Erkenntnis Volume Issue 2013 [Doi 10.1007%2Fs10670-013-9573-x] Andreas, Holger -- Carnapian...

8/13/2019 Erkenntnis Volume Issue 2013 [Doi 10.1007%2Fs10670-013-9573-x] Andreas, Holger -- Carnapian Structuralism

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O R I G I N A L A R T I C L E

Carnapian Structuralism

Holger Andreas

Received: 31 August 2012 / Revised: 4 June 2013 / Accepted: 27 September 2013 Springer Science+Business Media Dordrecht 2013

Abstract This paper aims to set forth Carnapian structuralism, i.e., a syntactic

view of the structuralist approach which is deeply inspired by Carnaps dual level

conception of scientific theories. At its core is the axiomatisation of a metatheo-

retical concept AE(T) which characterises those extensions of an intended appli-

cation that are admissiblein the sense of being models of the theory-element T and

that satisfy all links, constraints and specialisations. The union of axiom systems of

AE(T) (whereT is an element of the theory-netN) will allow us to present scientifictheories in an axiomatic fashion so that deductive reasoning in science can be

formalised. Thereupon defeasible and paraconsistent means of reasoning will be

introduced. The logical study of scientific reasoning is the key motivation of Car-

napian structuralism. A further motivation is to help overcome the syntactic-

semantic split in the philosophy of science.

1 Introduction: Micrologic versus Macrologic

The relationship between Sneedian structuralism and CarnapianWissenschaftslogik

is ambivalent. On the one hand, the structuralist framework aims to provide formal

means of rationally reconstructing scientific theories and as such continues the

Carnapian enterprise. On the other hand, structuralism breaks with the syntactic

view of scientific theories which is essential to Carnaps logic of science. As is well

known, the structuralist notion of a scientific theory is explicated in terms of

interrelated sets of models as opposed to interrelated sets of axioms.

In order to explain and to highlight the distinctive traits of the Sneed formalism,

Stegmuller (1976, p. 12) came to speak about the difference betweenmicrologicandmacrologic. The former concerns inferential relations between sentences and

H. Andreas (&)

Munich Center for Mathematical Philosophy, Munich, Germany

e-mail: [email protected]

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DOI 10.1007/s10670-013-9573-x


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formulas, whereas the latter explicates such metatheoretical concepts as an intended

application, a model of a theory-element, a global link etc. Needless to say,

Stegmuller considered the macrological perspective of scientific theories to be more

advanced and superior to the micrological one. It seems to be an implication,

moreover, of his account of structuralism that the two perspectives exclude oneanother. Holding macro- and micrologic to be incompatible is a telling case of the

general split into semantic and syntactic views in the philosophy of science.

As a consequence of introducing structuralism as a particular species of the

semantic view of scientific theories, research on the structuralist framework has

become largely disconnected from the analysis of scientific reasoning. The label

non-statement view, which was introduced by Stegmuller, emphasises this

dissociation from the logicians enterprise of studying the norms and the laws of

human reasoning. This characterisation was supported by the doctrine of the

indivisibility of a theory-elements and a theory-nets global empirical claim intosingle axioms. Not all structuralists, however, were happy about the non-statement

characterisation and the split into micro- and macrologic, let alone philosophers of

science outside the structuralist community. Nonetheless, the issue of scientific

reasoning in the structuralist framework has rarely been addressed in the literature.

Hence, the doctrine of the indivisibility of global empirical claims has remained

unrefuted.

The objective of the present paper is to reestablish the micrological perspective in

the macrologic of structuralism. This will be brought about by introducing a novel

metatheoretical relation AE(T) into the structuralist framework by means ofCarnapian postulates. More precisely, I will set forth axioms that characterise those

extensions of an intended application that are admissible in the sense of being

members of a system of extensions in which all links, constraints and specialisations

are satisfied and where all extensions are models of the respective theory-element.

The resulting system qualifies as a scientific theory in the sense of Carnaps good

old-fashioned dual level conception. Carnapian structuralismappears a fitting label

for logical and philosophical investigations being guided by these axioms.

Carnapian structuralism allows for the logical analysis of deductive and

defeasible reasoning in science. It retains the expressive power of the structuralist

framework without losing the expressive resources of predicate logic. This is its

chief motivation. Carnapian structuralism, thus, aims to contribute to the

understanding of how formal mathematical logic is related to actual scientific

reasoning. It should be of interest, therefore, to any logician who thinks that formal

logical systems are explanatory of scientific reasoning.

The present paper refines and summarises results obtained in Andreas (2010b,

2011, 2013). The major refinements concern, first, the formulation of links and

constraints, which is unified in the present account. This deviation from classical

structuralism achieves a significant simplification of structuralist theory presenta-

tion. Second, the inference system ofpreferred subtheories by Brewka (1989) now

introduces defeasible forms of reasoning into structuralism. This system is simpler

and behaves logically more nicely than prioritised default logic, which was used in

Andreas (2011).

H. Andreas

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2 The Formal Framework of Carnapian Structuralism

Structuralism makes essential use of set-theoretic predicates in the sense of Suppes

(1957). Such predicates have a semantic and a syntactic side such that a statement

view on set-theoretic predicates and their explicit definitions seems to remain aviable option. In fact, Suppes (2002, p. 30) himself considers the analysis of

scientific theories in terms of set-theoretic predicates as a proper way of

axiomatising such theories:

Although a standard formalization of most empirically significant scientific

theories is not a feasible undertaking for the reasons set forth in the preceding

section, there is an approach to an axiomatic formalization of such theories

that is quite precise and satisfies all standards of rigor of modern mathematics.

From a formal standpoint the essence of this approach is to add axioms of set

theory to the framework of elementary logic, and then to axiomatize scientific

theories within this set-theoretical framework. From the standpoint of the

topics in the remainder of this book, it is not important what particular variant

of set theory we pick for an axiomatic foundation. From an operational

standpoint what we shall be doing could best be described as operating in

naive set theory.

Both standpoints described by Suppes, the formal and the operational one, are in

play in the present investigation. As in classical structuralism, I will proceed in a

semiformal fashion using naive set-theory. However, the format of structuralisttheory presentation will be an axiomatic one as a theory-net is given by a set of

postulates and definitions. Moreover, the descriptive vocabulary of these postulates

and definitions is divided into theoretical and observational terms as in Carnaps

dual level conception. Unlike the latter conception, the key theoretical and

observational terms of Carnapian structuralism are set-theoretic predicates.

As has been pointed out by Suppes (2002, Ch. 2), the semiformal style of

defining set-theoretic predicates is precise enough to indicate how a complete

formalisation using axiomatic set theory can be obtained. Such a full formalisation

of the definitions of set-theoretic predicates allows for a full formalisation of the

axiomatic system of a theory-net to be developed in the present investigation. This

view upon the axioms of a theory-net amounts to the formal standpoint.

The formal standpoint requires an axiomatisation of set theory in first order

logic.1 The first order system that comprises some variant of axiomatised set theory

is assumed to satisfy the following conditions:

(1) There is one and only one domain of interpretation, which comprises empirical

objects and sets.

(2) The axiomatisation of set theory must make room forUrelements in order to

distinguish empirical objects from one another.

1 Using higher order logic without set theory would be an alternative, though. This strategy is not pursued

here.


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Several systems satisfying these two conditions have been developed in the

literature. For an extension of ZF that accounts for distinguishable Urelements see

Loewe (2006). (This system is favoured by Gerhard Schurz (this volume) in his

axiomatic approach to structuralism.) Another promising strategy is to use the

formal axiomatisation of naive set theory in a paraconsistent framework, such as

that of adaptive logics [see Verdee (2013)].

Throughout the paper I assume the reader to be familiar with the fundamental

metatheoretical concepts of the structuralist framework. So, for example, the

symbolsM(T),Mp(T)and r(T)(which stand for the set of models, potential models

and the restriction function) are understood in the standard way. Any structuralist

reconstruction of a scientific theory comprises, for any T2 N;explicit semi-formaldefinitions of these symbols. Carnapian structuralism builds upon such definitions.

The notion of a theory-net is understood here in the wider sense of capturing a set

of axioms coming from whatever scientific theories. This understanding conforms to

the formal definition of a theory-net (Balzer et al. 1987, p. 172) but does not require

such a net to have a tree-like structure. Theory-nets thus understood resemble

theory-holons as introduced in Balzer et al. (1987, Ch. VIII). Unlike the latter,

however, they do not capture relations of theoretisation, reduction and equivalence

between scientific theories. Two theory-elements T; T0 2N may well differ inregard to the type of their potential models.

3 Postulates for Structuralism

In the original exposition of the structuralist framework by Sneed (1979), the global

empirical claim of a tree-like theory-net was described in terms of T-theoretical

extensions of sets of intended applications. More precisely, this claim says that, for

all theory-elements T in the net N, the set of intended applications ofT has a set

E(T) of structures such that (i) any intended application of T has a T-theoretical

extension in E(T), (ii) the members ofE(T) are models ofT, (iii) any member of

E(T) is a model ofT0 wheneverT is a specialisation ofT0 and (iv) all constraints are

satisfied among the members of E(T). This formulation of the global empiricalclaim came out as a refinement of the Ramsey sentence, wherefore it was labelled

the Ramsey-Sneed sentence by Stegmuller.

In An Architectonic for Science by Balzer et al. (1987), the by now classical

exposition of structuralism, the global empirical claim of a theory-net is captured by

explicit set-theoretic constructions and complemented by the consideration of links

among theory-elements. The global empirical claim of a theory-element T, for

example, is the proposition that there is a set Y of models of T such that

(i) Ysatisfies all constraints, (ii) the members ofYare correctly linked to potential

models of other theory-elements T0

and (iii) the reduction of the members ofY topartial potential models yields the set I(T) of intended applications of T. For the

global empirical claim of a theory-net N, specialisation relations among theory-

elements are to be considered in addition to these conditions.

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Implicit in all variants of global empirical claims is the notion of an admissible

T-theoretical extension of an intended application. For theory-nets, this relation may

be explained as follows:

Explanation 1(

AdmissibleT-extension) We say that a structure

xis an admissibleT-theoretical extension, orT-extension, of a structureyand writex;y 2AET if

and only if (i) x is a T-theoretical extension ofy, (ii) xis a model ofT, (iii) yis an

intended application ofT, (iv) x satisfies all links and constraints to the admissible

T0-extensions of other intended applications, (v) all specialisation relations are

satisfied among the intended applications of theory-elements in N, and (vi) any

intended application ofT has an admissible T-extension.

The core idea of Carnapian structuralism is to formally introduce the relation

AE(T) as a theoretical concept by means of axioms that qualify as postulates in the

sense of Carnaps (1958) dual level conception of scientific language. To this end, weneed to axiomatise the above explanation of AE(T). Let us start with the first

condition. An admissible T-extensionx is required to be a T-extension of a structurey:

P1T 8x8yx;y 2AET ! y rTx

where r(T)(x) is the restriction function that cuts off the theoretical relations as

standardly defined in structuralism.

The second condition of the above explanation is that a theoretical structure must

be a model ofT in order to be an admissible T-extension of some other structure:

P2T 8x8yx;y 2AET !x2 MT

Third, it is required that the range ofAE(T)consists only of intended applications of

T:

P3T 8x8yx;y 2 AET ! y 2 IT

I(T), of course, denotes the set of intended applications.

To obtain a concise axiomatic formulation of links and constraints, I suggest a uni-

fication of these two notions along the following lines. Recall that links are relations

among potential models of different theory-elements, whereas constraints are relationsamong potential models of one and the same theory-element. Both links and constraints

constrain the theoretical extensions of intended applications that are admissible. If a tuple

of potential models ofT violates a constraint, the members of this tuple will not together

be members of a set Yof potential models that verifies the global empirical claim of

T. Links and constraints will always be expressible by some formula /x1;. . .;xn;wherex1;. . .;xnare potential models of possibly different theory-elements.

The simplest types of links and constraints are equality links and constraints, which

require the interpretations encoded by two structuresx andx0 to be consistent with one

another. For example, ifx

andx0

contain an interpretation of the mass function, thenthese two interpretationsof the mass function must agree for arguments that are in the

domains of bothx and x0:2

2 Admittedly, it is not obvious that the authors of Balzer et al. ( 1987) intended to have equality links. The

formal notion of a link, however, is expounded there in such a manner that it allows for equality links.


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Why do we need links and constraints other than equality ones? For a number of

theory-elements, there are constraints that express properties of compound objects.

For example, ifcis a particle composed of particles a and b, then it must hold that

m(c) = m(a) ? m(b), where m designates the mass function. The corresponding

constraint expresses that mass is an extensive quantity. Further kinds of constraintsfor compound objects are formulated in equilibrium thermodynamics. All these

properties, however, can be equally well expressed by separate theory-elements.

There is nothing formally wrong with establishing a theory-element whose

substantial claim is the extensivity constraint of the mass function. The set of

models of such a theory-element may be defined as follows:

Definition 1 (Models ofEXTm) x is a model of the extensivity law of the mass

function x2 MEXTm if and only if there exist P, C, m such that

(1) x hP; C; mi(2) x2 MpEXTm(3) 8u8v8wCu; v; w !mu mv mw:

C(u, v, w) is a relation with the intended interpretation that uis composed of the

objects v and w but not of further objects. This relation is considered to be non-

theoretical with respect to classical particle mechanics. By the definition of

Mp(EXTm) it is specified that the cardinality ofP is three.

The primary motivation for the introduction of (intertheoretical) links is to

account for the transfer of data between two theories. Such links encode a

correlation among relations of two theory-elements T and T0: Again, if this

correlation is not of the equality type, it can be expressed by a separate theory-

element. Let us therefore confine the consideration of links and constraints to

equality ones and express all non-equality links and constraints by separate theory-

elements. This will lead to a more concise formulation of the structuralist

representation scheme.

One further consideration is necessary to achieve a unified axiomatic formulation

of links and constraints in terms of admissible T-extensions. A typification of a

relation R in structuralism has the form R2 rD1;. . .;Dk; where rD1;. . .;Dk is

an echelon set over D1;. . .;Dk; which are base sets of a structure x. Alternatively,we may typify a relation R by a proposition of the form R rD1;. . .;Dk: In thisformulation, rD1;. . .;Dk is the total relation corresponding to R in a structurex whose base sets are D1;. . .;Dk: So, let rR(x) designate the total relation that

corresponds to R in a structure x.

This being said, the axiomatic formulation of equality links and constraints for a

relation R is almost straightforward:

P4T; T0;R 8x8y8x08y0x;y 2 AET ^ x0;y0 2AET0

! 8zz2 rRx \ rRx0

! z2 xR$z2 x0

Rwhere T and T0 need not be distinct. In less formal terms, this postulate says that,

wherever two structures xand x0 overlap in terms of their empirical domains and in

terms of a relation conceptR, the interpretation ofR by x must agree with that ofx0.

H. Andreas

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(x)R denotes the extension of the relation R in the structure x, as is standard in the

structuralist notation.

Let us now deal with specialisations. T0 being a specialisation ofT (in symbols:

rT0; T)) means that, first, all intended applications of T0 are also ones of T and,

second, that the admissible T 0-extensions ofy must also be admissibleT-extensionsofy. The first of these two conditions can explicitly be captured by the following

postulate schema:

P5T; T0 8xrT0; T ! x2 IT0 !x2 IT

The second condition is implied by this postulate schema and P1(T)P4(T) and

P6(T).

So far, the postulates do not express any empirical claim concerning the intended

applications of T. It might just be the case that no intended application of T is

successful in the sense of having an admissible T-extension. In this case, theextension of AE(T) would be the empty set. Carnapian postulates, however, have

the potential to also express some empirical content, in addition to specifying the

meaning of theoretical concepts. We shall therefore advance a postulate that

specifies a condition of success, i. e., the condition that T successfully applies to any

of its intended applications. Being successful as an intended application ofT means

that it can be extended to a T-theoretical structure that is a model of T and,

moreover, satisfies all the links, constraints and specialisations. Hence, any intended

application ofT must have an admissible T-extension:

P6T 8yy2 IT ! 9xx;y 2AET

The postulates P1(T) - P6(T) do capture the intended meaning of AE(T) com-

pletely. Henceforth, the set of these postulates is designated by P(T) and is referred

to as the system of postulates for the theory-element T. D(T) designates the set of

definitions ofM(T), M_p(T) and r(T). In the case of a theory-net N, P(N) desig-

nates the set that contains P1(T) - P6(T)for allT2 N:Likewise,D(N) designatesthe set that contains precisely the definitions of M(T), Mp(T) and r(T) for all

theory-elements T in N.

4 Semantics of Theoretical Terms

A formal study of scientific reasoning must have expressive resources for the

formulation of local empirical claims, i.e., claims about the particular properties of

particular empirical objects. To give a very simple example, asserting the

proposition that some object has a mass of such and such a value is a local

empirical claim. To prepare the formulation of local empirical claims in the present

system, it is necessary to spell out the semantics of theoretical terms and postulates

that underlies the postulates P(N). This semantics will also be used for the

formulation ofNs global empirical claim.

Following Carnap (1958), we shall understand the notions of a theoretical term

and of a postulate such that a theoretical term is a symbol whose meaning is

determined through the axioms of a scientific theory, wherefore such axioms are


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also called postulates. What is it for the meaning of a term to be determined by

axioms? Carnaps Foundations of Logic and Mathematics (Carnap 1939) contains

the seeds of a precise semantics of theoretical terms. There he develops the idea that

abstract, or theoretical, terms are indirectly interpreted through axioms of a

scientific theory. That means that the interpretation is given by one or severalsentences of the object language as opposed to an assignment of an extension or

intension in the metalanguage. Sentences indirectly interpreting symbols are to be

adopted as axioms in the calculus that is associated with the formal language and the

theory in question. This adoption does not rest on a prior understanding or

assumption of truth concerning the axioms. Only observation terms are directly

interpreted to the effect that only propositions of the observation language have a

direct factual content.

Carnaps notion of indirect interpretation in the Foundations of Logic and

Mathematics(1939) has a strong formalist flavour. In his Beobachtungssprache undtheoretische Sprache (1958) we can recognise the elements of a model-theoretic

notion of indirect explanation. There he proposes to divide a scientific theory TC

into the Ramsey sentence TCR and the following conditional:

TCR !TC

This conditional became labelled later on theCarnap sentenceof a scientific theory.

Recall that TCdesignates the conjunction of T- and C-postulates. T-postulates are

those axioms of the theory that contain only theoretical terms, whereas C-postulates

have occurrences of both theoretical and observation terms, thereby establishingconnections between the theoretical and the observational vocabulary of the

respective scientific theory. Let Vodesignate the set of observation terms and Vtthe

set of theoretical terms.L(Vo) andL(Vo, VT) designate the respective languages.PTCdenotes the set of T- and C-postulates.

The Ramsey sentence of a theoryTCin the language LVo; Vt is obtained by thefollowing two transformations of the conjunction of T- and C-postulates. First,

replace all theoretical symbols in this conjunction by higher-order variables of

appropriate type. Then, bind these variables by higher-order existential quantifiers.

As result one obtains a higher-order sentence of the following form:9X1. . .9XnTCn1;. . .; nk;X1;. . .;Xn TC

R

where X1;. . .;Xn are higher-order variables.

Now, Carnap instructs us to understand the Carnap sentence as follows: if there is

an interpretation of the theoretical terms that satisfies TC(in the context of the given

interpretation of L(Vo)), the theoretical terms should be understood as designating

such an interpretation. Notably, the Carnap sentence indirectly interprets the

theoretical terms as their interpretation is characterised by the condition thatTCR !TC is always true. Carnap, however, does not explicitly address the problem that

TCR !TCusually does not uniquely interpret the theoretical terms. Nor is the caseconsidered where the Ramsey sentence is false.3

3 The former problem is addressed in Carnap (1961), where he uses Hilberts -operator for expressing

that the Carnap sentence yields only an indefinite description of theoretical terms.

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Taking up this train of thought of Carnap, we can form the notion of an

admissible structure of the complete language L(Vo, Vt): an L(Vo, Vt) structure is

admissible if and only if it (i) extends the given L(Vo) interpretation to interpret the

Vtterms and (ii) satisfies PTC, provided there is such a structure. If there is no such

structure, all extensions of L(Vo) that interpret the Vt terms may be consideredadmissible. L(Vo, Vt) structures may interpret Vt terms in a theoretical domain Dtthat expands the observation domain Do. Henceforth, letAo designate the given orintended interpretation of the observation language and EXTAo; Vt;Dt the set ofextensions ofAo that interpret Vt, where Vtterms may be interpreted in Dt[ Do.MOD(A) designates, for a set A of sentences, the set of models ofA.

To make these ideas about admissible structures precise (Andreas 2010a):

Definition 2 (Admissible structures) Sa designates the set of L(Vo, Vt) structures

that are admissible under an interpretation of the Vtsymbols by the postulatesPTC. It

is defined as follows:

Sa : MODPTC \ EXTAo; Vt;Dt ifMODPTC \ EXTAo; Vt;Dt 6 ;;

EXTAo; Vt;Dt ifMODPTC \ EXTAo; Vt;Dt ;:

Understanding thus the notion of an admissible structure, the following truth-

rules are intuitive:

Definition 3 (Truth rules for theoretical sentences) m :LVo; VtS ! fT; F;Ig;

where L(Vo, Vt)S designates the set ofL(Vo, Vt) sentences:

(1) m(/) := Tif and only if for every structure A 2Sa; A /;(2) m(/) := Fif and only if for every structure A 2 Sa; A2 /;

(3) m(/) := I(indeterminate) if and only if there are A1; A22 Sasuch that A1/and A22/:

The idea lying behind these rules is rather simple. A theoretical sentence is true if

and only if it is true in every admissible structure. A theoretical sentence is false if

and only if it is false in every admissible structure. And a sentence has no

determinate truth-value if and only if it is true in, at least, one admissible structure

and false in, at least, another structure being also admissible.

The semantics of defined terms can be considered as a special case of theoretical

terms. For a set Vd of defined terms and a set D of corresponding definitions,

MODD \ EXTAo; Vt;Dt will be a singleton. Hence definition 2 remains to beapplicable to defined terms, where in such applications a set Ad of definitions

replaces the set PTCof postulates.

Things are slightly more complicated if the language of our theory TCcontains

both defined and theoretical terms. Then, the definition of admissible structures

needs to be adjusted as follows:

Definition 4 (Admissible structures of L(Vo, Vd, Vt)) Sa designates the set of

L(Vo, Vd, Vt) structures that are admissible under an interpretation of the Vtsymbols

by the postulatesPTCand an interpretation of the Vdsymbols by the definitionsDTC.

It is defined as follows:


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Sa :MODPTC[ DTC \ EXTAo; Vt[ Vd;Dt

ifMODPTC[ DTC \ EXTAo; Vt[ Vd;Dt 6 ;;EXTAo; Vt[ Vd;Dt \MODDTC otherwise:

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