the sciences, he preferredphysics because of its greater conceptual clarity. Healso perceived the distinction between ethical, evalua-tive, emotive and metaphysical language and scientificdoctrine.

From this position as an undergraduate just beforethe great war he moves forward in a period of aboutseven years to the point at which we can see the for-mulation of the central ideas which motivated his workthrough the rest of his life. The most significant new in-fluence in forming these ideas came from Bertrand Rus-sell. Carnap became acquainted with Principia Mathe-matica[WR13] and began to prefer its notation to thatof Frege. He also begins to feel that a concept is not

2.2. CARNAP’S PROGRAMME 9

clearly understood until he can see how to express it insymbolic language.

At the end of 1921 Carnap read Russell’s OurKnowledge of the External World [Rus21] and was in-spired by Russell’s characterization of “logico-analyticmethod” in philosophy. It is at this point that Car-nap self-consciously devotes himself to this philosoph-ical method and begins intensive reading of Russell’swritings on the theory of knowledge and the methodol-ogy of science.

Russell, in his work on Principia Mathematica hadundertaken with Whitehead a task similar in characterto Frege’s project. However, Frege’s focus had been onthe logicisation of mathematics. Russell had a broaderconception of the applicability of the methods, and ad-vocated a kind of philosophy in which such methodswere used exclusively. Carnap took up this challengeenthusiastically.

Carnap’s philosophical programme therefore involvedfirst the idea that philosophy in general should be an-alytic in the specific sense that its methods and resultsare logical, and should be obtained by the new logi-cal methods pioneered by Frege and Russell. SecondlyCarnap conceived of the role of the philosopher as be-ing primarily concerned with facilitating the progressof empirical science. The kind of facilitation he had inmind was analogous to the innovations in mathemat-ics undertaken by Frege and Russell, the establishmentof new languages in which scientific laws could be pre-cisely enunciated and the surrounding theoretical andmethodological framework for the formalization of sci-ence.

The work of Frege and Russell had been universalis-tic in the sense that they sought a single new languagein which the whole of mathematics could be developed.At this stage Carnap’s thinking was along similar lines,but since he was trying to carry forward the ideas intoempirical science, it did not seem feasible to stick withthe same purely logical systems. Carnap’s aims at thispoint may therefore be thought similar to Leibniz’s de-sire for a universal characteristic in which all of sciencecould be formalized. Later Carnap became more plu-ralistic, but the idea of using formal logic to encodescientific knowledge and of formalizing deductive rea-soning about science places his enterprise within thescope of Leibniz’s.

Like Leibniz, Carnap’s ideas were ahead of their time.He was working in the context of modern symbolic logic,which is (I shall argue), no longer shackled by the lim-

itations of Aristotelian syllogistic. But strict formalityin language and proof is arduous, and good mechan-ical support a prerequisite to widespread adoption ofsuch methods. Principia Mathematica formalized a sig-nificant part of mathematics, and was very influentialduring the formative years of the new academic disci-pline of mathematical logic. These new developmentsdid have a significant impact on Mathematics, whichwas made more rigorous through the systematic use ofaxiomatic set theory. But mathematicians did not fol-low the example of Russell and Whitehead by taking upstrictly formal notations or formal proofs. Not even inmathematical logic did this happen, for mathematicallogic became a meta-theoretic discipline in which for-mal systems were studied rather than applied. Proofsin mathematical logic may often be concerned with for-mal systems, but they are not themselves formal.

Not only was formalization rejected by mathemat-ics and science, it was soon to be rejected, togetherwith Carnap’s entire philosophical outlook, by philoso-phy. The first major assault on Carnap’s position wasby Quine, who had been sniping at some of the coredoctrines, notably the concept of analyticity, ever sincehis first exposure to Carnap’s philosophy as a Harvardpostgraduate visiting Carnap in Vienna. Quine’s cri-tique modulated into outright repudiation of the cen-tral tenets of Carnap’s philosophy in the his “Two Dog-mas of Empiricism”. It was not long before an attackcame on another flank from Saul Kripke, who disman-tled the identification of analyticity and necessity whichwere inter-definable in Carnap’s conceptual scheme andthereby invented a new kind of metaphysics.

The idea of Russell and Carnap, that philosophyshould consist of logical analysis and of Carnap thatit should also be conducted formally both disappeared.Symbolic or mathematical logic does play a substantialrole in analytic philosophy, but it is the meta-theoretictechniques of mathematical logic which are used in phi-losophy. There is no general adoption of formality as away of achieving rigour in philosophy, no general recog-nition that there is a deficit in rigour might requiressuch a remedy.

Another impediment to the realization of Leibniz’sproject was, during Carnap’s lifetime, beginning to bedissolved. The digital electronic computer was inventedand the first steps toward using this technology for theautomation of deduction were in progress. The torchwas about to pass from philosophy, not to mathematicallogic, but to Computer Science.


2.3 Information Technology

These two philosophers strove to realize ambitionswhich were soon to be made more feasible by the inven-tion of the stored program digital computer, and with itanother academic discipline, Computer Science, whichover the second half of the 20th century would conductan enormous amount of research relevant to both oftheir projects. The first home of logic, Philosophy, andthe new discipline of mathematical logic, treated logicin a meta-theoretically. They took formal logic primar-ily as something to be studied rather than directly used(by contrast with the earlier work by Frege and Rus-sell and Whitehead in which large parts of mathematicswere formally derived). Computer Science, as well asconducting theoretical and meta-theoretical investiga-tions relevant to computing, found reason to undertakeproofs formally, with the assistance of their computingmachinery.

For these reasons, the primary academic locus of re-search relevant to the ambitions of Leibniz and Carnapmoved from Philosophy to Computer Science. How-ever, the idea of using computers for science was morepredominantly pursued by purely computational ratherthan formally logical methods. I hope to make thisdistinction clearer, to make the case that logical meth-ods deserve to be given special consideration, to give aphilosophical context in which it make sense to do thaton a substantial scale, and to consider some aspects ofhow this might be progressed.

2.3.1 Alan Turing

Alan Turing was a mathematical logician, computa-tional engineer and a visionary thinker whose ideas onartificial intelligence have been influential on researchin the automation of reasoning.

For our present purposes it is primarily his writing onartificial intelligence which is of interest, as encompass-ing objectives even broader than our own, but Turingwas an important figure in a milestone in our under-standing of computation which was reached during the1930s.

One of the tenets of the philosopher and mathemati-cian David Hilbert in the early part of the 20th centurywas that all definite mathematical problems are sus-ceptible of solution. This thesis was held to be equiva-lent to the effective decidability of validity in first orderpredicate logic, which was called the entscheidungsprob-lem. To make the problem precise enough to be given

a definitive mathematical answer it was necessary tomake precise the notion of effective calculability. Thisresulted in several different logicians putting forwarddifferent notations in which arbitrary algorithms (meth-ods of solution) could be expressed. Church came upwith concise notation for functional abstraction knownas “the lambda calculus”, Kleene with a system fordefining numerical functions by recursion, Emil Postcame up with a system based on transforming stringsusing a set of “productions” which defined transforma-tions on the strings. All these systems provided ways ofdefining effectively computable functions over the nat-ural numbers, but the one which looked most like a de-scription of a computing machine was that invented byAlan Turing, since called the Turing machine [Tur36].

These different ways of describing effective computa-tional processes all turned out to be similarly expres-sive, and this remarkable discovery underpinned theidea that they did indeed all capture the notion of ef-fective calculability. The entscheidungsproblem havingby these means been made more definite, Church andTuring independently solved the problem by exhibit-ing problems which were provably unsolvable (by anyeffective procedure).

This result limits what could possibly be achieved byLeibniz’s calculus ratiocinator, as did Godel’s previousresult on the incompleteness of arithmetic. It seemsfrom Leibniz’s description that he probably did imag-ine that any definite question within the scope of thescientific knowledge codified in the lingua characteris-tica would be answerable using his calculus and a suffi-ciently rapid calculator. The Godel and Church-Turingresults show that this cannot be the case. I will looka little closer later at these and other limiting factorsand consider their significance for “the project”.

Turing also wrote on Artificial Intelligence, and hisconception of Artificial Intelligence provides an alter-native ideal for the automation of reason which is notpredicated on machines achieving the impossible. Hisseminal paper in this area was Computing Machin-ery and Intelligence[Tur50]. Turing’s essential ideawas that human intelligence, though very differentlyimplemented, is not fundamentally distinct from ormore capable than the kind of information processingwhich could be undertaken by a sufficiently complexand powerful Turing machine. In addressing the ques-tion whether a machine can think, Turing describedan “imitation game” to make this question more pre-cise, this later became known as the Turing test. The

2.3. INFORMATION TECHNOLOGY 11

Turing test involves human beings interacting with aperson and a machine through similar interfaces whichconceals which of the two was involved, i.e. using akeyboard to converse with someone or something in adistant or private place. If the observer cannot tell thedifference between the man and the machine when heinteracts with it in this way, then perhaps we shouldbe ready to acknowledge that the machine thinks, andhence exhibits some important aspects of human intel-ligence.

2.3.2 The Science of Computing

During the first half of the 20th century the technol-ogy of computation had moved rapidly. The Turingmachine provided a very simple abstract prototype foran important transition, from special purpose to gen-eral purpose computational engines. It anticipated thegeneral purpose stored program digital computer, andthese relatively complex calculating machines were justbecoming technically feasible. They first appeared inthe 1940s using technologies such as electromagneticrelays, and the electronic vacuum tube. Soon the tran-sistor emerged, and the technology of computation be-gan a long period of progressive miniaturization yield-ing ever smaller and faster building blocks for digitalcomputers.

With the invention of the digital computer came thenew academic discipline of Computer Science, and en-thusiasm for formal systems and the automation of rea-son passed from Philosophy and Mathematical Logic(itself just an infant) to Computer Science. Computerscientists were interested in formal languages becausethey allowed algorithms to be described precisely forexecution by a computer. The complexity of these al-gorithms rapidly increased and correctness became anissue. How could one be sure that the instructions forachieving some computational objective would realizetheir intended aim? Since formally defined algorithmscould be construed as mathematical entities, it seemsthat certainty might best be realized by a mathemati-cal (i.e. a deductive) proof. Unfortunately these proofsturned out themselves to be even more complex thanthe programs whose correctness they were intended toshow, and the questions then arose how one could facili-tate the construction of these proofs and how one couldbe sure that the proofs were correct? The computer it-self provided one answer to these questions. The com-puter could assist in the construction of the proofs, and

since it is in the nature of formal proofs that their cor-rectness can be checked mechanically, they could alsocheck the proofs. In this way, that part of ComputerScience which was particularly interested in ensuringthat computer programs are correct was drawn into thethe use of formal notations, formal deductive systemsand the automation of reason.

Meanwhile another branch of Computer Science de-veloped which sought to program computers so thatinstead of merely doing large scale drudgery, they ex-hibited some kind of intelligence. This was the field ofArtificial Intelligence, and later the related interdisci-plinary enterprise of Cognitive Science. Within thesedisciplines a variety of approaches to the problem werepursued, but despite this variety a single importantcleavage was captured by the contrast between “neats”and “scruffs”. The project around which this book isconstructed may be thought of leading towards an ex-tremely “neat” AI, so this distinction can be helpful ininitial sketches of the project.

The distinction between neats and scruffs may beseen through the contrast between “connectionist” AIand methods based on theorem proving. The connec-tionist approach to AI approximates the human brainby simulation of a neural network using highly paral-lel processing. The idea is to devise a network whichis capable of learning and then to teach it the requiredskills. If such a system achieved a competence in a com-plex deductive science it would do so indirectly. Suchcompetence would be an application of its general in-telligence, not a source or means of attaining that in-telligence.

“neats”, on the other hand, think more like Leib-niz. For them the intelligent capabilities are build on afoundation which includes a formal language for repre-senting propositional knowledge and a deductive infer-ential capability. This does not necessarily mean thatthey are aimed at different problem domains to the“scruffs”. Neats may seek systems capable of ordinarylanguage discourse and common sense reasoning, butstill approach this on logical foundations which wouldbe unfamiliar to most people.

The neat/scruffy distinction is concerned with meansrather than ends. There is a related distinction whichconcerns ends rather than means. This is the distinc-tion between taking the aim of a research programmeto be the replication of some aspect (or the whole of)human intelligence, of whether some other characteri-zation, independent of the methods or competence of


human beings, is give of the objective of the research.

2.3.3 Automation and Intelligence

We arrive then at my first characterization of theproject which this philosophical dissertation seeks tounderpin.

The aim of the project is a comprehensive formaliza-tion of the deductive aspects of philosophy, mathemat-ics, empirical science and engineering design.

2.3.4 Some Distinctions

Developments since Leibniz and Carnap give us betterinsights into the extent to which their projects are real-izable. The diversity of related research which has beenundertaken enables us to discriminate many differentsimilar kinds of project and to identify more preciselythe kind of projects which Leibniz and Carnap mighthave entertained if they were working today.

2.4 The Philosophy

Here are the bare bones of a philosophical frameworkto underpin “the project”.

The philosophy is positivistic, insofar as it delineatesa particular approach to empirical science and its ap-plication, which it is suggested may in some ways bepreferable to extant methods. In this I am not beingprescriptive, I do not assert that science should be con-ducted in this way.

The framework is, in the first instance, conceptual.This consists to a large extent in the choice of particularmeanings for terms which have a long history of shiftingusage. Some of the most fundamental of these come inpairs as dichotomies, i.e. disjoint concepts which to-gether exhaust some significant larger domain. Themost fundamental of these is the distinction betweenlogical and empirical truths. Simple though it mightappear to be, it is a source of endless philosophical con-troversy, and it will therefore be necessary to considerin detail the evolution of this distinction over its entirehistory and come to as precise an understanding of thisfundamental cleavage as possible.

Such fundamental concepts are not mere accidentsof language, and are one element of the philosophicalframework which we may think constitute a kind ofmetaphysics. 1

1On this my disagreement with Carnap lies only in the scope

of applicability of the term metaphysics. It does not constitutemetaphysics under either of the two principle categories whichconstitute metaphysics for Carnap. I take great care to make themeaning of these concepts definite and clear, so meaninglessness,I hope, is avoided. Nor does this kind of metaphysics constitute,in our framework, a priori justifications about synthetic claims.

Chapter 3

Fundamental Dichotomies

[In this chapter, as in all chapters which are primar-ily historical, it is particularly important not to slip intotoo purely chronological an account. All the life is inthe particular themes which the narrative is intended toilluminate, and it is the development of these themeswhich must be at all times in the foreground. I do notknow how to achieve this. Part of the difficulty in pro-ducing the story is that the detailed content of thesethemes has to be well understood in order to find a goodway of presenting them, but one feels that the requiredlevel of understanding can only be extracted from thehistory. ]

We saw in the last chapter how the most recent suc-cessor to Leibniz’s project, in the philosophy of RudolfCarnap, was derailed by the rejection of some of themost fundamental tenets of Carnap’s philosophy.

It is conceivable that the project could be revived ona different basis, for neither Quine nor Kripke, despitetheir devastating critiques, actually abandoned the ideaof deductive reason or the use of formal deductive sys-tems. If I was myself convinced of the soundness ofthose criticisms then I might follow that course, but Iam not. It is best instead to answer them.

In this chapter I therefore focus on the concept of an-alyticity and its relation to those of necessity and of thea priori. Before addressing specifically the criticisms ofCarnap’s position by Quine and Kripke, I propose tosketch the history of the evolution of these conceptsover the past two and a half millennia. I will presentthis as falling in two principal stages, first the devel-opment of predecessors of these fundamental conceptsand then the subsequent refinement of our understand-ing of the concept. The point of transition I suggest,is with Hume, though this is more an expository de-vice than a dogmatic claim. The idea that there existsany such definite point of transition is tenuous, but the

supposition helps to give structure to the narrative.It is with Hume, I suggest, that we find a first account

of the dichotomy which is not easily seen to be in somerespects defective. This is perhaps as much to do withHume not seeing himself as defining the distinction, butrather as drawing attention to it and placing it in acentral place in his philosophy.

Before entering into the history I will sketch the tech-nical content of the narrative, to provide a frameworkin which the evolving detail can be placed. This is doneusing the ideas of Hume.

3.1 Hume’s Fork

David Hume was a philosopher of the Scottish Enlight-enment. The enlightenment was a period of ascendencyin the place of reason in the discussion of human affairs,when science had secured its independence from the au-thority of church and state and had a new confidencein its powers derived substantially from the successesof Newtonian physics.

Hume looked upon the philosophical writings of hiscontemporaries and found in them two principal kinds,an “easy” kind which appealed to the sentiments of thereader, and a “hard” kind which trawled deeper and ap-pealed to reason. This latter kind, “commonly called”metaphysics, was preferred by Hume, but found nev-ertheless, by him, to be lacking, infested with religiousfears and prejudices. Hume’s feelings about these as-pects of philosophy were not vague misgivings. He hada specific epistemological criterion which he saw thesephilosophical doctrines as violating.

Hume’s project involves an enquiry into the nature ofhuman reason for the purpose of eliminating those partsof metaphysics which go beyond the limits of knowl-edge, and establishing a new metaphysics on a solid

13

14 CHAPTER 3. FUNDAMENTAL DICHOTOMIES

foundation limited to those matters which fall withinthe scope of human understanding.

David Hume wrote his philosophical magnum opus, ATreatise on Human Nature [Hum39] as a young man.He was disappointed to find his work largely ignoredand otherwise misunderstood, and thought perhapsthat his presentation had been at fault. To improvematters he wrote a much shorter work more tightly fo-cussed on the core messages which he thought of great-est importance, which he called An Enquiry into Hu-man Understanding [Hum48].

In a central place both logically and physically in thismore concise account of his philosophy he says:

“ALL the objects of human reason or enquirymay naturally be divided into two kinds, towit, Relations of Ideas, and Matters of Fact.”

We shall see that Hume is here identifying a singledichotomy which corresponds to all three of the distinc-tions which here concern us. In his next two paragraphshe expands in turn on the kinds he has thus introduced.

3.1.1 Relations of Ideas

“Of the first kind are the sciences of Geom-etry, Algebra, and Arithmetic; and in short,every affirmation which is either intuitivelyor demonstratively certain. That the squareof the hypotenuse is equal to the square ofthe two sides, is a proposition which ex-presses a relation between these figures. Thatthree times five is equal to the half of thirty,expresses a relation between these numbers.Propositions of this kind are discoverable bythe mere operation of thought, without de-pendence on what is anywhere existent in theuniverse. Though there never were a circleor triangle in nature, the truths demonstratedby Euclid would for ever retain their certaintyand evidence.”

Hume is distinctive here among empiricist philoso-phers in having a broad conception of the a priori(though he does not use that term here), allowing no-table the whole of mathematics. In this he may becontrasted for example with Locke who allowed onlycertain rather trivial logical truths to be knowable apriori. Nevertheless, Hume’s conception of the a pri-ori remains narrow by comparison with the rational-

ists, and in particular, as Hume will later emphasize,excludes metaphysics.

3.1.2 Matters of Fact

“Matters of fact, which are the second ob-jects of human reason, are not ascertained inthe same manner; nor is our evidence of theirtruth, however great, of a like nature with theforegoing. The contrary of every matter offact is still possible; because it can never im-ply a contradiction, and is conceived by themind with the same facility and distinctness,as if ever so conformable to reality. That thesun will not rise to-morrow is no less intelligi-ble a proposition, and implies no more con-tradiction than the affirmation, that it willrise. We should in vain, therefore, attemptto demonstrate its falsehood. Were it demon-stratively false, it would imply a contradic-tion, and could never be distinctly conceivedby the mind.”

The evolution of following three dichotomies are thesubject matter, though we will find other related di-chotomies which feature in the history.

The terms which I will use to speak of them, in thischapter are:

• analytic/synthetic

• necessary/contingent

• a priori/a posteriori

As I shall use these terms these are divisions of dif-ferent kinds of entity, by different means.

The first is a division of sentences, understood in suf-ficient context to have a definite meaning, and is a di-vision dependent upon that meaning.

The second is a division of propositions, which maybe understood for present purposes as meanings of sen-tences in context. The division is made according towhether the proposition expressed must under all cir-cumstances have the same truth value, or whether itstruth value varies according to circumstance. In thiswe are concerned with two particular notions of neces-sity, those of logical and of metaphysical necessity, thelatter being sometimes taken to be broader than theformer. A part of the role of Hume’s fork in positivistphilosophy is to banish metaphysical necessity insofaras this goes beyond logical necessity.

3.2. A CONTEMPORARY PERSPECTIVE 15

The third is for our purposes also a division of propo-sitions, on a different basis. It concerns the status ofclaims or of supposed knowledge of propositions. It isexpected that such a claim must in some way be justifiedif we are to accept it, and that the kind of justificationrequired depends upon the proposition to be justified.The justification is a priori if it makes no reference toobservations about the state of the world.

3.1.3 The Place of The Fork in Hume’sPhilosophy

The mere statement of the fork (which we shall see,is not original in Hume) is of lesser significance thanthe role which it plays in Hume’s philosophy, whichserves to clarify the distinction at stake and draw outits significance.

Hume’s philosophy, like Descartes’ comes in two partsof which the first is sceptical in character, and the sec-ond constructive. In both cases the sceptical part clearsthe ground for a new approach to philosophy which isthen adopted in the constructive phase.

For our present purposes we are concerned princi-pally with the first sceptical phase of Hume’s philoso-phy, because of the delineation of the scope of deductivereason, and hence of the analytic/synthetic dichotomywhich is found in Hume’s sceptical arguments. This de-lineation is baldly stated in Hume’s first description ofthe distinction between “relations between ideas” and“matters of fact”, for there Hume tells us that no mat-ter of fact is demonstrable.

This bald statement would by itself have little per-suasive force if it were not followed up with more detail,even though ultimately this detail does not so much un-derpin the distinction as depend upon it.

Hume’s further discussion begins with the considera-tion of what matters of fact can be known ‘beyond thepresent testimony of our senses or the records of ourmemory’. The inference beyond this immediate datais invariable causal, we infer from the sensory impres-sions or memories to the supposed causes of those im-pressions. But these are not logical inferences, causalnecessity is for Hume no necessity at all (even less theinference from effect to cause). Hume’s central thesisthat matters of fact are not demonstrable is in this wayreduced first to the logical independence of cause andeffect, and then to the distinction between deductive(and hence sound) inference and inductive inference(whereby we infer causal regularities and their conse-

quences).Given that Hume considers all inferences from senses

to be based on induction, and sees no validity in causalinference, it follows that from information provided di-rectly to us by the senses nothing further can be de-duced which is not simply a restatement, selection orsummary of the information itself. Further enlighten-ment from this sceptical doctrine is primarily the appli-cation of this principle to various kinds of knowledge.In the process Hume does a certain amount of

3.2 A Contemporary Perspec-tive

In tracing the history of the analytic/synthetic and re-lated distinctions I hope to show how their various his-torical manifestations relate, and to present a perspec-tive from which the various developments may be eval-uated perhaps as plain progress, perhaps as advancesin certain respects, perhaps as involving or constitutingregress.

Such evaluations can only be made from a particularpoint of view, and in this section that point of view isoutlined, and related, in the first instance to Hume’sfork.

3.3 Before Hume

The distinctive place of Hume in the history of “thefork” which bears his name is not a mark of his pri-ority in making the distinction, but rather of an im-portant point in its development, a clear identificationof exactly the right distinction (I now suggest) of thecentral place which he gave it in his philosophy, and ofthe evidence he gives (in what is are now regarded asHume’s various sceptical doctrines) of a sound intuitivegrasp of the scope and limitations of logical truth anddeductive inference.

To underpin the significance of Hume’s distinction,and to see that, notwithstanding its apparent simplic-ity, it is by no means simple to arrive at, I now look atsome of the most important stages in the earlier devel-opment of the ideas.

—————————The philosophers I will consider here are Socrates,

Plato, Aristotle, Descartes, Locke and Leibniz.Most of the elements which we find in Hume’s fork

can be found in the earliest of these philosophers.


The relevant ideas which these philosophers discussedinclude Plato’s world of ideals and of appearances, thenotions of essential and accidental predication, and ofnecessary and contingent truth, and Aristotle’s logicand the idea of demonstrative truth. Leibniz is impor-tant for his conception of the scope of logic and thepossibility of arithmetisation and mechanisation. Hisintended method for these depends upon a conceptualatomism which was to exert significant influence on thephilosophy of Russell and of the Early Wittgenstein,and thus indirectly upon Carnap’s conception of logi-cal truth.

The distinction between the logical and the empiricalinfluenced two major tendencies in modern philosophy,namely rationalism and empiricism of which Leibnizrepresented the first and Hume the second. The distinc-tive feature of these tendencies has been respectivelythe overestimation and underestimation of the scope ofdeduction, championed by metaphysically and scientif-ically inclined philosophers (or philosophically inclinedscientists). The dialogue between these has thereforeserved to refine the distinction between the logical andthe empirical. These two tendencies may be seen tohave been anticipated by Plato and Aristotle, of whomPlato seems like the rationalist, and Aristotle is closerto empiricism. Curiously the connections with Leibnizand Hume are crossed over, it is Plato the rationalistwho seems to provide the clearer anticipation of Hume’sfork, and it is Aristotle the less rationalistic of the twowho provides the logic on which Leibniz’s ideas werebuilt.

Hume’s principle description of his fork is in termsof subject matter, the distinction between knowledge ofideas and knowledge of the world. In Plato we see some-thing quite similar, he distinguishes between the eter-nally stable world of Platonic ideals, of which we canhave certain knowledge obtained by reason, and a worldof appearances, in constant flux, and of whose fleetingnature we can at best have tentative opion based on theunreliable testimony of our senses. Not only does thedistinction of subject matter match, but the key charac-teristics of these domains, that in the one we have solidprecise, reliable knowledge, and in the other, nothingof the kind, are agreed between them.

For an understanding of the philosophy of Socratesand Plato it may be helpful to consider first the pre-Socratic philosophers.

3.3.1 The Pre-Socratics

The history of the analytic/synthetic dichotomy is con-nected with that of deductive inference, which is gen-erally held to have begun with Greek mathematics.

It might be argued that the ability to undertake ele-mentary deductions is an essential part of competencein a descriptive language. For to know the meaning ofconcepts, one must also know at least the more obvi-ous cases of conceptual inclusion. One cannot be saidto know the meaning of the word “mammal” if onedoes not know that all mammals are animals, and hencethat any general characteristic of animals is possessedby mammals. The ability to draw inferences purelybased on an understanding of language does not how-ever entail the capability to discriminate between suchdeductive inferences and inferences which are based notmerely on meanings but also on supposed facts, so noclear grasp of what inferences are deductive need beinvolved. Furthermore, competence in drawing con-clusions may by entirely unwitting, there may be notawareness of the idea of inference.

It is in Ionia1in the 6th century BC that mathematicswas transformed into a theoretical discipline, a key fea-ture of which is the practice of proving general math-ematical laws (rather than simply recording them forgeneral use). At this time philosophy, the love of knowl-edge, encompassed mathematics and science as well aswhat we might now call philosophy. The use of reasonin mathematics was very successful, and Greek math-ematics grew into a substantial body of rigorously de-rived, knowledge of which much is known to us throughthe thirteen books of the Elements of Euclid (c300BC). In Euclid’s geometry, developed deductively usingan explicitly documented deductive axiomatic method ,Greek mathematics achieved a standard of rigour whichwas not to be surpassed for two thousand years.

The success of deduction in mathematics was not re-flected in other aspects of the work of the pre-Socraticphilosophers. It is distinctive of Greek philosophy, be-ginning with the Ionians, that religion and myth wasnot a dominant influence, and that philosophers soughtknowledge by rational means, rather than deferringto any kind of authority. Ionian philosophers, oftendescribed as cosmologists, sought unifying principles,as a way of understanding the diversity of the worldaround them. Different philosophers adopted differenthypothesis about the ultimate constituents of the uni-

1A part of classical Greece now belonging to coastal Anatolia,itself a part of Turkey.

3.3. BEFORE HUME 17

verse. Thales held that the world originated from wa-ter, Anaximenes from air. Empedocles held that allmatter was formed from air, water, earth and fire. Bycontrast with mathematics, Greek philosophy becamea great diversity of irreconcilable doctrines.

The philosophical search for unity and stability un-derlying apparent diversity and flux yielded no univer-sally accepted result. This is nowhere more conspic-uous than in the philosophies of Heraclitus and Pare-menides, the first asserting an external flux in whichnothing remained stable and the second that nothingchanges. The failure of reason to resolve differences inthis domain of enquiry was underlined by the argumentsof Zeno of Elea, who devised many paradoxical argu-ments2to reduce to absurdity the possibility of change,in support of the philosophy of Parmenides. In thisway it is made conspicuous that apparently the samemethod, reason, works very well for mathematics butfails miserably in metaphysics.

3.3.2 Two Kinds of Stability

We can understand the search for the pre-Socraticphilosophers for the unity and stability underlying thediversity of the world by analogy either with modernscience or with mathematics.

In the case of science we seek physical laws whichgovern the changes which take place in the world. Theworld changes, but the laws are immutable. In modernscience the preferred way of formulating scientific lawsis using mathematics. The scientific hypothesis is thatthe relevant aspects of the world correspond more orless exactly with the structure of some mathematicalmodel.

The unchanging fundamental truths may thereforebe sought either in empirical scientific laws or in math-ematics.

These two alternatives connect with tendencies inmodern philosophy, empiricism and rationalism, and toaspects of the two great philosophical systems of classi-cal Greece, those of Plato and Aristotle. The systems ofPlato and Aristotle in different ways anticipate Hume’sfork, and in the rationalist and empiricist tendencies wecan see a dialectic on the scope of deduction throughwhich the distinction evolved towards Hume’s formula-tion of the fork.

2Of which the best known that of Achilles and the Tortoise.

3.3.3 Plato’s Theory of Ideals

Plato sought to place philosophy on as rigorous a foot-ing as mathematics, and he did this with a synthe-sis of the ideas of Heraclitus and Parmenides whichrecognized two distinct domains of rational discourse.Mathematics succeeded because it reasoned deductivelyabout abstract ideas

In Plato Hume’s fork is anticipated as a distinctionbetween two worlds, the world of ideal forms and theworld of appearances. Thought of in this way the dis-tinction is about subject matter,

Distinction 3.1 the world of forms and world of ap-pearances

Distinction 3.2 the a priori (knowledge) and a pos-teriori (opinion)

Distinction 3.3 essential and accidental predication

Connection 3.4 definition and essence

3.3.4 Aristotle’s Logic and Metaphysics

3.3.5 Rationalist Philosophy

3.3.6 British Empiricism Before Hume

Hume was one of a line of British philosophers whoemphasized the role of experience in the acquisition ofknowledge. Important figures in this tradition were Ba-con, Hobbes and Locke.

3.3.7 Leibniz

Leibniz conceived of the project to which this book isdevoted. We are concerned here with just one aspect ofhis work, which is his contribution to the ideas leadingto Hume’s fork.

Through most of the intellectual history right up tothe 20th century Aristotle is a dominating influence onthinking about all aspects of Hume’s fork. Some of theinfluence of Aristotle is negative, his subject-predicateanalysis of propositions and his syllogistic conceptionof logic were both to narrow and adherence to Aristo-tle’s ideas may have inhibited the development of logicsufficient for our purposes.


Leibniz conceived of the idea that reason might beautomated as a young man, and pursued this projectfor the rest of his life. His insight was specifically abouthow the Aristotelian syllogism could be automatedthrough arithmetisation, thus anticipating a methodwhich was to become famous in logic when applied byGodel in his incompleteness results.

Within this context the logical and metaphysicalideas which underpinned and provided technical sub-stance to his project are important.

Like Aristotle, Leibniz did have the concepts of Nec-essary and Contingent truth, he also had the conceptsof a priori and a posteriori truth which were closelyconnected, as in Hume. There is also a connection withsemantics, similar to that in Aristotle, through theirrole in a priori proof and hence in the establishment ofnecessary truths. To this is added a precise descriptionof a mechanizable process of analysis whereby neces-sary truths might be established, which differs greatlyin character from any previous work in logic.

Though Leibniz’s work in this area (by contrast withhis work on the calculus) exerted little influence in histime, but proved an inspiration for some of the lead-ing figures in the development of logic in the 19th and20th centuries, including Gottlob Frege and BertrandRussell. His influence is conspicuous in Russell’s phi-losophy of logical atomism[Rus18] and Wittgenstein’sTractatus Logico-Philosophicus [Wit61].

Leibniz was a rationalist philosopher and his concep-tion of the division which concerns us sometimes soundsradical. He holds for example that, at least for God, alltruths are necessary. But he nevertheless does distin-guish between necessary and contingent truths and, forus mere mortals the dividing line between these twofalls more or less where Hume will put it. Mathemat-ics is necessary, science is contingent. He also closelyconnects, as will Hume, necessity and a priority.

In relation to the analytic/synthetic distinction,Leibniz still uses these concepts as they are used inclassical Greece, to describe two different approaches tological proof, contemporary usage of “synthetic” comesfrom Kant. Leibniz does have something like a semanticand a proof theoretic characterization of necessary anda priori truth. On the proof theoretic side it is of in-terest not just that Leibniz considers necessary truthsto be those susceptible of a priori proof, thus implic-itly distinguishing the method of verification from themanner of discovery. On the semantic side we have anexplanation of necessity and of proof via the analysis of

definitions.

Leibniz tells us that truth is a matter of concep-tual containment, the containment of the concept ofthe predicate in the concept of the subject. This kindof containment we would now describe as extensionaland would not accept as a characterization of logicalor necessary truth, but rather of truth in general, in-cluding contingent truths. There is another kind ofconceptual containment which applies only to necessarytruths, and that may variously described as intensional,essential or risking circularity, as necessary or analytic.This is the containment of meanings and descriptionsof the role of definitions in the process of proof are con-sistent with the view that logical truths correspond tocontainment of meanings rather than containment ofextensions.

This distinction does not feature in Leibniz, his dis-tinction between necessary and contingent propositionsfor us mortals is made in two other ways. The first is byappeal to the omniscience of God and the limitationsof man. Men are not capable of the kind of completecomprehension of meaning which God exhibits. Propo-sitions which are contingent for us are those whose anal-ysis is beyond the limits of our knowledge or analyticcapability, but will nevertheless, if true, be knowable byanalysis for God. The second way of making the dis-tinction is through the principle of “sufficient reason”.Logical necessities can be shown to be true by reduc-tion to the law of identity of which self-predication isan instance. The procedure is to analyze the predicateand subject by expanding their definitions and discard-ing components present in the subject but not in thepredicate, until the subject and predicate turn out tobe identical. The remaining necessities arise from “theprinciple of sufficient reason”, which tells us that noth-ing happens without sufficient reason, and more specif-ically that the world is the way it is because from thelogically possible alternatives available to God, he haschosen the best.

We seem to have here two kinds of regression from theposition occupied by Aristotle. We see a reduction inthe clarity of the concept of demonstrative truth, by in-troducing the idea that the definitions of concepts maybe beyond our ken, and intelligible only to God. Thisreplaces the distinction between essence and accident,which is closer to the idea that in one case meaning suf-fices, and in the other observation will prove necessary.

Leibniz’s ideas on mechanisation of the syllogism flowfrom a conceptual atomism. The word “term” is used

3.4. AFTER HUME 19

in Aristotelian logic for the subject and predicate of aproposition in subject/predicate form, both of whichmay be thought of in more modern language as ex-pressing concepts. Leibniz’s atomism is the idea thatthere are simple concepts from which all other conceptsare formed in specific ways, together with the retentionfrom Aristotle of the idea that all propositions havesubject/predicate form.

In this is it important to distinguish the simple con-cepts from simple expressions which may be used toexpress them. Leibniz atomism here is not about syn-tax, it is about those things which the syntax expresses,the underlying concepts. A complex concept may nev-ertheless have a simple name, a complex concept is onewhich is defined in terms of other concepts, a simpleconcept is one which has no such definition. The atom-istic thesis is that there are simple concepts, and thatall complex concepts have a definition directly or indi-rectly in terms of simple concepts.

More specifically, a complex concept can always beanalyzed as a finite conjunction of literals (in todaysterminology) where a literal is either a simple conceptor the negation of a simple concept.

Leibniz saw that if every distinct simple concept wererepresented by a unique prime number, then a conjunc-tion of concepts could be represented by the product ofprimes. The fundamental theorem of arithmetic tells usthat any such a product can be factorized to retrievethe primes corresponding to the simple constituents.He proposed to represent arbitrary complex conceptsby a pair of such products, i.e. a pair of numbers, ofwhich the first number is the product of the primes rep-resenting the concepts which occur positively in the def-inition of the complex concept, and the second numberis the product of the primes representing the conceptswhich occur negatively in the complex. The same con-cept cannot appear both positively and negatively, orelse the definition is inconsistent, so these two numberswill be relatively prime (having no common factors).If Leibniz were correct then once the simple conceptshave been identified and given unique prime numbers,all complex concepts are representable as pairs of co-prime numbers.

It is then possible to describe (quite simply) calcu-lations which determine the truth of any propositionof the four forms which Aristotle uses in his syllogis-tic, (when these propositional schemes, which includevariables instead of definite concepts in the subjectand predicate places, are instantiated with specific con-

cepts).

3.4 After Hume

Hume’s fork abolishes a certain conception of meta-physics, making a difficulty in establishing any other.

What it leaves is something that looks rather likescience, falling into two parts. One part contains onlymatters devoid of empirical content, albeit includingthe whole of mathematics. The other contains opinionsof the empirical world, obtained by guesswork based onsensory impressions.

Not many philosophers, or scientists find this a veryattractive picture. Much philosophical reaction to thisseeks to refute the general scepticism in Hume’s posi-tion, but the refutation of his empirical scepticism ismore of benefit to science than to philosophy. The sig-nificance to philosophy of Hume’s fork is more acutein relation to those special kinds of knowledge, belovedparticularly to Plato, which are the truest subject mat-ter of philosophy, and were later to be known as meta-physics. From this point of view positivism, first amplyexemplified in Hume, is an anti-philosophical philoso-phy, in which philosophy gives up its own true groundyielding knowledge to science.

It is understandable that many philosophers will re-act specifically against this central feature of posi-tivism, and in this section we will consider some of thesereactions.

These we consider in three aspects.

1. Firstly there arises in the ongoing dialectic, furtherrefinement of our knowledge of exactly where thefundamental line is drawn.

2. Secondly there are challenges to and reaffirmationof the idea that a single dichotomy is involved.

3. Finally, coupled with the previous two there arevarious ways of reviving and of dismissing the pos-sibility of some kind of metaphysics.

Alongside these philosophical themes there are twotechnical problems which are gradually addressed. Inorder for the dichotomies to be definite it is necessaryfor languages to have definite meaning. One way toachieve this is to define new languages, and for them,give a definite semantics and deductive system. Theability to do this appears on the scene very late, littlemore than a century ago. The history both before and


after that point of transition of ideas about semanticsand proof, is part of our present concern.

The clarity which I find in Hume’s description of hisfork is like a moment of lucidity in life of confusion.Not until Rudolf Carnap do we find a return to and apositive refinement of that distinction.

3.4.1 A Broad Sketch of the Develop-ment

It would be easy in a sketch of the subsequent historyof Hume’s fork to loose the central issues in the de-tail of the very considerable developments in logic andphilosophy since then.

To help make draw these out the headlines aresketched here before a little more detail is supplied.

Kant supplied the first challenge to Hume’s concep-tion of a single dichotomy. Kant rescued a limitedconception of metaphysics by separating the a priorifrom analytic truth (which is contrasted with synthetictruth), and declaring that our knowledge of arithmetic,of space and time were synthetic a priori . It is notclear here whether Kant’s conception of analyticity dif-fered materially from Hume’s relations between ideas,or whether the concept was the same but Kant dis-agreed about its extension.

3.4.2 Kant

The first challenge we consider came from Kant, whowas awoken from his dogmatic slumbers by Hume, andwas moved to reject the unity of the triple dichotomy.

Kant was the first philosopher to make prominent useof the notion of analyticity. He has a dual conception,in both following Leibniz in ideas going back to Aristo-tle. The first is apparently semantic, that an analyticsentence is one in which the subject contains the predi-cate. This is in Leibniz, but Leibniz’s conception of con-ceptual containment is extensional, and therefore cor-responds to truth not analyticity. The other is “prooftheoretic” (insofar as one can talk of proof theory at thistime), a sentence is analytic if it follows from the law ofnon contradiction. This latter corresponds to Hume’s“intuitively or demonstratively certain” which in turnis from Aristotle (the “intuitive” part corresponds tofirst principles which must be essential truths in whichagain the predicate is contained in the subject).

What is distinctive about Kant’s philosophy is not hisdefinition of analyticity, which adds nothing to what is

found in Hume and Leibniz, but his view on what thingsfall under the definition. He has reverted to somethingcloser to the earlier empiricist Locke, for whom the cor-responding notion is confined to trivia.

Notably Kant excludes two groups of a priori truthsfrom analyticity, those of arithmetic and of geometry.

3.4.3 Bolzano

With Bolzano we see a first approach to the definitionof concepts relevant to Hume’s fork which advance sig-nificantly beyond Aristotle.

The techniques used by Bolzano are a considerableadvance, and the conception of logical truth he promul-gated is quite close to the idea of logical truth whichhas dominated mathematical logic ever since. However,this is a narrower concept than the ones we have con-sidered so far, and depends upon the classification ofconcepts into logical and non-logical.

Though the technical apparatus which Bolzano de-ploys is a great advance, the concepts he defines withthis apparatus take us further away from a notion oflogical or analytic truth which is properly complemen-tary to the notion of empirical or synthetic truth.

3.4.4 Frege

Frege’s logical and philosophical work was in part aimedat overturning Kant’s critique of Hume, in particularKant’s refusal to accept that mathematics is analytic.To achieve this aim Frege invented a new kind of formallogical system of which the two most important featureswere the abandonment of the Aristotelian emphasis onthe subject/predicate form of propositions. Instead ofpredicates Frege worked more generally with functions,among which predicates are functions whose results arealways truth values. Crucially, logical sentences nowhave arbitrary complexity, and the universal quantifieris introduced to operate over an arbitrarily complexpropositional function.

For Frege the notion of logical and analytic truths areseparated. Logical truth similar to Bolzano’s in beingdefined through a separation of concepts into logicaland non-logical, but analytic truth in Frege representsthe most precise formulation of Hume’s truths of reasonto that date.

In the analysis of Frege’s contribution the notion offree logic becomes significant, (though this is not a termhe uses). This is so because in his Begriffsschrift or con-cept notation those aspects of deductive logic which do

3.4. AFTER HUME 21

not involve ontology, or questions of what exits, are sat-isfactorily addressed. However, when he later comes toapply essentially the same logical system to the devel-opment of arithmetic it is necessary to incorporate ax-ioms which suffice to establish an ontology sufficient formathematics, and at this stage Frege introduces prin-ciples which are not logically consistent.

3.4.5 Russell

Russell began his work on mathematical logic ratherlater than Frege, having been born in 1872, not longbefore the publication in 1879 of Frege’s Begriffsschrift.Furthermore, because Frege’s work was ignored, Russelldid not become aware of it until he was already wellprogressed with his own approach to the logicisation ofmathematics.

Prior to this work Russell had already studied andwritten his own account of the work of Leibniz, andRussell’s philosophy is significantly influenced by Leib-niz. Russell perceived Leibniz as having been handi-capped by too closely following the logic or Aristotle,most particularly his retaining the idea that all propo-sitions have subject/predicate form. Russell’s logicalideas build on the work of Pierce and Schroder on therelational calculus. The relational calculus provided amore general form for propositions. A sentence might ineffect have multiple subjects of which jointly some pred-icate is asserted (predicates involving multiple subjectsare called relations).

The divergence from Aristotle in this matter was in-dependently progressed by Frege and by Pierce andSchroder.

3.4.6 Wittgenstein

While studying engineering at Manchester, LudwigWittgenstein became interested in philosophy throughan interest in the logical foundations of mathematics.On advice from Frege, Wittgenstein went to Cambridgeto study under Russell at a time when Russell wasdeeply involved in the completion of Principia Math-ematica[WR13].

3.4.7 Tarski

3.4.8 Carnap

3.4.9 Quine

Having studied Russell’s Principia Mathematica evenas an undergraduate,

3.4.10 Kripke

The following table gives a chronology.

585 BC Thales Mathematics428-348 BC Plato The Theory of Forms384-322 BC Aristotle Essential v. Accidental

Necessary v. ContingentDemonstrative v. Dialectical

1632-1704 Locke on trifling propositions1646-1716 Leibniz1711-1776 Hume relations between ideas v.

matters of fact1724-1804 Kant the synthetic, a priori1781-1848 Bolzano the method of variations1845-1918 Cantor set theory1848-1825 Frege Begriffsschrift, sinn and be-

deutung1848-1825 Russell the theory of types1871-1953 Zermelo the cumulative hierarchy1889-1951 Wittgenstein logical truths as tautologies1901-1983 Tarski definition of truth1891-1970 Carnap logical syntax, the method of

intensions and extensions1908-2000 Quine against the ana-

lytic/synthetic distinctionand modal logics, holism

1940- Kripke separating analyticity, neces-sity and a priority via rigiddesignators

Table 3.1: Development of the Analytic/Synthetic dis-tinction

Chapter 4

Analyticity and Analysis

[ Having established the distinction between logicaland empirical truths we now turn to a closer consid-eration of logical truth.

There are two main considerations. One is the scopeand relevance of logical truth. This should include adiscussion of its place in analytic philosophy, in math-ematics, empirical science, and engineering. A princi-ple aim in this discussion is both to re-emphasize, asHume did, the severe limits to what can be establishedpurely by deduction, and to show that it neverthelessis of the greatest practical significance. The limitationsand potential are to be made real by illustrations of adifferent character to those of Hume but which are con-nected with the various concerns of my own positivephilosophical thinking, theoretical and practical. Specif-ically in relation to philosophy, some discussion of thenature of philosophical analysis making the distinctionbetween a claim being analytic and a claim being “aboutlanguage”, i.e. between subject matter and epistemicstatus.

I would like to introduce here the notion of an ana-lytic oracle, which can then be refined in the next chap-ter to the FAn oracle.

]

We have seen that the words “analytic” and “syn-thetic” acquired in the philosophy of Kant a sense dis-tinct both from their use outside academic philosophyand from previous philosophical and mathematical us-age.

In non-philosophical use the related terms analysisand synthesis have diverse application, generally con-cerned with taking apart or putting together the partsof some complex whole. In the special domain of logicalproof, the usage in classical Greece was similar, connot-ing two methods of proof. An analytic proof proceededby analysis of the proposition to be proven, ultimately

reducing it to principles which can be known withoutproof. A synthetic proof begins instead with axiomaticprinciples reasoning forward until the desired theoremis eventually reached. In contemporary automation ofreason, these different methods, which we will considerfurther in a later chapter, are sometimes known as back-wards and forwards proof respectively.

Kant introduced a new usage in which analytic andsynthetic are applied to propositions (in the Aris-totelian sense), classifying them along the lines ofHume’s fork. Kant gave two criteria for this classifi-cation, one proof theoretic (concerning the manner inwhich such a proposition might be established), andthe second semantic concerned with meanings. How-ever characterized, analyticity in this sense is closelyconnected with deductive reason, and it is our purposein this chapter to relate this precise technical conceptwith the very general notion of analysis, both in itsacademic and its more worldly applications.

4.1 Abstract Logical Analysis

Though we are concerned here with analysis in general,there is one particular kind of analysis which will havespecial place. This kind of analysis is closely coupledwith the notion of analyticity through the concept ofdeductive soundness.

The notion of soundness is applicable both to formallogical systems which are supplied both with a seman-tics (an account of the meanings of the sentences of thelanguage) and a formal notion of derivation or proof,or to particular inferences in any language for whichthe semantics is well-defined. In the former case thesystem is sound if the inference rules respect the se-mantics in such a manner that from true propositions

22

4.2. 23

only true conclusions can be derived. In the latter case,without reference to any particular deductive system wemay say that an inference is sound if the premises en-tail the conclusion. A set of sentences (the premises)entails some other sentence (the conclusion) if underall circumstances whenever the premises are true theconclusion will also be true.

The connection with analyticity is then that allclaims about entailments are true if and only if analytic.An alternative statement of the connection is that allpropositions derivable from the empty set of premises,the theorems, of a sound deductive system are analytic.

Because of this connection, Rudolf Carnap and thelogical positivist held that philosophy, which theythought of as an a priori science consisted of analytictruths. This involves a position on the demarcation ofphilosophy which we will not adopt here. Instead wesimply note the scope of this particular kind of analysis,and of the kind of philosophy which employs it. Philos-ophy falling within the scope of such methods can nowbe made as rigorous and reliable as mathematics by theuse of modern formal languages and computer softwarewhich assists in the construction of formal proofs andautomatically checks that the proofs are correct.

Abstract logical analysis consists in the applicationof deductive reasoning to analysis of arguments, of con-cepts, of theories or doctrines. The application to ar-guments may be considered to be primary, and in thiscase the idea is that in any domain in which reason isthought to be applicable, modern logical methods canbe used to improve the rigour of the reasoning. The ideahere is much as it was in Plato, it is that the standardsof rigour which are generally found in mathematics andgenerally lacking elsewhere, can be made more widelyavailable by appropriate methods. In particular, by fo-cussing on the concepts involved, by ensuring that wehave clear definitions of these concepts, we can reasonwithin the relevant domain rigorously.

Plato, Aristotle, and the many logicians who fol-lowed them until very recent times, failed to realizethis ideal of logically rigorous reasoning beyond math-ematics. The greater difficulty in pinning down non-mathematical concepts may have been a factor here,and it is not until the 19th century that logic was pro-gressed to the point at which the logic itself could beused in making precise definitions from which one canreason with formal rigour.

4.2

Other points of controversy important to this projectremain, concerned with the scope, applicability and im-portance of analytic truth. These are important to ushere for two distinct kinds of reason.

The theoretical core of Positive Philosophy, Meta-physical Positivism, is primarily an analytic philosophy,and it is essential in presenting a conception of analyticphilosophy to address some of the reasons why a purelyanalytic philosophy might be thought to be of narrowscope and limited value.

The project we are considering is of broader scope,just as were the projects of Leibniz and Carnap, en-compassing not merely an approach to philosophicalanalysis, but also important and substantial parts ofmathematics, empirical science, engineering and otheractivities in which deductive reasoning might play asignificant role.

There is a circle to be squared here. There is oneperspective from which the entire enterprise is withoutcontent, and represents a preoccupation with academictrivia, and another diametrically opposed perspectivein which it is of the greatest practical significance andmay be thought to warrant substantial and energeticprosecution.

4.3 The Scope of Analytic Truth

I have presented a history of the evolution of theconcept of analyticity and related concepts, and haveadopted in Metaphysical Positivism a conception whichis similar to that described by Hume “relations betweenideas”. A good recent account of this concept may befound in the writings of Rudolf Carnap, but more im-portantly analyticity is a characteristic of the theoremsof a large class of modern tools for constructing andreasoning in formal logical theories. Further sharpen-ing remains of interest from a philosophical point ofview, and will be discussed later, but for practical pur-poses, even its relevance to applications demanding thevery highest standards of rigour, the concept and ourtechnologies for checking when it applies are sufficient.

It remains a matter of controversy how significantanalyticity, analytic truths, and methods in which theyfigure prominently are or might be.

Analytic truths may be thought of as falling into twoprincipal groups.

The first group consists of true claims in languages

24 CHAPTER 4. ANALYTICITY AND ANALYSIS

whose subject matter is entirely “abstract”. For thiswe understand abstract entities as mere ideas, aboutwhich we reason by offering a “definition” of the rel-evant domain of entities and then reasoning logicallyfrom the definition to conclusions about that domainwhich will be true by definition. This group encom-passes the whole of mathematics. It also encompassesreasoning about abstract models of any domain what-ever, whether or not one considers the model to be“mathematical”, so long as the model is well-definedand the method is deductive.

This is a Platonic conception of the scope of deduc-tive knowledge, and can be broadened very broadly inthe direction which Aristotle took, to yield a body ofanalytic truths which might be thought to be about thematerial world rather than purely concerned with ab-stract entities. There are several ways in which this canbe done in the context of modern logic without embrac-ing the complexities of Aristotelian metaphysics.

Carnap’s approach is to adopt formal languageswhose subject matter is the material world, to definethe semantics of the languages by giving the truth con-ditions, and to define analyticity in terms or such atruth conditional semantics. The analytic truths arethen those sentences in such languages whose truth con-ditions are invariably satisfied.

4.4 Analytic Philosophy

The term analytic philosophy was first applied in the20th century to a kind of philosophy which began at theturn of the century as Bertrand Russell and G.E.Moore. These two men, though sharing the idea that phi-losophy should be in some sense analytic, had quitedifferent conceptions of the kind of analysis involved,and their differences remained significant as methodsof analysis evolved throughout the 20th century.

Russell conception of analysis was shaped by thenew methods in logic in the development of which heplayed an important part. He believed that failures ofrigour in philosophical reasoning resulted in some casesfrom imperfections in ordinary language, and could beavoided if a special ideal language were adopted alongthe lines of the Theory of Types which he devised withA.N. Whitehead for the formalization of mathemat-ics in Principia Mathematica. He did not advocate orpractice the adoption of such formality in philosophicalrather than mathematical reasoning, but did advocatethe adoption of similar logical methods. An example

the kind of method he envisaged is the use of logi-cal constructions. By such means Russell advocated a“scientific” philosophy in which logical methods trans-formed philosophy into a rigorous deductive disciplineprogressively advancing in a manner similar to that ofmathematics and science, rather a perpetual sequenceof conflicting theories and a lack of solid progress. Wemay recall Plato’s prescription here, in which the sub-ject matter of philosophy and the only place where trueknowledge might be attained is in the world of idealforms. Even though Russell was an empiricist, his con-ception of philosophy is as a deductive discipline.

On the other hand Moore’s conception of analysisconcerned natural languages and consisted in the clari-fication of concepts in those languages, yielding illumi-nation consistent with common sense. There is a con-nection here with Socratic method, but neither Socratesnor Plato conferred such authority upon ordinary lan-guage and common sense. Socrates sought the true na-ture of concepts such as justice and virtue, and thoughhe believed that ordinary men were in some sense pos-sessed of these true concepts, they nevertheless mightnot correctly grasp them until they are induced by aSocratic dialogue to properly recall this knowledge.

Is it the case that analytic philosophy in the 20th

century was narrower in its scope than philosophicaltradition from which it grew, and if so is this a necessaryconsequence of the conception of philosophy as analytic,or of some particular ideas of what kind of analysis wasat stake.

Two mid century conceptions of the nature of an-alytic philosophy illustrate the issue, those of RudolfCarnap, exhibited in his Philosophy of Logical Syn-tax and the subsequent developments of his ideas, andthose of the “linguistic philosophy”, exemplified byJ.L.Austin, which prevailed briefly in post war Oxford.

A central thesis of Carnap’s philosophy of logicalsyntax was that philosophy consists of logical analy-sis and yields results insofar as they are establishedtruths, which are themselves analytic. This apparentnarrowing of scope is confirmed by explicit exclusion offields such as ethics, not only from the domain of ana-lytic philosophy, but from scientific discourse altogether(bearing in mind that this kind of philosophy is itselfconceived of as scientific in character, though not em-pirical). The severity of the apparent narrowing here ismitigated by the predominance in the actual philosophyof Rudolf Carnap of work which is methodological, in-cluding work which ensues in some proposal for the use

4.5. ANALYTICITY IN SCIENCE 25

of language. Though philosophy has relinquished anyclaim to offer factual enlightenment about the world,it may nevertheless make material contributions to theadvancement of such knowledge by contributions to themethods of science.

Linguistic philosophy in its most extreme form con-ceives of philosophy as the analysis of natural language.

4.5 Analyticity in Science

Chapter 5

Computation and Deduction

[

This chapter will include some of the history of com-putation and logic, showing the relationship betweenthese two topics, showing the changes to them whichhave arisen from the advent of mathematical logic andthe digital computer, leading to the idea of sound com-putation.

However it is the forward looking aspects which arethe more important. This will include discussion of thestep forward from “sound computation” to “crisp AI”,making use of the idea of “FAn oracle” a refinement ofthe analytic oracle.

]

In the architecture which we later propose, proof andcomputation are intimately intertwined. Throughouttheir history these two concepts have frequently beentransformed, and have frequently impacted upon eachother. Far from stabilizing it seems that the pace ofchange in our ideas about how these concepts interre-late has accelerated.

The chapter begins historically, but ends in the con-temporary context with some technical points intendedto open up new possibilities for the relationship.

The history of computation goes back a long way.Our history begins with the idea of proof in mathe-matics, which we first know of in the beginnings ofGreek philosophy and mathematics with the philoso-pher Thales. Whether the reasoning at this pointcould properly be called deductive is moot. Two laterlandmarks which may be thought of as pinnacles inthe articulation and application of deductive methodswere the logical writings of Aristotle’s Organon and themathematical tour de force of Euclid’s Elements. Theseworks became the standard for two millennia in philo-sophical logic and in deductive rigour and axiomaticmethod in mathematics. Substantial advances beyond

these would not be made until the 19th century.

5.1 Before the Greeks

It is generally said that the ancient Greeks inventedmathematics and deduction.

This is probably true of the self conscious use of de-duction and the development of mathematics as a the-oretical discipline.

However, there are some points worth making aboutwhat happened before that. The man in the street,hearing the word “mathematics” thinks primarily ofarithmetic, and of practical skills of computation. Hemay have little conception of mathematics as a theo-retical discipline.

Mathematics in that sense preceded the Greeks. No-tation for numbers, numerical computation and elemen-tary geometrical capabilities probably go back as far as10,000 years BC. Before classical Greece, the princi-pal sources of expertise in these practical matters wereBabylonia and Egypt. There are at this stage meth-ods for computation and for solving certain kinds ofalgebraic problems, but these methods are presentedwithout any attempt at justification, there is neither atheoretical discipline establishing results by deductiveproof.

The ability to perform elementary steps of deductivereasoning is probably coeval with descriptive language,since one cannot be said to understand the languagewithout grasping and being able to apply elementaryconceptual inclusions. That is to say, for example,that if someone knows the meaning of “azure” then heknows that everything which is azure is also blue. How-ever, the first signs of deduction being used elaboratelyand systematically, and the idea or deductive proof are

26

5.2. SKETCH 27

found at the beginning of the classical period of ancientGreece at about 600BC.

The Greek civilization is thought to date back toabout 2,800 BC, but mathematics as a theoretical dis-cipline originates with the school of the philosopherThales in Ionia at around 600 BC.

5.2 Sketch

The idea here is to sketch the history of proof, and thehistory of computation.

Initially these two seem quite separate.The potential for connection begins with the ap-

proach to formality found in Aristotle’sAristotle syl-logistic, and with the axiomatic method for mathemat-ics documented in Euclid’s elements. Aristotle’s workdominated the philosophical study of logic until the19th century, but his logic was the object of philosoph-ical enquiry rather than a practical tool in deduction.The axiomatic method on the other hand had provedhighly effective in Greek geometry. The standards thenset were held as an ideal in mathematics, but were notmatched in subsequent developments in mathematics,even (or especially) when mathematical innovation andthe effectiveness of mathematics as a tool for scienceand engineering was scaling new heights, until we cometo the end of the 19th century.

Leibniz connects Aristotelian logic, by conceivinga scheme for arithmetisation of Aristotelian logic,which he thought would permit the truth of a sub-ject/predicate proposition when arithmetically codedin his universal characteristic. Calculating machineswere already in existence, and Leibniz designed such amachine contributing to the development of the tech-nology. Important though these ideas were, they wereneither applicable nor influential for hundreds of years.

The next important developments both occur inthe 19th century. The first was the design by Bab-bageBabbage of his analytical engine, the first concep-tion of a universal computational engine. The secondwas the idea of formal proof first exhibited in Frege’sBegriffsschrift [Fre67]. Frege’s idea was the first ad-vance in rigour beyond Euclid’s axiomatic method, ad-vancing that method by requiring not only that that theaxioms, postulates and definitions on which a proof isbased are made explicit, but also by requiring that therules by which conclusions are drawn at each step in aproof from these premises and previous conclusions, becarefully specified in advance. This required that the

language in which the proofs are conducted be formallyspecified.

There is a connection here, not explicitly made, be-tween these two developments. By designing a univer-sal analytic engine Babbage had implicitly defined anotion of computability. A function is computable if itcan be calculated by Babbage’s analytical engine. Byrequiring that the rules of inference used in a proof befully specified Frege was demanding that the correct-ness of such a formal proof be mechanically checkable,that in principle the correctness of such a proof couldbe verified by Babbage’s analytical engine.

Though we can now see this connection, it was notmade explicit for a while.

Subsequent to these two developments the notion ofproof and of the axiomatic method were refined as apart of an explosion of work in logic and its applica-tion to the derivation of mathematics. A key figure inthis was David Hilbert, who began by advancing theaxiomatic method, for the first time achieving levels ofrigour surpassing those of Euclid, following the initialinsights into the nature of proof by Frege. He did thisby returning to the axiomatic theory of geometry and inthe process found and resolved weaknesses in Euclideangeometry. By this time Frege’s the logical system inwhich Frege had sought to formally derive arithmeticusing his new logical methods had been found to be in-consistent (by Bertrand Russell ), precipitating a senseof crisis in the foundations of mathematics. Hilbertresponded to this with a program for putting mathe-matics on a firm foundation, and a major part of thiswas a new domain of research which he called meta-mathematics.

This new domain of meta-mathematics was the studyof the kinds of formal system in which mathematicsmight be developed, with a view to establishing theconsistency of such systems.

Bertrand Russell came up with a new logical systemwhich he called the “theory of types” for use in hiscollaboration with A.N.Whitehead on Principia Math-ematica. With this formal logical foundation they car-ried through the formal development of a significantpart of mathematics, starting (after more logical andabstract considerations) with arithmetic. It is notori-ous that in that monumental work they prove the ele-mentary arithmetic theorem that 1+1=2, do not reachthat point until page 362. This emphasises an impor-tant difference between computation and proof as itwas then understood. It is possible to prove theorems

28 CHAPTER 5. COMPUTATION AND DEDUCTION

which express the results of arithmetic computations,but the effort and complexity involved in such proofsoutstrips by many orders of magnitude the difficultyof simply performing the calculation, which would beaccepted in an informal proof which depended upon itwithout further justification. Formal proofs are so de-tailed that their complexity compares badly with theinformal proofs which were then and still are the normin mathematics, and in the case of elementary equa-tions with the difficulty in performing arithmetic bythe usual methods.

5.3 The Status of Proof

Early in the 20th century radical changes have takenplace in the conception of proof and of the role of proofin mathematics.

There were at this time several distinct attitudes to-wards proof in mathematics which are of interest here.

The simplest is that a formal deductive system con-sists of a collection of axioms which are simple enoughto be self-evidently true, together with some rules ofinference which likewise may be thought self-evidentlyto preserve truth. In consequence all theorems deriv-able from the axioms using the rules of inference willbe true.

The discovery of the inconsistency of Frege’sGrundgesetze undermined this position by showing thata system whose soundness in the above sense seemedself-evident, might nevertheless prove inconsistent.

In devising a new logical system intended to avoidthe kinds of difficulty found in Frege’s, Russell wastherefore not able wholeheartedly to claim soundnesson these grounds, and instead suggested a partly post-hoc rationale. Russell thought that the success of alogical system in deriving mathematics without provingany contradictions provides evidence that the system issound.

The developments in mathematics which renderedmathematical analysis rigorous by eliminating the useof infinitesimals lead on to a liberal conception ofthe notion of function as an arbitrary graph whichproved unacceptable to some mathematicians, notablyLeopold Kronecker . Kronecker’s scepticism about thenew mathematics particularly as it is found in Cantor’sSet Theory, was an early example of constructivismin mathematics in which mathematical objects are re-quired to be constructible by limited means resultingin an ontology radically less expansive than that of set

theory. Associated with this more frugal ontology thereis a completely different attitude to proof. The effect isnot just that mathematical proofs are expected to usemore limited means, but that the subject matters ofmathematics are modified because of ontological reser-vations, and in effect certain domains of enquiry arebanished (of which again, set theory as conceived byCantor is a principal exemplar).

Against this “impoverishment” of mathematics themathematician David Hilbert reacted. He devised aprogram intended to establish the soundness of thewhole of classical mathematics to a standard equiva-lent to that of constructive mathematics, and intro-duced the concept of meta-mathematics to that end.Meta-mathematics is the study of the kinds of formaldeductive system which can be used in the derivationof mathematics. Hilbert’s aim was by developing meta-mathematical techniques (now know as proof theory) toestablish the consistency of the whole of mathematicsby means which would be acceptable to constructivists.

We have now three attitudes toward proof.

The first consists in choosing a deductive systemcarefully, adopting some underlying philosophical ra-tionale (in Russell’s case this was the avoidance of “vi-cious circles”), choosing axioms and rules guided by thechosen rationale, and then seeing whether the systemproves to be adequate for the mathematics and seemsto be immune from paradox.

The second rejects the idea of a post hoc justifica-tion, is ontologically conservative, and expects a closeconnection between the demand for a constructible on-tology and the kind of proofs which are acceptable.

The third seeks a bridge between the two, with theaim of allowing mathematics the scope which it mightassume under the first approach, while achieving thestandards of constructive rigour sought in the secondapproach. This promising idea turned out to be unreal-izable, but its failure spawned another, in which relativeconsistency proofs provide a partial ordering of formaldeductive systems providing a measure of strength andof risk.

5.4 Incompleteness and Recur-sion Theory

The next important connection between proof and com-putation comes from Godel in connection with Hilbert’sprogram, and is the first of a series of advances relevant

5.5. COMPUTING MACHINERY AND PROOF 29

to proof and computation which appeared in the 1930sand 40s. Godel’s famous incompleteness theorems wereobtained by the technique of arithmetisation of whichwe saw a precursor in Leibniz’s calculus ratiocinator .The result demonstrates a limitation on what can bedone with certain kinds of consistent formal deductivesystem, viz that no such system can prove every state-ment of arithmetic. The precise definition of the kindof system within the scope of the proof provides for-mally the connection implicit in the idea of Frege thatthe notion of proof be computable.

We are now on the verge of an explicit theory ofcomputability. The idea of computability first appear-ing implicitly in the design of Babbage’s analytical en-gine, now comes under theoretical scrutiny in the newdiscipline of mathematical logic. Several different for-mal conceptions of effective process are enunciated (byChurch, Kleene, Post, and Turing) and are shown tobe equivalent in expressiveness, leading Alonzo Churchto put forward Church’s thesis that these equivalentformal ideas capture the informal notion of effectivecalculability.

5.5 Computing Machinery andProof

In the meantime, the technology of computation hasmoved on. The analytic engine designed by Babbagewas a mechanical computer, and its realization was be-yond his resources. Computation was by the 1940s intransition from electro-mechanical to electronic tech-nology, universal computers were now within reach.

When they arrived, though predominantly appliedto the purposes of businesses or for scientific calcu-lations, they were soon also applied in academia forthe construction and checking of formal proofs. Thenew academic discipline of Computer Science and therapidly growing computer industry became principalusers and developers of formal languages, in which thealgorithms to be executed by computers were described.The theory behind computing was also studied, and thetechniques of mathematical logic extensively adopted.Mathematical methods were advocated for proving cor-rectness of programs, and even of the “hardware” whichexecuted the programs, and the expected complexity ofthese proofs encouraged the development of software toassist in constructing and checking the proofs.

These developments spawned new kinds of logical

languages, new approaches to reasoning in those lan-guages, and new applications for proof. Some of theseare significant here because the notion of proof forwhich our project is to support and exploit is distantfrom those which predate the computer and computerscience.

A principal influence in shaping a more diverse con-ception of proof is the development of automatic andsemi-automatic or interactive theorem provers. Threedifferent kinds of such software systems are of interest.

The first kind is the one closest to the new concep-tion of formal proof which came from Frege. In such asystem a proof is a sequence of theorems each of whichis either an axiom of the logical system or is derivedin a specified way according to an inference rule of thelogic from one or more theorems appearing earlier inthe list. The end result of the proof is the last sentencein this sequence.

A second approach to theorem proving simply leavesthe details to the software, which is not required to con-struct or verify a detailed formal proof, but instead iswritten to check reliably for derivability in the relevantformal system and to deliver an informal proof of theresult.1

The third approach which is of greatest relevancehere began again at Edinburgh University as a theoremprover for a Logic for Computable Functions (LCF).This approach to proof later became known as “theLCF paradigm”. It makes use of a feature of cer-tain kinds of programming language which is calledan “abstract data-type”, whereby a new kind of struc-ture is defined which can only be constructed by theuse of a limited set of functions defined for that pur-pose. The implementation of such a theorem proverbegins with the specification of a formal deductive sys-tem, and proceeds by the implementation of an abstractdata type which corresponds closely to the deductivesystem. Each of the available constructors in for theabstract data type is required to correctly implementone of the axioms or inference rules of the logic deliv-ering only theorems which could be obtained using therelevant axiom or inference rule.

By this means it is ensured that values which can becomputed using this abstract data type must be theo-rems of the logic, for the structure of the computation

1The classic example of such a prover is NQTHM, a theoremprover first developed at the University of Edinburgh by RobertBoyer and Jay Moore, others in this genre include PVS fromthe Stanford Research Institute, and Eves, developed by a smallCanadian company.

30 CHAPTER 5. COMPUTATION AND DEDUCTION

yielding the value corresponds precisely to the structureof a proof in the logic.

In this approach to the automation of proof, the proofitself as an object has disappeared, instead of a se-quence of theorems of which the result demonstrated isthe last, we have a computation which by means guar-anteed sound delivers as its result the desired theorem.Here the proof is a computation.

The implementation of the abstract data type pro-vides a logical kernel, which ensures that any valuewhich can be computed of type thm will be a theo-rem of the logic. This is by itself too primitive to beuseful for a user of the theorem prover. More com-plex software is developed which partly automates thecomputation of the theorem (which is analogous to andserves as the discovery of a proof). These additionallayers of software are then made available to the userthrough an interactive programming interface throughwhich he can undertake proofs, possibly augmentingthe proof capability by further programming of proofdiscovery methods.

Several further steps along this direction are signifi-cant for us here. As with the kind of formal proof foundin Principia Mathematica complexity remains an issue.Even with a computer to do the work, the search fora formal proof, or even the checking of the proof canbe onerous. For the sake of efficiency it is thereforeusual to implement in the logical kernel of a theoremprover following the LCF paradigm a number of de-rived rules, the use of which significantly improves thecomputational efficiency of the theorem prover withoutimpairing its soundness.

Taking this a step further we may consider the ad-dition of any inference rule which is known only toperform inferences which could have been done in theprimitive logic. Such enhancements in principle im-prove computational efficiency without alteration to thetheorems which are provable in the system.

5.6 Sound Computation as Proof

The difference between computation and proof may beseen as a difference in perception or interpretation.

A computation is an operation on data yielding data,an inference is an operation on propositions yielding aproposition. Propositions are interpreted data. Thedata transformed by a computation may have multiplepossible propositional interpretations, relative to whichthe computation might be seen to be a sound logical

inference.A computable function becomes a sound inference

rule if can be given a suitable specification. This isalways possible, though this nominal possibility doesnot necessarily translate into something useful.

5.7 Oracles and the Terminator

We know from recursion theory that there are unsolv-able problems. One of these is the Turing machine halt-ing problem, the problem of determining for any givenspecification of a Turing machine and its tape whetherthe Turing machine will halt.

Recursion theory studies degrees of unsolvability us-ing the notion of an Oracle. An Oracle is a hypotheticalmachine which answers a problem which is not in facteffectively decidable, such as the Turing machine halt-ing problem. This idea can be used to give an upperbound on what one could conceivably achieve by wayof problem solving capability in a program running ona digital computer.

The halting problem is equivalent to the problem ofdeciding whether a sentence is a theorem in the kindof deductive system which is suitable as a formal foun-dation for mathematics. Automatic proof methods forfirst order logic can be arranged to deliver with theproof of an existential theorem a term which is a wit-ness for that theorem, which satisfies the body of theexistential. If a design problem is expressed as existen-tial proposition in which the body stipulates the condi-tions required of a solution to the design problem and asolution to the problem is any term which satisfies thebody of the existential, then degree of recursive solv-ability corresponds to a design capability.

5.8 Self Modifying Procedures

Because of their intelligence, people can solve problemswhich have no feasible algorithmic solution. It doubtfulthat they do this because intelligence can draw uponresources which go beyond the limits of Turing com-putability, that possibility and its consequences are out-side the scope of this book.

Let us assume that, as Turing himself believed, thereis nothing in human intelligence which might not berealized in a Turing machine. That entails, that forany particular definite problem domain, there exists analgorithm which performs as well in that domain as an

5.8. SELF MODIFYING PROCEDURES 31

above average appropriately trained and experiencedhuman being.

The algorithm might not be one which any of us willever be able to code. It is possible that its complexity,like that of the human brain, might exceed anythingwhich a human beings could design and implement.It might be that such an algorithm could only arisethrough a process of evolution.

Our project is intended to support the use of evo-lutionary processes to realize machine intelligence insolving crisp problems. Some details of how that mightbe realized are touched upon here.

Let us suppose that some implementation or exten-sion of our architecture makes provision for its ownevolution over time. Because the notion of proof is as-surance sensitive, some assurance may be sought thatsuch changes do not render the system unsound andthus degrade the assurance which we could attach toits results.

It is possible to implement a system which allowsfor self modification in a qualified way, permitting onlymodifications which preserve the trustworthiness of thesystem. This can be done by admitting only modifica-tions which preserve that level of assurance, which theywill do if they are conservative.

For any formally precise property of systems, therewill be a system which undertakes self modification inways which are guaranteed to preserve that property.

The implementation of such a system is not difficult,for any proof tool which implements a reflection princi-ple of the kind discussed above will be a self-modifyingproof system which preserves the property of infallibil-ity in asserting theorem-hood. No simplistic approachto realizing intelligence in such a way would be likelyto succeed.

Chapter 6

Rigour, Scepticism and Positivism

[ The main point of this chapter is to lead us into twofeatures of positive philosophy and of the architecturefor crisp AI.

These are:

Epistemic Retreat At its simplest this is the ideathat, instead of asserting a claim, one retreats todescribing the evidence in favor of it. For science itis the idea that a scientific theory is to be presentedas an abstract model, and rather than asserting the“truth” of the model (whatever that might mean)one makes statements about its fidelity and util-ity in various circumstances. Part of this is theidea that the assertion of a logical truth is to bedone formally, whereas empirical claims are as-serted only informally.

Graduated Scepticism Not sure whether to call thisscepticism, but the idea is that instead of choosingbetween two theories and then calling one true andthe other false, one suspends judgement about truthand confines oneself to comparative evaluation (ofvarious kinds).

Assurance and Authority The second principleidea, which is applicable primarily to logical truth,is that these truths are asserted by authorities inwhich we may think of an authority as anythingwhich may wish to formally express an opinionabout the truth of some conjecture. Authoritiesmay do so baldly, or may come to such an opinionin the light of the opinions of other authorities.In the latter case, the assertion will mention thoseother authorities, and the combination of the set ofauthorities thus cited and the authority expressingthe opinion is regarded as a level of assurancewhich fits into a lattice structure providing apartial ordering of such assurance levels.

The chapter also needs to connect to contemporaryreasons for scepticism and to the reasons for doubtabout philosophy arising from the work of Quine andTarski.

]

In defending key elements of the philosophy withwhich I propose to underpin the automation of reasonI have found it necessary to reject wholesale two of themost potent influences on analytic philosophy since themid 20th century.

If philosophy can go so badly astray, even after themodern revolution in logic transformed it, what hopeis there that the ideas I champion can be better. Is thedescent into nihilism not inevitable?

This line of thought is not new, it recurs throughoutthe history of Western Philosophy. In this chapter Itrace some of this history. I do this partly to make mynegative conclusions seem less startling and more plau-sible, but principally because key features of PositivePhilosophy (the positivism!) have evolved from moreextreme scepticisms over a long period of time, and arebest understood in that context.

In each historical era different sources of dogma pro-vide new principal targets for the sceptic. In the firstwholly sceptical philosophies, those of the Pyrrhonistsand the sceptics of Plato’s academy, a principal tar-get was the dogmatic metaphysics of the pre-Socraticphilosophers, and the great syntheses of Plato and Aris-totle. When pyrrhonism first reappeared in Europe itstarget was the dogmas of Catholicism. Today we havemany sources of dogmatic disinformation, but the oneof greatest concern here is academia in general and an-alytic philosophy in particular.

The thought of the radical Greek sceptics was passedinto history principally in the comprehensive scepticalwritings of Sextus Empiricus[Emp33], which began to

32

6.1. SYSTEMATIC SKEPTICISM 33

influence modern European thought after being trans-lated from Greek into Latin in the sixteenth century.

These revived sceptical arguments figured at first inthe religious controversies of the reformation, and weredeployed against Catholic dogma. Influential amongthese new pyrrhoneans was Michel de Montaigne.

Once these sceptical arguments were rediscoveredthey were not so much adopted as moderated. Thiswas particularly important for science, and it was scien-tific philosophers such as Gassendi and Mersennes whosought a constructive scepticism compatible with thenew science. The most important enduring traditionwhich emerged from this moderation was positivism,which rejects pseudo-science and promotes higher stan-dards for “positive” science.

In modern times the arch sceptic and the target ofmuch anti-sceptical argument in epistemology has beenDavid Hume. Greek scepticism, despite providing mostof the ammunition for Hume, figures less prominently.Hume’s philosophy was a sequel to a variety of scepti-cal thought through the sixteenth and seventeenth cen-turies, which begins with the religious controversies ofthe reformation.

6.1 Systematic Skepticism

Systematic, wholesale skepticism, the doubt that thereis any true knowledge, was found in two importantphilosophical traditions in post-classical Greece. Thebest known of these was that associated with the nameof Pyrrho of Ellis (though it is not known how many oftheir ideas actually came from Pyrrho). The other isthe Academy of Plato, which, after the death of Platounderwent periods of radical scepticism.

The pyrrhoneans offered a comprehensive range ofskeptical arguments which were later documented bySextus Empiricus, which we need not examine in detail.We need only consider a single kind of very generalargument. which is argument by regress of justification.In its simplest form it asserts that:

1. to count as knowledge a belief must be justified

2. a justification of a proposition consists of a num-ber of known premises, together with an argumentknown to be conclusive which shows that the con-clusion must be true if the premises are.

3. a justification therefore depends on prior knowl-edge, either of the supposed evidential support or

of the validity of some form of inference, and wetherefore have an infinite regress and no knowledgeis possible.

Of course, this argument undermines itself, and socannot itself yield the knowledge that knowledge is im-possible.

Philosophies which are skeptical about knowledgerisk incoherence, and are often accused of inconsistency.The most naive incoherence would be to claim to knowthat nothing can be known. This accusation has beenlevelled, but it is doubtful that any skeptic on recordmade that mistake. The closest is the doctrine that wecan know nothing but this single proposition, which isconsistent if a little ad hoc.

There are two subtler kinds of inconsistency whichare more significant for us.

The first is equivocation about the meaning of theverb “to know’ We are concerned with “knowing that”,knowledge of true propositions, rather than “knowinghow”. This generally involves a true belief, and usuallydepends on the knower having adequate grounds for hisbelief, thus the formula: knowledge is justified true be-lief. However, the requirements in terms of justificationvary widely. In some discourse this requirement of jus-tification lapses, and someone may be said to “know” afact simply because he has been apprised of it. On theother extreme the requirement for justification can beexhaustive, in some contexts the required justificationmust be conclusive. What counts as conclusive is alsoup for grabs, In the context of a philosophical discussiona justification may need to be the possession of evidencewhich logically entails the candidate proposition.

Arguments in favor of radical scepticism will oftenimplicitly assume the need for the highest standardsof justification. However, once established in this way,the resulting scepticism may be applied to all kinds of“knowledge” even in contexts where the required stan-dards of justification is weak or nugatory.

One way of preventing inconsistency is to oppose dog-matism with doubt. Thus, the skeptic responds to aclaim to knowledge, not by contradicting that claim,but merely casting doubt upon it. This leads to scep-tics who seek to establish “equipollence” a situation inwhich the evidence for and against a proposition areperfectly balanced.

Pyrrhonean scepticism advocates reserving judge-ment, keeping an open mind, rather than the dogmaticassertion of some proposition. The pyrrhonean is some-times portrayed as someone seeking knowledge who fails

34 CHAPTER 6. RIGOUR, SCEPTICISM AND POSITIVISM

to find the certain knowledge that he seeks. There aretwo elements of inconsistency or equivocation which canbe observed in an account at this level. The first isthe contrast between the two objectives, on the onehand that of seeking knowledge (albeit unsuccessfully),and on the other of seeking equipollence in respect ofany particular judgement. Surely if equipollence is anobjective, then this represents a prejudice against thepossibility of knowledge, rather than a positive attitudeof seeking knowledge? This accusation can perhaps bedeflected by presenting the search for equipollence as away of testing a conjecture, and hence establishing theproposition by failing to show equipollence.

A second point of apparent equivocation is in theword dogmatic. In ordinary parlance a dogmatist issomeone who holds onto a fixed belief in the face ofcontrary evidence. In Pyrrhonean usage one becomes adogmatist in virtue of even the smallest departure fromcomplete doubt. The mere expression of a tentativebelief counts as dogmatism. The sceptical argumentshowever, very often depend on the demand for con-clusive justification, and hence have force only againstdogmatism in its more ordinary usage, rather than inthe more liberal interpretation which embraces mereopinion.

When we come to positivism we find a kind of mit-igated scepticism in which the systematic doubt is apart of scientific method, is a way of testing hypotheses.Elements of skepticism provide a basis for positivismand its notion of positive science, and will have an in-fluence on our proposed “architecture of knowledge”,so we seek here a way of removing inconsistencies andequivocations to obtain a defensible position on whichto build.

In this connection we leave the Pyrrhonists to con-sider the graduated scepticism of the Academic skepticCarneades.

—————- Extract something from Skepticism —————-

6.2 positivism

• Positivism as a moderation of skepticism

• Positivism as scientific method

• Kolakowski’s characterization of positivism

• Hume’s fork

The term positivism was coined by Auguste Comteand refers to the whole of his broad ranging philosophi-cal thought. Typically however the term has been usedfor a narrower position centering around his conceptionof positive science, and has been construed as relating toa philosophical tradition many elements of which werefirst clear in the philosophy of David Hume. This is theperspective on positivism which is found in the historyby Kolakowski [?], and which we work with here.

Comte held that the human mind progresses throughdistinct phases, the theological the metaphysical andthe positive. In the theological stage the hidden natureof things is connected with the belief in supernaturalbeings, in the metaphysical stage the hidden nature ofthings is sought without resort to the supernatural, andin the positive stage scientific understanding supersedesboth theology and metaphysics through the formulationof universal laws about the phenomena which do notdepend on hidden entities. He offered his conception ofpositive science not as an innovation, but rather as anaccount of the scientific method already established byhis predecessors, men such as Bacon and Galileo.

6.3 Rudolf Carnap

6.3.1 Tolerance, Pluralism, Meta-physics

These three topics are intimately interwoven in Car-nap’s philosophy. I’m going to attempt an elucidationelements of Carnap’s philosophy by discussing some in-terpretations of Carnap on pluralism which seem to meto be, in various degrees, mistaken.

I discuss the views on Carnap’s pluralism of threefictitious philosophers, R, A and K. Their positions aresuggested to me by the writings of three actual philoso-phers, but it is not necessary for my present purposes toresolve the question of what those real philosophers ac-tually meant, it suffices to discuss positions they mightpossibly have meant.

First a brief preliminary statement of some key as-pects of Carnap’s notion of pluralism and his principleof tolerance. According to Carnap’s intellectual auto-biography [Car63a, Car63b] his tolerance was antici-pated in his student days by a willingness to discussissues with friends in a variety of “philosophical lan-guages”, which corresponded to distinct and incompat-ible metaphysical stances. The principle of tolerancedid not appear until much later, in “logical syntax”

6.4. 1 35

[Car34, Car37]. It is understandable that people willcome away from this book with diverse and incompat-ible ideas of what Carnap’s pluralism is. Carnap doesnot actually use the word pluralism in the book. Hedoes enunciate his “principle of tolerance” in

6.4 1

7:

It is not our business to set up prohibitions,but to arrive at conventions.

Which is further explained in that section as:

Everyone is at liberty to build up his ownlogic. i.e. his own form of language, as hewishes.

I begin with R. R claimed that he was more pluralis-tic than Carnap. I reacted against that, since I didn’tsee that Carnap’s pluralism actually excluded anything.The feature of his pluralism which he thought beyondCarnap’s was that he would accept that the same sen-tence could have different meanings and different truthvalues. But this is also the case for Carnap. For Car-nap this would be possible by having that same sentencein two different “language frameworks” (which we mayperhaps think of as being both the grammar of the lan-guage and the semantics in some form). I mention thishere because it contrasts with the next one.

K took a more substantial interest in Carnap’s philos-ophy and wrote quite a bit about it. He recognizes twointerpretations of Carnap’s pluralism, on the one handas a substantial thesis, and on the other as a proposal.However, his critique is concerned with the former. Inthis case he takes Carnap to be taking a radical viewentailing the indeterminacy of truth in languages likearithmetic and set theory, so that Carnap is to be un-derstood as saying that these truths are not completelydeterminate but can be chosen arbitrarily.

585 BC Thales deductive mathematics6thC BC Ionia metaphysical cosmology

Zeno paradoxes of motion5thC BC sophists man is the measure of all things470-399 BC Socrates384-322 BC Aristotle the organon and the metaphysics428-348 BC Plato the theory of formsc300 BC Euclid the axiomatic methodc365-270 BC Pyrrho scepticismc180-110 BC Carneades graduated scepticismc12881348 William

of Occam nominalism

Table 6.1: Rigour, Scepticism, Positivism

Chapter 7

Epistemic Retreat

This chapter is philosophically epistemological, pro-viding a kind of constructive epistemology in the formof some architectural principles in relation to the rep-resentation of knowledge in networked storage systemsand its processing by distributed processing systems in-volving both natural and artificial processing elements.

The chapter is therefore a conflation of aspects ofphilosophical epistemology with abstract architecturaldesign for knowledge processing systems. This neces-sitates (or flows from) some novelty in my conceptionof both these enterprises, so I will begin with some dis-cussion of the innovations involved.

Epistemology may most briefly be characterized asthe theory of knowledge. It has typically been an-thropomorphic, i.e. concerned specifically with humanknowers, and sometimes linguistic, concerned with thespecifics of the meaning of the concept of knowledge.

Here I am interested in knowledge in information sys-tems, not in human brains, and seek to avoid givingany deep scrutiny to the meaning of the word know.The conduct of epistemology without attachment to theconcept of knowledge is analogous to the preference inscience for avoiding vague and relative concepts such as“hot” and “cold”, in favor of objective and precise prop-erties such as “temperature in degrees Kelvin”. Insteadof claiming knowledge, we will be aiming to providemore objective descriptions of grounds for the truthof propositions, or of evidence showing the reliabilityand fidelity of abstract models of the real world. I donot prescribe particular measures, but provide instead acontext in which a plurality of comparative evaluationscan be accommodated.

The first comparison is the one provided by Hume’sfork, and this has a major impact on the epistemologyand the knowledge architecture. Because of the greatprecision with which “relations between ideas” can be

expressed, such propositions we regard as assert able,and in connection with such assertions “epistemic re-treat” involves a form of assertion in which the groundsfor asserting truth are made explicit. On the other sideof Hume’s fork, in relation to “matters of fact”, epis-temic retreat in the first instance reflects the impreci-sion in our knowledge of Plato’s fleeting world of ap-pearances. Scientific knowledge is regarded as embod-ied in abstract models of the real world, which are ap-plied by deduction (this is our version of a nomological-deductive scientific method).

A principle effect of the status of Hume’s fork in thisepistemology is that our formal knowledge base assertsonly logical or analytic truths. In it, scientific theoriesare represented through abstract models, which will ingeneral have concrete interpretations, but which havethe logical status of definitions rather than assertions.

Epistemology has been concerned with the refutationof scepticism. It here embraces an open scepticism (butnot a dogmatic negative scepticism), and is thereforeoriented not with the refutation of scepticism but withthe establishment of a viable constructively scepticalsystem.

The first step in the moderation of a pyrrhonean scep-ticism is the recognition that, though no proposition isabsolutely certainly true, some appear to be more cer-tain than others, and among the more doubtful thereare also degrees of doubt. Can we know with abso-lute certainty that some proposition is more certainthan another? No, but we can form opinions aboutrelative certainty which are themselves more solid thanour opinions about truth (again, abstaining from abso-lutes). Among these comparative assurance judgementsare the relations between a proposition and the evidencewe have for it. It will generally be the case that ourconfidence in the evidence on which we base an opinion

36

37

will by stronger than that in the proposition we inferfrom the evidence. Similarly, it will be the case that astatement effectively describing the evidential supportfor a theory will can be more confidently asserted thanthe theory itself.

This is the principal way in which Metaphysical Pos-itivism interprets the positivistic principle that scienceshould not go beyond presentation of observationaldata. It is not taken to impede the formulation of gen-eral scientific theories which go beyond the content ofany possible body of experimental evidence. It is takeninstead to impede the assertion of such empirical gener-alizations as truths. The positive scientist instead com-piles various bodies of evidence which provide a basisfor decisions about when the scientific generalizationmight be applied.

The experimental data obtained will provide infor-mation about the fidelity and accuracy of the theory asmodelling the real world in a variety of circumstances,so that someone contemplating use of the theory willbe in a position to form an opinion about whether thetheory will provide a sufficiently reliable and accuratemodel for his purposes.

The mitigated scepticism of positivistic philosophy,following David Hume, accepts a priori truths of rea-son as certain, but expects of positive science that itdoes not go beyond what is entailed by the experimen-tal or observational evidence. Since most scientific the-ories involve empirical generalizations which go beyondany finite collection of particular observations, positivescience would seem by this doctrine to be eviscerated.

Metaphysical positivism recognizes two principleswhich constitute a kind of foundationalism. The first isconnected with the sceptical doctrine that all we knowis that “appearances appear”. However, what we countas an “appearance” is not confined to sensory impres-sions. Any impression which we may form on the truthof some proposition, whatever its source, is counted asan appearance. Such appearances we accept as whatthey are, and the body of science is considered to con-stituted just an organized presentation of a great deal ofsuch material. The enterprise is not solipsistic, it is col-laborative, and we therefore recognize as significant thesource of the impression, the identity of the person orentity who was the subject of the impression. Becauseof the broadening of the notion of appearance, proposi-tions expressing such appearances are called “opinions”and are to be tagged or digitally signed by the entitywhose opinion they are. The kind of entity which has

opinions is called an “authority”.Because of the collaborative nature of the enterprise,

an opinion will normally be formed by an authorityon the basis of (as entailed or otherwise supported by)some collection of opinions of other authorities. It istherefore normal for an opinion to be expressed, on theassumption that various authorities can be trusted, ormore specifically, on the assumption that all their pre-vious opinions are true. The collection of authorities onthe basis of whose opinions a further opinion is formed,together with the authority forms the new opinion, givea measure of the risk associated with the opinion whichwhich I call a degree of assurance. These degrees ofassurance are partially ordered. The more authoritieswhose opinions are involved, the lower the level of trust.The partial ordering becomes a lattice when we allowthat an opinion may be endorsed by several authoritieson the basis of distinct collections of other opinions.Adding more independent opinions increases the degreeof assurance.

Chapter 8

Language Planning

In his intellectual autobiography [Car63a] RudolfCarnap provides, as well as a sketch of the develop-ment of his thinking, a section covering ten importanttopic headings under which his ideas aspirations andaccomplishments in these areas were discussed. One ofthese topic areas is “language planning”.

Under this heading Carnap mentions two differentbut related problems, the problem of constructing “anauxiliary language for international communication”,and that of constructing “language systems in symboliclogic”. In both cases he credits Leibniz’s previous con-tributions.

Perhaps because different aspects of the latter prob-lem were a major part of his work and were addressedunder different headings (such as logical syntax and se-mantics) he devoted most of the space under this head-ing to non-symbolic languages.

He mentions the shift in his perspective from the uni-versalist conceptions of logic found in Frege and Russellto his own pluralism, and his concern with the difficul-ties arising in the choice of languages for science andthe systematization of science in a pluralistic context.

Though Carnap devoted much time to devising anddescribing methods for defining languages, their seman-tics and deductive systems, the problems which mightarise from the use in science of a plurality of such formallanguages were not addressed.

Since his day as a result of the invention of the digi-tal computer there has been an explosion in the use offormal languages, and other kinds of representation ofdata or knowledge.

If we wish to reason using the information heldin such representations, then we embrace a pluralismbroader than Carnap could have conceived, and some-thing like Carnap’s problem of “language planning” inits formal aspect, becomes pressing.

This problem is considered here as a prelude to thediscussion of architectures for formal analysis in whichthis diversity of representation can be encompassedwithout compromise to logical coherence.

8.1 Languages, Notations andRepresentations

The support of rigorous deductive reasoning is of theessence both in the projects of Leibniz and Carnap andin the successor we are considering.

Leibniz’s conception of how propositions should beexpressed in order for his calculus to operate was radicaleven by todays standards. The form of proposition wasthe subject predicate form, of which there were fourvariants, and in which the subject and predicate wereeach represented arithmetically as a pair of co-primenatural numbers. His calculus involved quite simplearithmetic operations on these numbers to determinewhether the proposition was true of false.

The merits and limitations of this proposal are notour present concern, it is mentioned to emphasize thevery broad limits on the kinds of representation andproof which are open to consideration.

By contrast with Leibniz, our aim is to admit intoour architecture a very broad range of representationsrather than devising one representation suitable for thecalculus and expecting all knowledge to be coded usingthat method of representation so that the calculus couldbe applied.

The general plan is that any representation be ad-missible provided only that it has a definite significance,preferably a formally specified significance or semantics.

38

8.2. UNIVERSALISM AND PLURALISM 39

8.2 Universalism and Pluralism

Frege and Russell, pioneers of the formalization ofmathematics, each conceived of a single language inwhich the whole of mathematics might be expressedand proven.

Frege’s formula has been paraphrased:

Mathematics = Logic + Definitions

Which may be read as asserting that given a suit-able logic, the whole of mathematics can be developedsimply by writing down formal definitions of the con-cepts of mathematics, and then proving the requiredmathematical theorems involving those concepts.

There is no reason in this scheme even to contemplatethe possibility of making use of more than one language,but if we do, problems immediately arise. The simplestis that the idea of a formal deductive system is such thatdeductions take place exclusively in one language. Thededuction rules which govern the structure of proofs insuch a system only allow premises within a single lan-guage to be used in a proof. The effect of this is toensure that if more than one language is used, the re-sults obtained in one language cannot be used in theproof of results in another language. This kind of awk-wardness is not a feature of informal mathematics andwould represent a serious and avoidable flaw in a formalapproach to mathematics.

Rudolf Carnap, inspired by Bertrand Russell toadopt a “scientific” approach to philosophy, and hopingto do for science what Russell had done for mathemat-ics, i.e. to establish methods suitable for the formaliza-tion of science. This was to be done by adapting thenew logical methods to the domain of empirical science.

There was in this an immediate difficulty, arisingfrom the necessity of making empirical claims. Con-cepts applicable to the material world cannot in generalbe defined in terms of purely logical contexts. It doesnot seem that one can proceed by analogy with Fregeand plausibly argue that:

Empirical Science = Logic + Definitions

Something more seems to be needed, and it is naturalto suppose that this is a language which goes beyondlogic by including concepts which are empirically sig-nificant.

So it appears that Carnap’s enterprise compelled himto consider new languages. Even before this, Carnap al-ready was, by his own autobiographical account, a plu-ralist with respect to language (though he might not at

that time have described himself in those words). Hispluralism at that time was primarily in relation to dis-tinct ways of talking about the world which were asso-ciated with different metaphysical postures. The mostsolid example of this is the distinction between the ma-terialistic language of science, in which material objectsare the subject matter of the language, and the phe-nomenalistic language of empiricist philosophers goingback to Hume, who confine themselves to talking notabout material objects but about the sensory evidencewhich we have of that world.

It is characteristic of Carnap’s quite original atti-tude to metaphysics that he was happy to talk in ei-ther of these languages even though their proponentstook them to be representative of incompatible viewson metaphysics. Carnap regarded the underlying meta-physical issues as meaningless, and took a pragmaticattitude to the use of the languages, eschewing anysupposed metaphysical commitment that might be sup-posed to entail. His pluralism, to be made explicit laterin his “principle of tolerance”, was to the effect that anylanguage can be adopted, subject to pragmatic con-siderations, rejecting the metaphysical dogmatists whoasserted the legitimacy of only that language which em-bodied their preferred metaphysics.

At the same time as Carnap enrolls himself into whathe thinks of as Russell’s program, in the 1920’s, thehigh noon of the Vienna circle, Hilbert at the center of agroup of mathematicians in Berlin is moving forward ona different conception of how to formalize mathematics,and Carnap, eager to absorb all the new developmentsin logic which might be relevant to his project, is payingattention.

Hilbert’s approach to formal mathematics, thoughfully embracing the new logical methods and pressingthem forward, is distinct from the universalistic con-ceptions of Frege and Russell and harks back to the ax-iomatic method first enunciated in Euclid’s elements.Hilbert had already applied the new methods to theaxiomatic theory of geometry, achieving standards ofrigour which for the first time surpassed those foundin Euclid’s geometry. This method envisages a speciallanguage for each branch of mathematics in which theprimitives were “implicitly” defined by systems of ax-ioms.

This is a substantial break with the ideas of Frege,who was very fussy about definitions. It is essential inthe development of mathematics to pay close attentionto definitions, since making use of a definition which

40 CHAPTER 8. LANGUAGE PLANNING

is incoherent invites paradox and invalidates all proofswhich depend on those definitions. Risks are also asso-ciated with undertaking definitions on a piecemeal ba-sis (defining a function first over one domain, and thenover some other domain), because of the possibility thatthese partial definitions might conflict, or that the par-tiality itself might result in unsound reasoning (Fregehad no methods for reasoning about partial functions).

8.3 On the Need for SyntheticPropositions

Carnap’s pluralism had several sources. One was hismetaphysical agnosticism, which made him disinclinedto reject languages which embodied objectionable on-tologies. A second was the gradual emergence afterFrege’s Begriffsschrift, first of direct alternatives (serv-ing essentially the same purpose) such as Russell’s The-ory of Types and then of a diversity of formal languagesserving a variety of purposes, such as modal logics. Thethird was the desire to have formal languages in whichsynthetic propositions could be expressed.

We will consider language planning here without ad-dressing the last of these three concerns, confining our-selves to the problem in relation to languages in whichonly analytic propositions can be expressed.

[There is a problem of order here. I had materialin the chapter on the architecture of knowledge whichexplains why and how we can do without formalizationof synthetic propositions. I have moved it here, but itprobably doesn’t fit yet. ]

To give a relatively concrete starting point to an oth-erwise highly abstract process I begin by describing anarchitecture which meets many of the more basic re-quirements, and in terms of which further requirementsmight be made intelligible. This is an architecture forthe support of interactive proof in a logical foundationsystem. In many respects this architecture answers tothe needs of Carnap’s project, but falls short of whatLeibniz envisaged. Before describing that architecturewe describe the idea of a logical foundation system formathematics.

8.4 Logical Foundation Systems

In describing the kind of architecture here proposed, itis useful to begin by describing certain kinds of systemwhich have been already in use and which provide a

starting point from which the proposed system may beregarded as a further development.

The kind of system I have in mind here is an “interac-tive theorem prover” for a “logical foundation system”.In this section I describe the idea of a logical founda-tion system. In the next, the structure of a typicalinteractive theorem prover working with a logical foun-dation system, and then the proposed architecture isapproached as a transformation to that starting point.

The idea of a logical foundation system begins withFrege, and refers to the kind of logical system which isneeded for his logicist project of reducing mathematicsto formal logic.

A logical foundation system is a formal language,preferably with a well defined semantics, and with adeductive system, which has sufficient semantic expres-siveness and deductive strength that in it the conceptsof mathematics may be defined and the theorems ofmathematics derived.

From the point of view of establishing Frege’s logicistthesis the system has to be in some sense “purely logi-cal”, and many contemporary philosophers doubt thatany such system can serve as a foundation for mathe-matics. From out present point of view however, thisaspect is inessential, it is not necessary for us to en-ter here into the question of what is a logical truth.The important part is that the system, whether purelylogical or not, be suitable for defining mathematicalconcepts and for deriving mathematical theorems fromthose definitions.

In speaking here of “definition” we do not intenddefinitions as mere syntactic abbreviation, but as theintroduction of new constants into the language satis-fying certain defining conditions. This is not to includeimplicit definition by arbitrary axiomatic extension asadvocated in Hilbert’s modern conception of axiomaticmethod, though the primitive constants of the systemwill effectively be defined in that manner. It is requiredthat the development of mathematics be possible bydefinitions which are conservative extensions of the log-ical system, and for which it is possible for the systemto reliably confirm that the proposed extension is con-servative, and hence that the extended system will beconsistent if the system being extended was.

The axiomatic set theory ZFC is a logical foundationsystem in the terms described above (though the meansof conservative extension are not normally regarded aspart of ZFC itself). Higher order logic as formulated byAlonzo Church in his Simple Theory of Types (STT)

8.6. THEORY HIERARCHY AS KNOWLEDGE BASE 41

is another foundation system which has proved popu-lar for applications of formal mathematics in ComputerScience, because of its successful implementation in anumber of interactive theorem provers.

8.5 Interactive Theorem Provers

There has been continued research on the use of com-puters for proving theorems since shortly after the in-vention of digital computers. Software for theoremproving can be divided into two kinds which are called“automatic theorem provers” and “interactive theoremprovers respectively.

An automatic theorem prover is one which is sup-plied a description of the proof required, and searchesfor a proof without further involvement of its user. Thismight be a theorem prover for first order logic which issupplied with a goal to be proven and some premisesfrom which to prove it and is then expected to comeup with a proof in first order logic. Generally, theuser would provide additional information influencingthe way in which the prover undertakes the search forthe proof, but the search would not itself involve anyfurther interaction with the user.

From the point of view of general mathematics, andespecially from the point of view of applications in in-formation systems engineering (say in software or hard-ware verification), an automatic prover may not be suit-able because of the complexity of the context in whichthe proof must be conducted, or simply because the re-quired theorem itself may be so complex that no com-pletely automatic proof is feasible within the currentstate of the art.

For these situations the use of an “interactive the-orem prover” may be preferable. Such a system pre-sumes that overall control of the construction of theproof is in the hands of its user, who will himself deter-mine the gross structure of the proof in which the com-puter will assist by filling in detail. Interactive theoremprovers usually incorporate automatic proof techniques,and may interface with automatic provers to obtain thekinds of proof which are within their scope. They in-creasingly act as integration platforms for diverse morenarrowly focussed proof automation facilities permit-ting a single problem to be solved by an appropriatemixture of methods.

The fact that the user has control over the proof ar-chitecture means that problems on a greater scale ofcomplexity can be tackled by an interactive theorem

prover, typically limited only by the amount of humaneffort needed to direct the proof and progressively re-duce the problem to portions modest enough to be un-dertaken automatically.

Because of the complexity of the specifications inthe context of which proofs with an interactive theo-rem prover, whether these be simply an aggregation ofbackground mathematical concepts relevant to the par-ticular branch of mathematics in which the advancesare to be made, or the specification of complex math-ematical models of computer software or hardware, aninteractive theorem prover must be able to organized anon-trivial collection of interdependent constant speci-fications. With typical theorem provers for variants onChurch’s STT this will be as a collection of “theories”.

A theory in this context is like a hybrid between theconcept of theory in mathematical logic (which is a setof sentences in some logical language) and the idea ofa module in a modular programming language. On thelogical side it is the purpose of the theory to collecttogether the definitions (axioms conservatively extend-ing the logical system) which determine some particularcontext in which reasoning will take place. The theory(as a data structure managed by an interactive theo-rem prover) may act as a repository for the theoremswhich are proven in the context of those definitions.The organization of definitions into a hierarchy of the-ories also serves to record interdependencies betweenconstant definitions.

On the modularity side theories serve to control thescope of relevance of definitions, so that a constant isonly in scope when all the constants upon which itsdefinition depends are also in scope, and may controlvarious aspects of the presentation of the formal mate-rial to the user, details of concrete syntax.

8.6 Theory Hierarchy as Knowl-edge Base

The interactive theorem prover and its theory hierarchyrepresents for us the status quo ante. That from whichwe seek to move forward in our successor to the projectsof Leibniz and Carnap.

To see this we consider how this relates to the ideasof Leibniz and Carnap.

Digital computers sewn together already acts in somerespects as the repository of all human knowledge, andthus serve a role similar to that for which Leibniz pro-

42 CHAPTER 8. LANGUAGE PLANNING

moted the development of academic journals and ency-clopaedia. They do so in the rather trivial sense thatall the very many academic journals which have ap-peared since the time of Leibniz are available online (fora price), but they also provide completely new waysof aggregating and disseminating scientific and otherknowledge, which threaten to displace the now tradi-tional academic journals. Beyond this computers storean ever growing volume of information about all aspectsof our lives and the world around us.

However, this information is mainly stored as data,the significance of which is not itself systematicallyrecorded. It is a database, not a knowledge base.

Though the theory hierarchy of an interactive the-orem prover was conceived of and is implemented tosupport only the accumulation of relatively small num-bers of definitions and theorems rather than data on alarge scale, it is of a very general character. Logicallyit would suffice for all the purpose for which computersstore and manipulate data.

To illustrate this in intermediate ground let us con-sider how these systems are in fact used, and make mod-est extrapolations in the direction of the ambitions ofLeibniz and Carnap.

The predominant uses of interactive theorem proversare in the development of purely mathematical theories,or of the kinds of applied mathematics which appearsin theoretical computer science and in the applicationof such theories to reasoning about various kinds ofcomputer systems, be they software or hardware. Thesoftware side is less significant because software mayplausibly be regarded as consisting of abstract entities,computer programs as analogous with mathematicalfunctions. The hardware is significant because we hereventure into reasoning about something concrete ratherthan abstract. We are here using logic and mathemat-ics to reason about the behavior of large scale electronicartifacts.

This is significant for our conception of Carnap’s pro-gram. One of the particular problems which Carnapgrasped in order to support the application of modernlogical methods in science was the extension of thoselogical methods to languages which speak of the con-crete world rather than simply to languages in whichone could speak only of abstract entities. This was thespur to Carnap’s pluralism, to his interest in the prob-lem of defining languages and their semantics, and inlanguages in which synthetic propositions could be ex-pressed. He departed from the universalistic perspec-

tive of Frege and Russell who sought a single logicallanguage for all purposes, because he saw these univer-sal logical languages as confined to the demonstrationof analytic propositions, and hence insufficient for thepurposes of empirical science.

But here we see, in the communities developing andapplying interactive theorem provers based on logicalfoundation systems, the application of those tools toreal world problems without any adaptation of the un-derlying logical system for that purpose.

Two considerations contribute to the possibility, eventhe desirability, of sidestepping this problem addressedby Carnap. The first is that computer scientists con-sidered first the use of these kinds of system for theverification of software, and it is reasonable to thinkof software as something abstract rather than as somekind of physical entity. For the verification of softwarethere is no apparent incentive to work with languagesin which synthetic propositions can be expressed.

Chapter 9

The Architecture of Knowledge

It is a thesis of this work that the advancement of in-formation technology renders choice of analytic method,not only in philosophy but wherever analysis mightprove useful, dependent upon the software available tosupport the method, and that the architectural designof such software should take place in the context of anexplicit (if generic or pluralistic) conception of analyticmethod.

This is an interdependency which may usefully beconsidered at the very earliest stages and at the highestand most abstract levels in the development of methodand of information or knowledge architecture. The in-terdependency is such that we may be tempted to iden-tify a certain kind of architectural design with a certainkind of fundamental philosophy or meta-philosophy.

In this chapter I undertake an analysis of knowledgearchitectures based the ideas about knowledge whichhave been so far presented. It is not desirable that anarchitectural discussion in a book of philosophy enterinto much specific detail, so the aim here will be theanalysis of certain ideas about the structure of knowl-edge as represented in information systems and the in-teraction between such conceptions of structure withthe kinds of functionality which the information sys-tems might then support, the methods which they facil-itate, and the directions of future development to whichthey are sympathetic.

Rather than attempting wholly to effect the kind ofintegration between architectural design and construc-tive philosophical analysis, I will present these as twodifferent perspectives upon a single enterprise, in thischapter the architectural design, in the next the philo-sophical perspective.

In this chapter the discussion will fall into two parts.An architecture is an abstract high level description.The analysis or evaluation of an architecture, must be

undertaken against some prior conception of the aimswhich the architecture is intended to realize. In en-gineering terms these are high-level requirements. Inphilosophical terms, these requirements correspond toa delineation of the problem domain.

The first level at which positive philosophy departsfrom being purely analytic is in the choice of subjectmatter. A substantive statement about some practi-cal matter, perhaps in politics or economics, may beclothed in a pure analysis, implicit in the choice of sys-tem to be studied. To promulgate ideas about howsociety might be organized, it would suffice to proceedby analysis, considering a class of realizations of theideas and examining their relative merits.

This is the manner in which I proceed here. My in-terest is in certain approaches to the development ofknowledge as a collaborative enterprise (which usuallyis) making effective use of our developing informationand network capabilities. I begin the architectural dis-cussion by setting out the domain of enquiry as a state-ment of requirements, and then proceed to consider andcompare some of the ways in which those requirementsmight be met.

These two stages, statement of requirements, re-sponse to requirements, will not be monolithic. Therequirements will be stated little by little. To eachstage architectural responses are considered, and therequirements may then be augmented in the light ofthe analysis.

9.1 Requirements from Leibnizand Carnap

I have already identified the projects of Leibniz andCarnap as points of departure, so I begin with some

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44 CHAPTER 9. THE ARCHITECTURE OF KNOWLEDGE

key features of those projects.The principal elements of Leibniz’s project were:

• A Universal Language.

• An Encyclopaedia encompassing all scientificknowledge.

• A Calculus Ratiocinator for deciding truth.

Carnap’s project was more narrowly scoped, butshared the first two items recast in pluralistic terms.

Here we adopt all three, adjusting the statement inthe direction of pluralism, and thinking of informationtechnology.

Thus we are interested here in:

RA1 A system for the representation of propositionalknowledge.

RA2 A substantial online knowledge base represented inthat system.

RA3 Software supporting the further development andapplication of the knowledge.

Carnap attached considerable importance to the dis-tinction between analytic and synthetic propositions.Though he did not acknowledge the influence of Hume,he agrees with Hume in characterizing essentially thesame dichotomy, Hume’s fork, in three distinct ways.Carnap sought to adapt methods similar to those ofFrege and Russell in the formal derivation of analyticpropositions to languages in which synthetic proposi-tions could be expressed and used in formal derivations.

Carnap’s approach to the meta-theory of such em-pirical languages is not from our point of view entirelysatisfactory. The approach envisaged here to the con-nection of our languages with the empirical world isentirely different. Whereas

............The central idea of interest concerns the role of de-

duction, in the acquisition and application of knowl-edge, and the possible contribution of information tech-nology in support of deduction.

Deduction is relevant in two general ways. The sepa-ration of deductive from other kinds of reason permitsthe deductions themselves to be checked more rigor-ously, and exposes the premises (which might otherwisehave been obscured) on which the deductions depend.We are here concerned with a machine supported injec-tion of formality into reasoning the intended effect ofwhich is to improve precision, rigour and reliability.

Formality and deduction also potentially enable newkinds of functionality to be realized. This is becausethe search processes involved in finding proofs of logi-cal conjectures can serve to discover witnesses for exis-tential claims, and hence solutions to design problems.The ability to demonstrate compliance of such a solu-tion underpins and sanitizes the application of exoticand possibly unreliable methods during the search fora solution, hence allowing solutions to be discoveredwhich might never be found by less exotic algorithms.

9.2 Epistemic Retreat

The positivistic idea of epistemic retreat influence therequirement in various ways.

Firstly we distinguish between analytic and syntheticjudgements, and between formal and informal claims.The system is primarily concerned with the formal side.

As in Hume, we accept analytic propositions as ex-hausting those which can be known with certainty. Infact we go one further, taking empirical claims to be atbest approximations, best thought of and formally rep-resented as models of aspects of reality. As such theyshould be assessed or affirmed not as true or false butin more complex and informative terms. Thus, what weaffirm of a theory is not its truth but its applicabilityto certain aspects of reality and the accuracy and relia-bility with which it models those aspects under variouscircumstances.

Formally, we do not assert an empirical claim inconnection with such a model, we instead formulatethe theory as an abstract model, and on the basis ofthis definition we can then undertake theoretical elab-oration of the theory, draw consequences in relationto its application in hypothetical situations, and for-mally evaluate the theory against experimental datapresented in the terms of the abstract model.

• RE1 Only analytic propositions are formally as-serted.

• RE2 Empirical theories are represented as abstractmodels.

• RE3 Judgements are qualified by a measure of as-surance.

Chapter 10

Metaphysical Positivism

Metaphysical Positivism is the name I have given tothe theoretical aspects of Positive Philosophy, which isitself of broader scope.

The reason for naming a philosophical manner in thiscase is primarily for convenience of reference. Meta-physical positivism provides certain background mate-rials in the context of which positive philosophy mustbe understood. Positive philosophy itself is simply phi-losophy conducted in the context of that background.The boundary between the two is not sharp but approx-imates to the traditional division between theoreticalphilosophy and practical philosophy.

In theoretical philosophy we consider those aspects ofphilosophy which have greatest impact on philosophi-cal method, viz. metaphysics, logic, epistemology, andcertain aspects of scientific method.

Metaphysical positivism is so called firstly because ofits close connection with logical positivism, particularlywith the philosophy of Rudolf Carnap, and because ofthe single most striking difference between it and log-ical positivism, which is in its use of the term meta-physics. It may therefore be helpful to begin with someremarks about the similarities and differences betweenmetaphysical positivism and its predecessor.

10.1 First Base

Metaphysical Positivism is principally concerned withthe foundations of knowledge. It therefore places epis-temology, the theory of knowledge, to a central place intheoretical philosophy.

If we seek to build an enduring structure, it is best tobuild on a solid foundation. Critics of foundationalismshave taken foundationalism as demanding that such afoundation be immune to doubt, that it be, as a foun-

dation for knowledge, absolutely solid. The foundation-alism of Metaphysical positivism is not predicated onthe existence of such foundations. It is rather the morepragmatic aim, given that we must start somewhere, tofind the best place to start from. A foundation is there-fore not to be absolutely solid, but just solid enough;fit for purpose.

When I speak here of a foundation as a place to start,this should not be taken too strictly. When we builda house, we begin with the foundations. The construc-tion of the foundations may take perhaps one third ofthe time required for completing the house. In thiscase, the foundation is not something which we simplyidentify and use as a starting point. It is a stage in theconstruction, after which the character of the enterprisechanges.

The foundation provides something one which ahouse can be built. The reason why the house stands se-curely is because it is built on a solid foundation. Thisis not the reason why a foundation is solid. A foun-dation is not evaluated in the same way as the house.Sometimes a foundation is solid because it consists ofconcrete laid on solid rock. Sometimes a foundation issolid because it is a concrete raft laid on something amuch less rigid, perhaps clay. Sometimes a foundationis solid because it is made by driving piles into groundwhich is rather soft.

Ultimately, in these cases, the criteria are pragmaticand based on experience. The best foundation for abuilding is chosen taking into account the nature of theland on which it is to be build, the kind of buildingit is to support, and a great deal of experience andscientific knowledge of how different kinds of foundationwill behave in these circumstances.

The foundationalism of Metaphysical Positivism issimilarly pragmatic. It consists of ideas about what

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46 CHAPTER 10. METAPHYSICAL POSITIVISM

ways of establishing, evaluating and applying differentkinds of knowledge have proven effective, and on whatmethods we may expect to be effective as informationtechnology and other factors transform the way we workwith knowledge in the future.

The foundationalism of Metaphysical Positivism isthus self-consciously futuristic, it is oriented towardsways of working which will make the most of futureadvances in information engineering. The pragmaticaspect brings with it an epistemological pluralism. Thefoundationalisms I here espouse are not offered to theexclusion of other approaches.

There are three major stages in the foundations,closely connected with Hume’s two forks. The stagesare addressed in sequence, each building on the earlierfoundations.

An important part of the foundations proposed issimply conceptual. It is in the adoption of certain con-cepts with relatively definite meanings. The first con-cepts to consider are those which distinguish the threekinds of foundation.

The most fundamental of these comes from Hume’sfork, and the principal concept which we associate withthis dichotomy which Hume identified is that of analyt-icity. Our first foundations are therefore foundationsfor analytic truth, and this is the main focus of thediscussion here.

In relation to analytic truth, we do not advocate thatany proposition be regarded in as true in a completelyunqualified way. The suggestion is that our system in-volves only the expression of opinions on analyticity,and that such opinions are in general expressed as basedon certain other opinions. In very many cases these willbe very solid opinions. Often the opinion will be the“opinion” of a an interactive proof tool which has con-structed and checked a formal proof of the propositionand made use of no other opinion.

The second kind of foundationalism concerns syn-thetic propositions, the other side of Hume’s first fork.Let us think of scientific laws as typical of this kindof knowledge. In respect of such laws I do not advo-cate that these should be considered in terms of truthand falsity. Experience tells us that scientific law aregenerally no more than approximations to “the truth”.Whether or not this is always the case, there are suffi-ciently many useful scientific laws which are known notto be strictly true that an epistemology which recog-nizes the merits of these is desirable.

Scientific laws are therefore considered as models of

aspects of reality which are not regarded as either trueor false, but as more or less accurate and reliable modelsof various aspects of the real world. The constructionof such models is a purely logical matter, and the theo-retical aspects of science which consist in the analysis ofsuch logical models are covered under the foundationalproposals for analytic truth. The new epistemologicalproblems which arise concern the relationship betweenthese abstract theories and those aspects of the worldwhich they model.

10.2 By Comparison with Logi-cal Positivism

Here and throughout, whenever I speak of logical posi-tivism this should be understood to refer specifically tothe philosophy of Rudolf Carnap whenever it concernsa matter on which the logical positivists may not havebeen unanimous.

The headline contrast with logical positivism is in re-lation to the word metaphysics, which I use in a mannerquite distinct from the way in which it is used by Car-nap. The best known feature of Carnap’s philosophy ishis repudiation of metaphysics, around which, it is easyto suppose, his entire philosophy revolves. Metaphysi-cal positivism embraces metaphysics, but the kinds ofmetaphysics which are accepted are not the kinds whichwere rejected by Carnap.

Metaphysics for Carnap is construed in very specificways, and rather more narrowly than is usual in thepositivistic tradition. Positivism is usually associatedwith nominalism, and involves the denial that abstractentities exist. Carnap on the other hand, was an on-tological pragmatist, it sufficed for him that reasoningabstract entities was convenient for science to justifytheir use. His paper Empiricism, Semantics and On-tology is an exposition of his liberal attitudes in thesematters. The metaphysics which Carnap did reject fellprimarily under two headings. The first is the synthetica priori, the second heading covers claims which haveno definite meaning.

The first of these categories in Carnap’s conception ofmetaphysics is void likewise in Metaphysical Positivism.It is so partly because of the definitions (which areadopted in metaphysical positivism) of the concepts,and partly as an adopted epistemological criterion. Ofthese, more later.

So far as those which fail to be synthetic because

10.3. PRINCIPAL FEATURES 47

they are meaningless, the position of metaphysical pos-itivism is softer. It is in the nature of philosophy that ituses or investigates concepts whose meaning may be un-certain, or difficult to articulate. Dogmatic scepticismabout meaning in various degrees is common amongacademic logicians and philosophers, and is not a fea-ture of metaphysical positivism. In this context bydogmatic scepticism we mean the movement from in-comprehension to rejection. Our position in relation todoubt about meaning is to reserve judgement. How-ever, if the proposition in doubt were offered as syn-thetic a priori, in the sense in which these terms areunderstood in metaphysical positivism, then a firmerrejection would be called for.

Since Kripke the rejection of the synthetic a priori hasgenerally been supposed to have been refuted. How-ever, it can be seen that insofar as the relevant ar-guments are sound, then they must relate to conceptsdistinct from those adopted in metaphysical positivism(and distinct from these concepts as used by Carnap).The first step in showing this is to note that Carnapdefines necessity as in terms of analyticity.

................

Having seen the historical development of philosoph-ical positivism, having reconsidered positivism in thelight of the principal criticisms which were levelled atits most recent manifestation in Logical Positivism, andtaking account of certain ideas on about how informa-tion technology may transform the nature of knowledge,it is now time to draw these themes together in a con-cisely stated positivistic synthesis.

Metaphysical Positivism is a graduated, constructivescepticism. In describing it as sceptical the emphasis isplaced upon an open minded suspension of judgement.

This suspension is graduated, and does not deny ap-parent and sometimes quite radical differences in ourconfidence of working hypotheses. The most fundamen-tal of such differences are associated with that betweenlogical and empirical knowledge associated with the an-alytic/synthetic distinction, and this leads to quite dif-ferent ways of evaluating and affirming analytic andsynthetic hypotheses. The constructive side of thisscepticism leads us into an epistemology which is cou-pled with architectural principles for the the future ex-pansion of our knowledge in the context of a globallyshared information infrastructure.

..............

10.3 Principal Features

Metaphysical positivism is primarily concerned withanalytic method, and with the conceptual frameworkin which such methods can be articulated and evalu-ated and applied.

It is both linguistically and methodologically plural-istic, as in Carnap’s pluralism language is adopted onthe basis of pragmatic considerations. We are howeveraware, as Carnap was, that choice of vocabulary is im-portant, and there is no suggestion that these choicesare arbitrary.

Chapter 11

Digging Deeper

11.1 Foundations for Knowledge

How are we to judge claims to scientific knowledge?

In metaphysical philosophy and natural philosophythere have been ideas on this topic which may be calledfoundational. In this chapter we present some ideasalong these lines.

Principally this concerns foundations for logical andmetaphysical knowledge, knowledge a priori, but I willtouch upon foundational aspects of empirical knowl-edge.

Before entering into a positive account of foundationsI want to say a few words about the role which suchfoundations are intended here to fulfill.

It may be useful to draw an analogy with the use ofthe term ‘foundation’ in the construction of buildings.In the context a foundation provides a base solid enoughfor the construction of the desired building, so that thebuilding will stand firm and will survive the stresses towhich it may reasonably be expected to be subject.

For this a foundation does not need to be absolutelysolid.

The construction of a foundation does not itself pro-ceed in the same way as that for the building. Onedoes not, in order to obtain a solid foundation, seek ayet more solid foundation on which to build the founda-tion (though sometimes bedrock serves this function).There does not arise in this way, an unsolvable problemof regress in the foundations of buildings.

There are two reasons

11.2 Logical Foundations

In keeping with the positivist tendency to which it be-longs, our account of metaphysical positivism has been

concerned primarily with underpinning and articulat-ing methods and tools suitable not only for rigorousphilosophical reasoning but for application in scienceand engineering.

In Aristotle’s conception of first philosophy, utility isregarded with some scorn, and the insistence of posi-tivists that philosophy should facilitate positive scienceleads to the idea that positivism is an anti-philosophicalphilosophy (which is consistent with seeing it as contin-uous with academic and pyrrhonean scepticism).

Metaphysics is the name by which those topics atthe apex of philosophy as conceived by Aristotle is nowknown, and the name “metaphysical positivism” maytherefore be read as hinting that the pragmatic orien-tation of our positivism does not involve a rejection ofthose more remote regions of philosophy whose connec-tion with life seems most tenuous.

Nevertheless, in metaphysical positivism, locating aplace for metaphysical investigation is not easy. Twokinds of defect which may be found in metaphysical(and other controversy) from a positivistic standpointare meaningless claims and purely verbal disputes. Itis characteristic of positivist to reject metaphysics asmeaningless, and in metaphysical positivism I retain aconcern for precision and clarity in language, which mo-tivates some of the deeper concerns which we addresshere. However, it is the business of philosophy to ad-dress problems whose articulation is difficult, and thatone philosopher does not find a conjecture or a defini-tion meaningful does not suffice to establish that it isnot.

In keeping with the graduated scepticism in meta-physical positivism meaningfulness is not taken to bean all or nothing affair. Languages (or idiolects) maybe compared on two related kinds of scale. First theymay be compared according to their expressiveness. A

48

11.2. LOGICAL FOUNDATIONS 49

language A is as expressive as language B if everythingwhich can be said in language B can also be said in lan-guage A. To compare precision of definiteness of a lan-guage we have to consider languages as having multiplepossible meanings or interpretations and then comparethe range of interpretations of two languages.

An easy and fundamental illustration of this kind ofcomparison may be found in axiomatic set theory. Ifwe consider a specific theory. say ZFC, the axioms ofthe theory provide an implicit definition of the conceptof set which is the subject matter of the theory. Thetruth conditions of sentences of ZFC can be made verydefinite by stipulating that a sentence is true in ZFC iffit is true in every model of the axioms. Truth will thencorrespond to provability, in consequence of the com-pleteness of first order logic. It is also reasonable in thisdomain to take the meaning to be the truth conditions,so that ZFC becomes as definite in its meaning as firstorder logic is. Unfortunately when we look at the in-tended applications of set theory, of which the first isto the theory of arithmetic, we find that this concep-tion of the meaning of set theory is unsatisfactory. Thenormal procedure in reasoning about arithmetic in settheory is to define the natural numbers as some con-venient countably infinite set of representatives. Themost popular has been the scheme for representation ofordinal numbers under which the natural number zerois represented by the empty set, and every other natu-ral number is represented by the set of its predecessors,i.e. the set of all natural numbers which are less thanthat number.

Using this definition, together with definitions of theusual arithmetic operations over these representatives,we can derive the usual theorems of arithmetic in ZFC.More true theorems of arithmetic are provable in thisway than is possible in the usual direct axiomatiza-tion of arithmetic in first order logic (known as PA, forPeano Arithmetic). However, it is known, as a resultrelated to the incompleteness results proved by KurtGodel that not all the truths of arithmetic are provablein this way.

However, because of the completeness of first orderlogic, all the statements of arithmetic as expressed inthe way indicated in ZFC which are true under that se-mantics. The arithmetic truths which are not provablein ZFC, are, under that semantics, with that mannerof representation of numbers, not even true. We havefailed to produce an adequate definition of the naturalnumbers.

This is not an avoidable defect in the Von Neumannrepresentation of ordinals. It is easy to see that underthe proposed semantics for ZFC the truths of set theorywill be recursively enumerable, and therefore any de-cidable subset of those truths (the truths of set theorywhich happen to correspond to sentences of first orderarithmetic) will also be effectively enumerable, what-ever definition of natural number we start out with.But it is know that the truths of arithmetic are notrecursively enumerable. It follows that the concept ofnatural number is not representable in ZFC under thegiven semantics.

Though the semantics is definite, it is not expressive.

We can make the semantics more expressive, at thecost of making it less definite, in the following way. Thesemantics we have been discussing involves acceptanceof all models of ZFC, and its (semantic) incompletenessreflects the existence of models of ZFC in which theset defined to be the natural numbers is not what thedefinition is intended to give.

The definition of the natural numbers is intended togive a set whose members are all the sets obtainablefrom the empty set by repeated application of the suc-cessor function (the function s(x) = x+1). The idea“obtainable by repeated application” cannot be directlyexpressed in first order logic, so the definition instead isgiven in terms of closure under the successor function.A set is closed under the successor function if for everymember of the set, its successor is also a member. Thenatural numbers are then defined as the intersectionof all sets which contain the empty set and are closedunder the successor function.

Unfortunately, if we take an interpretation in whichthe intended set of natural numbers does not exist, inwhich every set which contains all the natural numbersalso contains some other set, then when you take theintersection you get a set which contains that other set.This is a model with non-standard natural numbers,and such an interpretation will not get the truths ofarithmetic right. Because our semantics allows thesenon-standard models, as well as models in which arith-metic is standard, statements of arithmetic which aretrue in the standard models but which are violated insome non-standard model will come out under the se-mantics as false.

To get the semantics on the nose for arithmetic state-ments we need to eliminate these non-standard modelsfrom the semantics. We cannot do this by adding an-other axiom, because the required constraint on the

50 CHAPTER 11. DIGGING DEEPER

models is not expressible in first order logic. But wecan add an informal stipulation to the semantics. Wecan specify the truth condition for sentences in ZFCas truth in all models of ZFC with standard naturalnumbers.

We now have a version of ZFC which is more expres-sive than the previous one. One in which the naturalnumbers really are definable (though not in the sense ofthis term which is used by mathematical logicians) andin which the sentences of arithmetic have the correcttruth values. Though the semantics of this languageare defined in part informally, the language can nowbe used to define the semantics of other languages in aformal way (relatively), whose semantics would not bedefinable in first order logic.

In this way we can define variants of the language offirst order set theory which have progressively greaterexpressive power. This is done by using informal con-straints on the class of intended interpretations of thetheory. Such informal constraints can be placed in or-der of strength and the stronger the constraint is themore expressive the resulting language will be.

Beyond the constraint to models with standard nat-ural numbers the following stronger constraints can beapplied:

• Constraining interpretations to be well-founded.

• Requiring full power-sets.

Constraining interpretations to be well-founded isstrictly stronger than requiring standard natural num-bers. It is as strong because in any model in whichthe set of natural numbers is non-standard it is notwell-founded. It is strictly stronger because the sameconsideration applies to all limit ordinals (all transfinitenumbers). In the absence of well-foundedness we canhave models which give standard natural numbers buthave a non-standard ordinal somewhere higher in thehierarchy. Well-founded models not only have standardarithmetic, but they also have standard ordinals all theway up. So under this semantics, but not under theprevious one, the ordinals are definable.

Yet greater strength is obtained by requiring the fullpower set. All models of ZFC are closed under the for-mation of power sets. That means, that for every setin the domain of discourse the collection of its subsets(in the domain of discourse) is also a set in the domainof discourse. However, it is not the case that the powerset is the same in every model, since not all subsets of

the set are bound to be in the domain of discourse. Thepower set will include all the subsets which exist, butthere may be subsets which don’t exist (in this particu-lar model). The constraint to full power sets eliminatesany model in which there are missing subsets.

If we define truth in ZFC as truth in every model ofZFC in which the power set is full, then we get anotherlanguage which is again strictly more expressive thanthe ones we have previously considered. Why is this?To understand this it is helpful to address the questionwhy there exist non-well-founded models of ZFC.

There is an axiom in ZFC which is intended to assertthat all sets are well-founded. This is called the axiomof regularity. Its intended effect is to deny that thereare any infinite descending chains in the membershiprelationship, but this, like the obvious informal defini-tion of the natural numbers, cannot be directly statedinformally. The axiom of regularity states instead thatevery set has a minimal element. A minimal element ofa set A is a member B of A which contains no memberof A, such that A intersection B is empty.

This definition does ensure well foundedness if thereare enough sets in the domain of discourse, which is thecase if we have full power sets. Otherwise it does not,and there are non-well founded models of ZFC despiteit having an axiom intended to deny their existence.Well foundedness is not expressible in first order logic,and so cannot be fully incorporated into the axioms,which is why the previous semantics relies on an infor-mal constraint to well-foundedness.

If instead of requiring well-foundedness we stipulatethe semantics in terms of models with full power sets,then since these are all well founded the resulting se-mantics is as expressive as the semantics based on ar-bitrary well-founded models.

That the semantics is strictly stronger can be seenfrom consideration of cardinal numbers. Cardinal num-bers may be represented in ZFC as initial ordinals. Aninitial ordinal is an ordinal which has a greater car-dinality than any previous ordinal. Two sets have thesame cardinality if there exists a bijection between theirelements. Such a bijection pairs up the elements in thetwo sets in a one-one manner so that they can be seento have the same size. A difficulty with this definition ofcardinality arises from the existence of models in whichnot all subsets of every set are present, in which wedo not have full power sets. This is because the non-existence of a bijection might arise not because the twosets really are of different size, but because the bijection

11.3. EMPIRICAL FOUNDATIONS 51

between their elements is just not in the domain of themodel. The effect of this is that not all models of ZFCagree about which ordinals are initial, and consequentlythey do not agree about cardinal arithmetic.

Constraining the truth conditions to involve truthonly in models with full power sets eliminates thissource of disagreement between intended models of ZFCabout cardinal arithmetic. Under this semantics butnot under any of the previous semantics we can definethe notion of cardinal number, so it gives a strictly moreexpressive language.

11.3 Empirical Foundations

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[VH67] Jean Van Heijenoort, editor. From Fregeto Goedel - A Source Book in MathematicalLogic, 1879-1931. 1967.

[Wit61] Ludwig Wittgenstein. Tractatus Logico-Philosophicus. Routledge Kegan and Paul,1961.

[WR13] Alfred North Whitehead and Bertrand Rus-sell. Principia Mathematica. Cambridge Uni-versity Press, 1910-1913.

52

List of Tables

3.1 Development of the Analytic/Synthetic distinction . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6.1 Rigour, Scepticism, Positivism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

53

Positions

3.1 the world of forms and world of appearances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 the a priori (knowledge) and a posteriori (opinion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 essential and accidental predication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 definition and essence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

54

Index

a priori, 20analysis, 22analytic, 22analytical engine, 27appearances, 17axiomatic method, 16

Bacon, 17Begriffsschrift, 27

calculus ratiocinator, 7, 29Carnap, Rudolf, 13Church, 29Church, Alonzo, 40Comte, Auguste, 34

Empiricism, 17Empiricus, Sextus, 33equipollence, 33Euclid, 27

Foundationsof knowledge, 45

Heraclitus, 17Hilbert, David, 27, 28Hobbes, 17Hume

enquiry, 14treatise, 14

ideal forms, 17

Kant, 22Kleene, 29Kolakowski, 34Kripke, Saul, 6Kronecker, Leopold, 28Kurt Godel, 49

Leibniz, 13, 29

Locke, 17logical foundation system, 40

meta-mathematics, 27Metaphysical Positivism@Metaphysical Positivism,

45Moore, G.E., 24

Parmenides, 17positivism, 34Post, 29Principia Mathematica, 21

Russell, 18Russell, Bertrand, 24, 27

Simple Theory of Types, 40STT, 40sufficient reason, 18synthesis, 22synthetic, 22

Thales, 26the LCF paradigm, 29Turing, 29

universal characteristic, 7, 27

Von Neumann, 49

Whitehead, A.N., 24Wittgenstein, 18Wittgenstein, Ludwig, 6

Zeno of Elea, 17ZFC, 40, 49

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