Non-classical logicFrom Wikipedia, the free encyclopedia

Contents

1 Naive set theory 11.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Cantors theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Axiomatic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.4 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.5 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Sets, membership and equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Note on consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.4 Empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Specifying sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Universal sets and absolute complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Unions, intersections, and relative complements . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 Ordered pairs and Cartesian products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.8 Some important sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.9 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Name 102.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 In religious thought . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Biblical names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Talmudic attitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Names of names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Naming convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1 Brand names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Name used by animals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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2.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Narrative logic 153.1 Example: Kill Bill: Volume 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Natural deduction 164.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Judgments and propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Introduction and elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4 Hypothetical derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.5 Consistency, completeness, and normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.6 First and higher-order extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.7 Dierent presentations of natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.7.1 Tree-like presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.7.2 Sequential presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.8 Proofs and type-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.9 Classical and modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.10 Comparison with other foundational approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.10.1 Sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Natural kind 275.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Natural language 316.1 Dening natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2 Native language learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.3 Origins of natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.4 Controlled languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.5 Constructed languages and international auxiliary languages . . . . . . . . . . . . . . . . . . . . . 326.6 Modalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.6.1 Sign languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.6.2 Written languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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6.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7 Necessity and suciency 357.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.2 Necessity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.3 Suciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.4 Relationship between necessity and suciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.5 Simultaneous necessity and suciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.6.1 Argument forms involving necessary and sucient conditions . . . . . . . . . . . . . . . . 397.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8 Neutrality (philosophy) 408.1 Criticisms and views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.2 In popular culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9 Nirvana fallacy 429.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.2 Perfect solution fallacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

10 Nixon diamond 4410.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

11 No true Scotsman 4611.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.2 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

12 Nominal identity 4812.1 Social sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

12.1.1 Nominal identity in ethnicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.2 Linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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13 Non-Aristotelian logic 5013.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.2 Use in science ction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

14 Non-classical logic 5314.1 Examples of non-classical logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.2 Classication of non-classical logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5514.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 56

14.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 1

Naive set theory

This article is about the mathematical topic. For the book of the same name, see Naive Set Theory (book).

Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics.[1] Unlikeaxiomatic set theories, which are dened using a formal logic, naive set theory is dened informally, in naturallanguage. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagramsand symbolic reasoning about their Boolean algebra), and suces for the everyday usage of set theory concepts incontemporary mathematics.[2]

Sets are of great importance in mathematics; in fact, in modern formal treatments, most mathematical objects(numbers, relations, functions, etc.) are dened in terms of sets. Naive set theory can be seen as a stepping-stone tomore formal treatments, and suces for many purposes.

1.1 MethodHere and below, a naive theory is considered to be a non-formalized theory, that is, a theory that uses a naturallanguage to describe sets and operations on sets. The words and, or, if ... then, not, for some, for every are nothere subject to rigorous denition. It is useful to study sets naively at an early stage of mathematics in order to developfacility for working with them. Furthermore, a rm grasp of set theorys concepts from a naive standpoint is a stepto understanding the motivation for the formal axioms of set theory. As a matter of convenience, usage of naive settheory and its formalism prevails even in higher mathematics including in more formal settings of set theory itself.Sets are dened informally and a few of their properties are investigated. Links to specic axioms of set theorydescribe some of the relationships between the informal discussion here and the formal axiomatization of set theory,but no attempt is made to justify every statement on such a basis. The rst development of set theory was a naiveset theory. It was created at the end of the 19th century by Georg Cantor as part of his study of innite sets[3] anddeveloped by Gottlob Frege in his Begrisschrift.Naive set theory may refer to several very distinct notions. It may refer to

Informal presentation of an axiomatic set theory, e.g. as in Naive Set Theory by Paul Halmos.

Early or later versions of Georg Cantor's theory and other informal systems.

Decidedly inconsistent theories (whether axiomatic or not), like a theory of Gottlob Frege[4] that yieldedRussells paradox, and theories of Giuseppe Peano[5] and Richard Dedekind.

1.1.1 Paradoxes

As it turned out, assuming that one can form sets freely without restriction leads to paradoxes. For example, theassumption that one can collect together, as a set, all (mathematical) objects that have a given property is false. Inother words, the statement that

1

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S = fx : P (x)g

(where P(x) should be read as "x has property P") is a set will lead to paradoxes, in particular Russells paradox.

1.1.2 Cantors theory

Some believe that Georg Cantor's set theory was not actually implicated by these paradoxes (see Frpolli 1991); onediculty in determining this with certainty is that Cantor did not provide an axiomatization of his system. By 1899,Cantor was aware of some of the paradoxes following from unrestricted interpretation of his theory, for instanceCantors paradox,[6] the Burali-Forti paradox,[7] and did not believe that they discredited his theory.[8] Cantors para-dox can actually be derived from the above (false) assumption using for P(x) "x is a cardinal number". Frege explicitlyaxiomatized a theory in which a formalized version of naive set theory can be interpreted, and it is this formal theorywhich Bertrand Russell actually addressed when he presented his paradox, not necessarily a theory Cantor, who, asmentioned, was aware of several paradoxes, presumably had in mind.

1.1.3 Axiomatic theories

Axiomatic set theorywas developed in response to these early attempts to understand sets, with the goal of determiningprecisely what operations were allowed and when. Today, when mathematicians talk about set theory as a eld,they usually mean axiomatic set theory. Informal applications of set theory in other elds are sometimes referredto as applications of naive set theory, but usually are understood to be justiable in terms of an axiomatic system(normally ZermeloFraenkel set theory).

1.1.4 Consistency

A naive set theory is not necessarily inconsistent, if it correctly species the sets allowed to be considered. This canbe done by the means of denitions, which are implicit axioms. It is possible to state all the axioms explicitly, as inthe case of Halmos Naive Set Theory, which is actually an informal presentation of the usual axiomatic ZermeloFraenkel set theory. It is naive in that the language and notations are those of ordinary informal mathematics, andin that it doesn't deal with consistency or completeness of the axiom system.Likewise, an axiomatic set theory is not necessarily consistent, i.e. not necessarily free of paradoxes. It follows fromGdels incompleteness theorems that a suciently complicated system, in rst order logic, which includes mostcommon axiomatic set theories, cannot be proved consistent from within the theory itself, provided it actually isconsistent. However, the common axiomatic systems are generally believed to be consistent, and they do exclude, viathe axioms, some paradoxes, like Russells paradox. It is just not known, and never will be, if there are no paradoxesat all in these theories or in any rst-order set theory.The term naive set theory is still today also used in some literature to refer to the set theories studied by Frege andCantor, rather than to the informal counterparts of modern axiomatic set theory.

1.1.5 Utility

The choice between an axiomatic approach and other approaches is largely a matter of convenience. In everydaymathematics the best choice may be informal use of axiomatic set theory. References to particular axioms typicallythen occurs only when tradition demands it, e.g. the axiom of choice is often mentioned when used. Likewise, formalproofs occur only when exceptional circumstances warrant it. This informal usage of axiomatic set theory can have(depending on notation) precisely the appearance of the naive set theory outlined below, and is considerably easier,both to read and to write, including in the formulation of most statements and proofs and lines of discussion, and isprobably less error-prone for most people than a strictly formal approach.

1.2. SETS, MEMBERSHIP AND EQUALITY 3

1.2 Sets, membership and equalityIn naive set theory, a set is described as a well-dened collection of objects. These objects are called the elements ormembers of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of theset of all even integers. Clearly, the set of even numbers is innitely large; there is no requirement that a set be nite.

Passage with the original set denition of Georg Cantor

The denition of sets goes back to Georg Cantor. He wrote 1915 in his article Beitrge zur Begrndung der transnitenMengenlehre:

Unter einer Menge verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiede-nen Objekten m unserer Anschauung oder unseres Denkens (welche die Elemente von M genanntwerden) zu einem Ganzen. Georg Cantor

A set is a gathering together into a whole of denite, distinct objects of our perception or of ourthoughtwhich are called elements of the set. Georg Cantor

First usage of the symbol in the work Arithmetices principia nova methodo exposita by Giuseppe Peano.

1.2.1 Note on consistencyIt does not follow from this denition how sets can be formed, and what operations on sets again will produce aset. The term well-dened in well-dened collection of objects cannot, by itself, guarantee the consistency andunambiguity of what exactly constitutes and what does not constitute a set. Attempting to achieve this would be therealm of axiomatic set theory or of axiomatic class theory.The problem, in this context, with informally formulated set theories, not derived from (and implying) any particularaxiomatic theory, is that there may be several widely diering formalized versions, that have both dierent sets and

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dierent rules for how new sets may be formed, that all conform to the original informal denition. For example,Cantors verbatim denition allows for considerable freedom in what constitutes a set. On the other hand, it is unlikelythat Cantor was particularly interested in sets containing cats and dogs, but rather only in sets containing purelymathematical objects. An example of such a class of sets could be the Von Neumann universe. But even whenxing the class of sets under consideration, it is not always clear which rules for set formation are allowed withoutintroducing paradoxes.For the purpose of xing the discussion below, the term well-dened should instead be interpreted as an intention,with either implicit or explicit rules (axioms or denitions), to rule out inconsistencies. The purpose is to keep theoften deep and dicult issues of consistency away from the, usually simpler, context at hand. An explicit rulingout of all conceivable inconsistencies (paradoxes) cannot be achieved for an axiomatic set theory anyway, due toGdels second incompleteness theorem, so this does not at all hamper the utility of naive set theory as comparedto axiomatic set theory in the simple contexts considered below. It merely simplies the discussion. Consistency ishenceforth taken for granted unless explicitly mentioned.

1.2.2 Membership

If x is a member of a set A, then it is also said that x belongs to A, or that x is in A. This is denoted by x A. Thesymbol is a derivation from the lowercase Greek letter epsilon, "", introduced by Giuseppe Peano in 1889 andshall be the rst letter of the word (means is). The symbol is often used to write x A, meaning x is not inA.

1.2.3 Equality

Two sets A and B are dened to be equal when they have precisely the same elements, that is, if every element of Ais an element of B and every element of B is an element of A. (See axiom of extensionality.) Thus a set is completelydetermined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal tothe set of all prime numbers less than 6. If the sets A and B are equal, this is denoted symbolically as A = B (as usual).

1.2.4 Empty set

The empty set, often denoted and sometimes fg , is a set with no members at all. Because a set is determinedcompletely by its elements, there can be only one empty set. (See axiom of empty set.) Although the empty set hasno members, it can be a member of other sets. Thus {}, because the former has no members and the latter hasone member. It is interesting to note that, in mathematics, the only sets with which one needs to be concerned canbe built up from the empty set alone (Halmos (1974)).

1.3 Specifying setsThe simplest way to describe a set is to list its elements between curly braces (known as dening a set extensionally).Thus {1, 2} denotes the set whose only elements are 1 and 2. (See axiom of pairing.) Note the following points:

The order of elements is immaterial; for example, {1, 2} = {2, 1}. Repetition (multiplicity) of elements is irrelevant; for example, {1, 2, 2} = {1, 1, 1, 2} = {1, 2}.

(These are consequences of the denition of equality in the previous section.)This notation can be informally abused by saying something like {dogs} to indicate the set of all dogs, but this examplewould usually be read by mathematicians as the set containing the single element dogs".An extreme (but correct) example of this notation is {}, which denotes the empty set.The notation {x : P(x)}, or sometimes {x | P(x)}, is used to denote the set containing all objects for which the conditionP holds (known as dening a set intensionally). For example, {x : x R} denotes the set of real numbers, {x : x hasblonde hair} denotes the set of everything with blonde hair.

1.4. SUBSETS 5

This notation is called set-builder notation (or "set comprehension", particularly in the context of Functional pro-gramming). Some variants of set builder notation are:

{x A : P(x)} denotes the set of all x that are already members of A such that the condition P holds for x.For example, if Z is the set of integers, then {x Z : x is even} is the set of all even integers. (See axiom ofspecication.)

{F(x) : x A} denotes the set of all objects obtained by putting members of the set A into the formula F. Forexample, {2x : x Z} is again the set of all even integers. (See axiom of replacement.)

{F(x) : P(x)} is the most general form of set builder notation. For example, {x's owner : x is a dog} is the setof all dog owners.

1.4 SubsetsGiven two sets A and B, A is a subset of B if every element of A is also an element of B. In particular, each set B isa subset of itself; a subset of B that is not equal to B is called a proper subset.If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A.In symbols, A B means that A is a subset of B, and B A means that B is a superset of A. Some authors use thesymbols and for subsets, and others use these symbols only for proper subsets. For clarity, one can explicitly usethe symbols and to indicate non-equality.As an illustration, let R be the set of real numbers, let Z be the set of integers, let O be the set of odd integers, andlet P be the set of current or former U.S. Presidents. Then O is a subset of Z, Z is a subset of R, and (hence) O is asubset of R, where in all cases subset may even be read as proper subset. Note that not all sets are comparable in thisway. For example, it is not the case either that R is a subset of P nor that P is a subset of R.It follows immediately from the denition of equality of sets above that, given two sets A and B, A = B if and only ifA B and B A. In fact this is often given as the denition of equality. Usually when trying to prove that two setsare equal, one aims to show these two inclusions. Note that the empty set is a subset of every set (the statement thatall elements of the empty set are also members of any set A is vacuously true).The set of all subsets of a given set A is called the power set of A and is denoted by 2A or P (A) ; the "P" is sometimesin a script font. If the set A has n elements, then P (A) will have 2n elements.

1.5 Universal sets and absolute complementsIn certain contexts, one may consider all sets under consideration as being subsets of some given universal set. Forinstance, when investigating properties of the real numbers R (and subsets of R), Rmay be taken as the universal set.A true universal set is not included in standard set theory (seeParadoxes below), but is included in some non-standardset theories.Given a universal set U and a subset A of U, the complement of A (in U) is dened as

AC := {x U : x A}.

In other words, AC ("A-complement"; sometimes simply A', "A-prime" ) is the set of all members of U which are notmembers of A. Thus with R, Z and O dened as in the section on subsets, if Z is the universal set, then OC is the setof even integers, while if R is the universal set, then OC is the set of all real numbers that are either even integers ornot integers at all.

1.6 Unions, intersections, and relative complementsGiven two sets A and B, their union is the set consisting of all objects which are elements of A or of B or of both (seeaxiom of union). It is denoted by A B.The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A B.

6 CHAPTER 1. NAIVE SET THEORY

Finally, the relative complement of B relative to A, also known as the set theoretic dierence of A and B, is the setof all objects that belong to A but not to B. It is written as A \ B or A B.Symbolically, these are respectively

A B := {x : (x A) or (x B)};A B := {x : (x A) and (x B)} = {x A : x B} = {x B : x A};A \ B := {x : (x A) and not (x B) } = {x A : not (x B)}.

Notice that A doesn't have to be a subset of B for B \ A to make sense; this is the dierence between the relativecomplement and the absolute complement (AC = U \ A) from the previous section.To illustrate these ideas, let A be the set of left-handed people, and let B be the set of people with blond hair. ThenA B is the set of all left-handed blond-haired people, while A B is the set of all people who are left-handed orblond-haired or both. A \ B, on the other hand, is the set of all people that are left-handed but not blond-haired, whileB \ A is the set of all people who have blond hair but aren't left-handed.Now let E be the set of all human beings, and let F be the set of all living things over 1000 years old. What is E Fin this case? No living human being is over 1000 years old, so E F must be the empty set {}.For any set A, the power set P (A) is a Boolean algebra under the operations of union and intersection.

1.7 Ordered pairs and Cartesian productsIntuitively, an ordered pair is simply a collection of two objects such that one can be distinguished as the rst elementand the other as the second element, and having the fundamental property that, two ordered pairs are equal if and onlyif their rst elements are equal and their second elements are equal.Formally, an ordered pair with rst coordinate a, and second coordinate b, usually denoted by (a, b), can be denedas the set {{a}, {a, b}}.It follows that, two ordered pairs (a,b) and (c,d) are equal if and only if a = c and b = d.Alternatively, an ordered pair can be formally thought of as a set {a,b} with a total order.(The notation (a, b) is also used to denote an open interval on the real number line, but the context should make itclear which meaning is intended. Otherwise, the notation ]a, b[ may be used to denote the open interval whereas (a,b) is used for the ordered pair).If A and B are sets, then the Cartesian product (or simply product) is dened to be:

A B = {(a,b) : a is in A and b is in B}.

That is, A B is the set of all ordered pairs whose rst coordinate is an element of A and whose second coordinateis an element of B.This denition may be extended to a set A B C of ordered triples, and more generally to sets of ordered n-tuplesfor any positive integer n. It is even possible to dene innite Cartesian products, but this requires a more reconditedenition of the product.Cartesian products were rst developed by Ren Descartes in the context of analytic geometry. IfR denotes the set ofall real numbers, thenR2 :=R R represents the Euclidean plane andR3 :=R R R represents three-dimensionalEuclidean space.

1.8 Some important setsNote: In this section, a, b, and c are natural numbers, and r and s are real numbers.

1. Natural numbers are used for counting. A blackboard bold capital N ( N ) often represents this set.

1.9. PARADOXES 7

2. Integers appear as solutions for x in equations like x + a = b. A blackboard bold capital Z ( Z ) often representsthis set (from the German Zahlen, meaning numbers).

3. Rational numbers appear as solutions to equations like a + bx = c. A blackboard bold capital Q ( Q ) oftenrepresents this set (for quotient, because R is used for the set of real numbers).

4. Algebraic numbers appear as solutions to polynomial equations (with integer coecients) and may involveradicals and certain other irrational numbers. A blackboard bold capital A ( A ) or a Q with an overline ( Q )often represents this set. The overline denotes the operation of algebraic closure.

5. Real numbers represent the real line and include all numbers that can be approximated by rationals. Thesenumbers may be rational or algebraic but may also be transcendental numbers, which cannot appear as solutionsto polynomial equations with rational coecients. A blackboard bold capital R ( R ) often represents this set.

6. Complex numbers are sums of a real and an imaginary number: r + si. Here both r and s can equal zero; thus,the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers, whichform an algebraic closure for the set of real numbers, meaning that every polynomial with coecients in Rhas at least one root in this set. A blackboard bold capital C ( C ) often represents this set. Note that sincea number r + si can be identied with a point (r, s) in the plane, C is basically the same as the Cartesianproduct RR (the same meaning that any point in one determines a unique point in the other and for theresult of calculations it doesn't matter which one is used for the calculation).

1.9 ParadoxesMain article: Paradox

The unrestricted formation principle of sets referred to as the axiom schema of unrestricted comprehension,

If P is a property, then there exists a set Y = {x : P(x)} (false),[9]

is the source of many of the early appearing paradoxes:

Y = {x : x is an ordinal} leads 1897 to the Burali-Forti paradox, the rst published antinomy. Y = {x : x is a cardinal} produced the Cantors paradox in 1897.[6]

Y = {x : {} = {}} yielded Cantors second antinomy in the year 1899.[8] Here the property P is true for allx, whatever x may be, so Y would be a universal set, containing everything.

Y = {x : x x}, i.e. the set of all sets that do not contain themselves as elements gave Russells paradox 1902.

If the axiom schema of unrestricted comprehension is weakened to the axiom schema of specication or axiomschema of separation,

If P is a property, then for any set X there exists a set Y = {x X : P(x)},[9]

then all the above paradoxes disappear.[9] There is a corollary. With the axiom schema of separation as an axiom ofthe theory, it follows, as a theorem of the theory:

The set of all sets does not exist.

Proof: Suppose that it exists and call it U. Now apply the axiom schema of separation with X = U and for P(x) use x x. This leads to Russells paradox again. Hence U can't exist in this theory.[9]

Related to the above constructions is formation of the set

Y = {x : (x x) {} {}}, where the statement following the implication certainly is false. It follows, fromthe denition of Y, using the usual inference rules (and some afterthought when reading the proof in the linkedarticle below) both that Y Y {} {} and Y Y holds, hence {} {}. This is Currys paradox.

8 CHAPTER 1. NAIVE SET THEORY

It is (perhaps surprisingly) not the possibility of x x that is problematic here, it is again the axiom schema ofunrestricted comprehension allowing (x x) {} {} for P(x). With the axiom schema of specication instead ofunrestricted comprehension, the conclusion Y Y doesn't hold and, hence {} {} is not a logical consequence.Nonetheless, the possibility of x x is often removed explicitly[10] or, e.g. in ZFC, implicitly,[11] by demanding theaxiom of regularity to hold.[11] One consequence of it is:

There is no set X for which X X,

in other words, no set is an element of itself.[12]

The axiom schema of separation is simply too weak (while unrestricted comprehension is a very strong axiomtoostrong for set theory) to develop set theory with its usual operations and constructions outlined above.[9] The axiomof regularity is of a restrictive nature as well. Therefore one is led to the formulation of other axioms to guarantee theexistence of enough sets to form a set theory. Some of these have been described informally above and many othersare possible. Not all conceivable axioms can be combined freely into consistent theories. For example, the axiom ofchoice of ZFC is incompatible with the conceivable every set of reals is Lebesgue measurable. The former impliesthe latter is false.

1.10 See also Algebra of sets Axiomatic set theory Internal set theory Set theory Set (mathematics) Partially ordered set

1.11 Notes[1] Concerning the origin of the term naive set theory, JeMiller says, "Nave set theory (contrasting with axiomatic set theory)

was used occasionally in the 1940s and became an established term in the 1950s. It appears in Hermann Weyls review ofP. A. Schilpp (ed) The Philosophy of Bertrand Russell in the American Mathematical Monthly, 53., No. 4. (1946), p. 210and Laszlo Kalmars review of The Paradox of Kleene and Rosser in Journal of Symbolic Logic, 11, No. 4. (1946), p. 136.(JSTOR). The term was later popularized by Paul Halmos' book, Naive Set Theory (1960).

[2] Mac Lane, Saunders (1971), Categorical algebra and set-theoretic foundations, Axiomatic Set Theory (Proc. Sympos.Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967), Amer. Math. Soc., Providence, R.I., pp. 231240, MR 0282791. The working mathematicians usually thought in terms of a naive set theory (probably one more orless equivalent to ZF) ... a practical requirement [of any new foundational system] could be that this system could be usednaively by mathematicians not sophisticated in foundational research (p. 236).

[3] Cantor 1874[4] Frege 1893 In Volume 2, Jena 1903. pp. 253-261 Frege discusses the antionomy in the afterword.[5] Peano 1889 Axiom 52. chap. IV produces antinomies.[6] Letter from Cantor to David Hilbert on September 26, 1897, Meschkowski & Nilson 1991 p. 388.[7] Letter from Cantor to Richard Dedekind on August 3, 1899, Meschkowski & Nilson 1991 p. 408.[8] Letters from Cantor to Richard Dedekind on August 3, 1899 and on August 30, 1899, Zermelo 1932 p. 448 (System aller

denkbaren Klassen) and Meschkowski & Nilson 1991 p. 407. (There is no set of all sets.)[9] Jech 2002 p. 4.[10] Halmos (1974), Naive Set Theory See discussion around Russells paradox.[11] Jech 2002 Section 1.6.[12] Jech 2002 p. 61.

1.12. REFERENCES 9

1.12 References Bourbaki, N., Elements of the History of Mathematics, John Meldrum (trans.), Springer-Verlag, Berlin, Ger-many, 1994.

Cantor, Georg (1874), Ueber eine Eigenschaft des Inbegries aller reellen algebraischen Zahlen, J. ReineAngew. Math. 77: 258262, doi:10.1515/crll.1874.77.258, See also pdf version:

Devlin, K.J., The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd edition, Springer-Verlag, NewYork, NY, 1993.

Frpolli, Mara J., 1991, Is Cantorian set theory an iterative conception of set?". Modern Logic, v. 1 n. 4,1991, 302318.

Frege, Gottlob (1893), Grundgesetze der Arithmetik 1, Jena 1893. Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books,2011. ISBN 978-1-61427-131-4 (Paperback edition).

Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.

Kelley, J.L., General Topology, Van Nostrand Reinhold, New York, NY, 1955. van Heijenoort, J., From Frege to Gdel, A Source Book in Mathematical Logic, 1879-1931, Harvard UniversityPress, Cambridge, MA, 1967. Reprinted with corrections, 1977. ISBN 0-674-32449-8.

Meschkowski, Herbert; Nilson, Winfried (1991), Georg Cantor: Briefe. Edited by the authors., Springer, ISBN3-540-50621-7

Peano, Giuseppe (1889), Arithmetices Principies nova Methoda exposita, Turin 1889. Zermelo, Ernst (1932),Georg Cantor: Gesammelte Abhandlungen mathematischen und philosophischen Inhalts.

Mit erluternden Anmerkungen sowie mit Ergnzungen aus dem Briefwechsel Cantor-Dedekind. Edited by theauthor., Springer

1.13 External links Beginnings of set theory page at St. Andrews Earliest Known Uses of Some of the Words of Mathematics (S)

Chapter 2

Name

Names redirects here. For other uses, see Names (disambiguation).For other uses, see Name (disambiguation).A name is a word or term used for identication. Names can identify a class or category of things, or a single thing,

A cartouche indicates that the Egyptian hieroglyphs enclosed are a royal name.

either uniquely, or within a given context. A personal name identies, not necessarily uniquely, a specic individualhuman. The name of a specic entity is sometimes called a proper name (although that term has a philosophicalmeaning also) and is, when consisting of only one word, a proper noun. Other nouns are sometimes called "commonnames" or (obsolete) "general names". A name can be given to a person, place, or thing; for example, parents cangive their child a name or scientist can give an element a name.Caution must be exercised when translating, for there are ways that one language may prefer one type of name overanother. A feudal naming habit is used sometimes in other languages: the French sometimes refer to Aristotle asle Stagirite from one spelling of his place of birth, and English speakers often refer to Shakespeare as The Bard",

10

2.1. ETYMOLOGY 11

recognizing him as a paragon writer of the language. Also, claims to preference or authority can be refuted: theBritish did not refer to Louis-Napoleon as Napoleon III during his rule.

2.1 Etymology

The word name comes from Old English nama; akin to Old High German (OHG) namo, Sanskrit (nman),Latin nomen, and Greek (onoma),[1] possibly from the Proto-Indo-European (PIE) *nomn-.[2] Also note thesimilarity to the Tamil "namam".

2.2 In religious thought

Further information: Names of God

In the ancient world, particularly in the ancient near-east (Israel, Mesopotamia, Egypt, Persia) names were thought tobe extremely powerful and to act, in some ways, as a separate manifestation of a person or deity.[3] This viewpoint isresponsible both for the reluctance to use the proper name of God in Hebrew writing or speech, as well as the commonunderstanding in ancient magic that magical rituals had to be carried out in [someones] name. By invoking a godor spirit by name, one was thought to be able to summon that spirits power for some kind of miracle or magic (seeLuke 9:49, in which the disciples claim to have seen a man driving out demons using the name of Jesus.) Thisunderstanding passed into later religious tradition, for example the stipulation in Catholic exorcism that the demoncannot be expelled until the exorcist has forced it to give up its name, at which point the name may be used in a sterncommand which will drive the demon away.

2.2.1 Biblical names

Main article: List of biblical names

In the Old Testament, the names of individuals are meaningful, and a change of name indicates a change of status.For example, the patriarch Abram and his wife Sarai are renamed "Abraham" and "Sarah" when they are told theywill be the father and mother of many nations (Genesis 17:4, 17:15). Simon was renamed Peter when he was giventhe Keys to Heaven. This is recounted in the Gospel of Matthew chapter 16, which according to Roman Catholicteaching,[4] was when Jesus promised to Saint Peter, the power to take binding actions.[5]

Throughout the Bible, characters are given names at birth that reect something of signicance or describe the courseof their lives. For example: Solomonmeant peace, and the king with that name was the rst whose reign was withoutwar. Likewise, Joseph named his rstborn son Manasseh (Hebrew: causing to forget)(Genesis 41:51); when Josephalso said, "God has made me forget all my troubles and everyone in my fathers family.Biblical Jewish people did not have surnames which were passed from generation to generation. However, they weretypically known as the child of their father. For example: (David ben Yishay) meaning, David, son ofJesse.(1 Samuel 17:12,58).

2.2.2 Talmudic attitudes

The Babylonian Talmud maintains that names exert a mystical inuence over their bearers, and a change of name isone of four actions that can avert an evil heavenly decree, that would lead to punishment after ones death. Rabbinicalcommentators dier as to whether the names inuence is metaphysical, connecting a person to their soul, or bio-socio-psychological, where the connection aects his personality, appearance and social capacities. The Talmud alsostates that all those who descend to Gehenna will rise in the time of Messiah. However, there are three exceptions,one of which is he who calls another by a derisive nickname.

12 CHAPTER 2. NAME

2.3 Names of names

2.4 Naming conventionFor Wikipedias own naming conventions see Wikipedia:Article titles

A naming convention is an attempt to systematize names in a eld so they unambiguously convey similar informationin a similar manner.Several major naming conventions include:

In astronomy, planetary nomenclature In classics, Roman naming conventions In computer programming, identier naming conventions In computer networking, computer naming schemes In the sciences, systematic names for a variety of things

Naming conventions are useful in many aspects of everyday life, enabling the casual user to understand larger struc-tures.Street names within a city may follow a naming convention; some examples include:

In Manhattan, roads that cross the island from east to west are called Streets. Those that run the length ofthe island (North-South) are called Avenues. Most of Manhattans streets and avenues are numbered, with1st Street being near the southern end of the island, and 219th Street being near the northern end, while1st Avenue is near the eastern edge of the island and 12th Avenue near the western edge.

In Ontario, numbered concession roads are East-West whereas lines are North-South routes. In San Francisco at least three series of parallel streets are alphabetically named, e.g. Irving, Judah, Kirkham,

Lawton, Moraga, Noriega, Ortega, Pacheco, Quintara, Rivera, Santiago, Taraval, Ulloa, Vicente, Wawona. The same tendency is seen in the Back Bay neighborhood of Boston, Massachusetts, where Arlington Streetis followed by roads to the west running parallel to it and named Berkeley, Clarendon, Dartmouth, Exeter,Faireld, Gloucester, and Hereford.

In Washington, DC, East Capitol Street runs east-west through the Capitol. East-west streets moving awayfrom Capitol Street toward both the south (toward the Potomac River) and the north are lettered A, B, C,...,omitting J to avoid confusion on street signs and addresses, but after these are exhausted to the north, the streetsare named with simple words in alphabetical order, omitting a few letters such as x. The rst cycle of namesconsists all of one-syllable words; then followed by a cycle of two-syllable words; then followed by a cycleof three-syllable words, and before these are exhausted, Maryland is reached. (Washington has north-southstreets that are numbered, increasing to either side of North Capitol which likewise runs through the Capitol.)Suxes (NE, SW, etc.) are used to distinguish between (up to four) duplicate addresses. For example 140 DStreet SW, to indicate the 140 D Street location southwest of the Capitol

In Montgomery, Alabama, the old major avenues are named for the Presidents of the United States, in theirorder of entering oce, omitting John Quincy Adams. Hence, these streets are Washington Ave., Adams Ave.,Jeerson Ave., Madison Ave., Monroe Ave., Jackson Ave., but not much farther than that. This was just theold plan from a long time ago, and it was eventually dropped. For example, there is not a Buchanan Ave., aLincoln Ave., or a Johnson Ave.

In Brampton, Ontario, dierent sections of town all have streets startingwith the same letter and the alphabeticalorder reects chronology.

In Phoenix, Arizona, roads east of Central Avenue are termed streets while those west are avenues. A similarsystem applies in Nashville, Tennessee, but only to the numbered avenues and streets, west and east of theCumberland River respectively, all of which run roughly north-south.

2.5. NAME USED BY ANIMALS 13

Large corporate, university, or government campuses may follow a naming convention for rooms within the buildingsto help orient tenants and visitors. Otherwise, rooms may be numbered in some kind of a rational scheme.Parents may follow a naming convention when selecting names for their children. Some have chosen alphabeticalnames by birth order. In some East Asian cultures, it is common for one syllable in a two syllable given name to bea generation name which is the same for immediate siblings. In many cultures it is common for the son to be namedafter the father or a grandfather. In certain African cultures, such as in Cameroon, the eldest son gets the family namefor his given name.In other cultures, the name may include the place of residence, or the place of birth. The Roman naming conventiondenotes social rank.Products may follow a naming convention. Automobiles typically have a binomial name, a make (manufacturer)and a model, in addition to a model year, such as a 2007 Chevrolet Corvette. Sometimes there is a name for the carsdecoration level or trim line as well: e.g., Cadillac Escalade EXT Platinum, after the precious metal. Computersoften have increasing numbers in their names to signify the next generation.Courses at schools typically follow a naming convention: an abbreviation for the subject area and then a numberordered by increasing level of diculty.Many numbers (e.g. bank accounts, government IDs, credit cards, etc.) are not random but have an internal structureand convention. Virtually all organizations that assign names or numbers will follow some convention in generatingthese identiers. Airline ight numbers, space shuttle ight numbers, even phone numbers all have an internal con-vention.

2.4.1 Brand names

Main article: Brand

The process of developing a name for a brand or product is heavily inuenced by marketing research and strategy tobe appealing and marketable. The brand name is often a neologism or pseudoword, such as Kodak or Sony.

2.5 Name used by animalsThe use of personal names is not unique to humans. Dolphins also use symbolic names, as has been shown by recentresearch.[6] Individual dolphins have distinctive whistles, to which they will respond even when there is no otherinformation to clarify which dolphin is being referred to.

2.6 See also Personal name - names of people

Anthroponymy - the study of personal names

List of adjectival forms of place names

Nickname

Numeral (linguistics)

Onomastics - the study of proper names

Popular cat names

Proper name

Title (publishing)

14 CHAPTER 2. NAME

2.7 References[1] , Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus project

[2] Online Etymology Dictionary. Retrieved 2008-09-20.; The asterisk before a word indicates that it is a hypotheticalconstruction, not an attested form.

[3] Egyptian Religion, E. A. Wallis Budge, Arkana 1987 edition, ISBN 0-14-019017-1

[4] Catechism of the Catholic Church, para 881: The episcopal college and its head, the Pope

[5] The Routledge Companion to the Christian Church byGerardMannion and Lewis S.Mudge (Jan 30, 2008) ISBN0415374200page 235

[6] Dolphins Name Themselves With Whistles, Study Says. National Geographic News. May 8, 2006.

2.8 Further reading Names by Sam Cumming, Stanford Encyclopedia of Philosophy (SEP), a philosophical dissertation on thesyntax and semantics of names

Matthews, Elaine; Hornblower, Simon; Fraser, PeterMarshall,Greek Personal Names: Their Value as Evidence,Proceedings of The British Academy (104), Oxford University Press, 2000. ISBN 0-19-726216-3

Name and Form - from Sacred Texts Buddhism

2.9 External links Lexicon of Greek Personal Names, Oxford (over 35,000 published names) Behind The Name, The etymology of rst names KateMonks Onomastikon (Dictionary ofNames) or KateMonks OnomastikonNames over theworld through-out the history

What is a Name?

Chapter 3

Narrative logic

In the broadest sense, narrative logic is any logical process of narrative analysis. Narrative logic is a tool throughwhich the audience may create events and explanations or otherwise elucidate details not included in the narrative.It is used to build a logical argument based upon the content of a narrative, using its events and rhetoric as evidenceto support the argument. This is done to ensure that ones argument does not contradict or alter the narrative itself.Problems and disagreements may arise from this xity of the narrative because it should also preclude alteration ofthe artistic statement being conveyed, something that is open to subjective interpretation and may be paradoxical orillogical in itself. Thus, this process is generally imperfect since, as with all narrative analysis and most forms of logic,dierent applications and interpretations can lead to diering conclusions.Narrative logic is most often employed to create continuity where there is a plot hole or some intentional gap in anarrative, or to explain other unresolved issues within a narrative (i.e. questions such as Did this character die orsimply disappear?" or Why did two instances under the same circumstances lead to dierent results?"). It may alsobe used for other purposes, such as answering theoretical questions derived from the narrative (i.e. What wouldhappen if...?" or Who would win in a battle between...?"). In a broader sense it is used in devices such as characterdevelopment, since a character is dened by the interpretations of its actions and the rhetoric used to describe it.

3.1 Example: Kill Bill: Volume 2

Consider the question of what happens to the character of Elle Driver (Daryl Hannah) in this movie. At the end of themovie a shot is shown of each of the characters on the Brides kill list with a caption telling the fate of the character.For Elle Driver the caption reads "???, director Quentin Tarantino seemingly leaving it up to the audience to decide.Using narrative logic, we take all relevant information from the narrative and come to a conclusion about Driversfate. Driver is last depicted thrashing about in a trailer, having been blinded by the Bride. This trailer is also occupiedby a deadly black mamba, a fact that is emphasized when it is shown poised to strike as the Bride leaves the trailer.Logically, a thrashing, blind woman and threatened snake occupying a conned space would lead most to concludethat Driver falls prey to the snake and is dead at the end of the lm. This conclusion may be further supported bytaking into consideration the lms rhetorical and artistic devices. The lm is largely a tale of deadly vengeance, andthe other targets of the Bride are known to be dead at the lms end, thus the ending is better rounded by assumingDriver to likewise be dead. Also, the Brides codename is Black Mamba, and this same snake killed another of theBrides targets, so if one presumes the snake is a rhetorical extension of the Brides wrath, it would most likely strikeDriver down while she is vulnerable.However, one may also argue that Driver survived. Again, this conclusion can be supported with logic and analysis ofthe movies narrative. The strongest evidence may be the very fact that Drivers fate is left open to question. Anotherpoint is that, despite blindness, Driver is still a highly trained professional who would not easily succumb to death bya small reptile.This example shows how the application of narrative logic may lead to dierent conclusions using the same evidence.However, these conclusions are similar in that they do not impact the narrative itself, but only build upon it.

15

Chapter 4

Natural deduction

In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed byinference rules closely related to the natural way of reasoning. This contrasts with the axiomatic systems whichinstead use axioms as much as possible to express the logical laws of deductive reasoning.

4.1 MotivationNatural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common tothe systems of Hilbert, Frege, and Russell (see, e.g., Hilbert system). Such axiomatizations were most famously usedby Russell andWhitehead in their mathematical treatise Principia Mathematica. Spurred on by a series of seminars inPoland in 1926 by ukasiewicz that advocated a more natural treatment of logic, Jakowski made the earliest attemptsat dening a more natural deduction, rst in 1929 using a diagrammatic notation, and later updating his proposal ina sequence of papers in 1934 and 1935.[1] His proposals led to dierent notations such as Fitch-style calculus (orFitchs diagrams) or Suppes' method of which e.g. Lemmon gave a variant called system L.Natural deduction in its modern form was independently proposed by the German mathematician Gentzen in 1934,in a dissertation delivered to the faculty of mathematical sciences of the University of Gttingen.[2] The term naturaldeduction (or rather, its German equivalent natrliches Schlieen) was coined in that paper:

Ichwollte nun zunchst einmal einen Formalismus aufstellen, der demwirklichen Schlieenmglichstnahe kommt. So ergab sich ein Kalkl des natrlichen Schlieens.[3]

(First I wished to construct a formalism that comes as close as possible to actual reasoning. Thusarose a calculus of natural deduction.)

Gentzen was motivated by a desire to establish the consistency of number theory. He was unable to prove the mainresult required for the consistency result, the cut elimination theorem the Hauptsatz directly for Natural De-duction. For this reason he introduced his alternative system, the sequent calculus, for which he proved the Hauptsatzboth for classical and intuitionistic logic. In a series of seminars in 1961 and 1962 Prawitz gave a comprehensivesummary of natural deduction calculi, and transported much of Gentzens work with sequent calculi into the naturaldeduction framework. His 1965 monographNatural deduction: a proof-theoretical study[4] was to become a referencework on natural deduction, and included applications for modal and second-order logic.In natural deduction, a proposition is deduced from a collection of premises by applying inference rules repeatedly.The system presented in this article is a minor variation of Gentzens or Prawitzs formulation, but with a closeradherence to Martin-Lf's description of logical judgments and connectives.[5]

4.2 Judgments and propositionsA judgment is something that is knowable, that is, an object of knowledge. It is evident if one in fact knows it.[6] Thus"it is raining" is a judgment, which is evident for the one who knows that it is actually raining; in this case one may

16

4.3. INTRODUCTION AND ELIMINATION 17

readily nd evidence for the judgment by looking outside the window or stepping out of the house. In mathematicallogic however, evidence is often not as directly observable, but rather deduced from more basic evident judgments.The process of deduction is what constitutes a proof; in other words, a judgment is evident if one has a proof for it.Themost important judgments in logic are of the form "A is true". The letterA stands for any expression representing aproposition; the truth judgments thus require a more primitive judgment: "A is a proposition". Many other judgmentshave been studied; for example, "A is false" (see classical logic), "A is true at time t" (see temporal logic), "A isnecessarily true" or "A is possibly true" (see modal logic), "the program M has type " (see programming languagesand type theory), "A is achievable from the available resources" (see linear logic), and many others. To start with, weshall concern ourselves with the simplest two judgments "A is a proposition" and "A is true", abbreviated as "A propand "A true respectively.The judgment "A prop denes the structure of valid proofs of A, which in turn denes the structure of propositions.For this reason, the inference rules for this judgment are sometimes known as formation rules. To illustrate, if wehave two propositionsA and B (that is, the judgments "A prop and "B prop are evident), then we form the compoundproposition A and B, written symbolically as " A ^B ". We can write this in the form of an inference rule:A prop B prop

(A^B) prop ^Fwhere the parentheses are omitted to make the inference rule more succinct:A prop B prop

A^B prop ^FThis inference rule is schematic: A and B can be instantiated with any expression. The general form of an inferencerule is:J1 J2 Jn

J namewhere each Ji is a judgment and the inference rule is named name. The judgments above the line are known aspremises, and those below the line are conclusions. Other common logical propositions are disjunction ( A _ B ),negation ( :A ), implication ( A B ), and the logical constants truth ( > ) and falsehood ( ? ). Their formationrules are below.A prop B prop

A_B prop _FA prop B prop

AB prop F > prop >F ? prop ?FA prop:A prop :F

4.3 Introduction and eliminationNow we discuss the "A true judgment. Inference rules that introduce a logical connective in the conclusion areknown as introduction rules. To introduce conjunctions, i.e., to conclude "A and B true for propositions A and B,one requires evidence for "A true and "B true. As an inference rule:A true B true

(A^B) true ^IIt must be understood that in such rules the objects are propositions. That is, the above rule is really an abbreviationfor:A prop B prop A true B true

(A^B) true ^IThis can also be written:A^B prop A true B true

(A^B) true ^IIn this form, the rst premise can be satised by the ^F formation rule, giving the rst two premises of the previousform. In this article we shall elide the prop judgments where they are understood. In the nullary case, one canderive truth from no premises.

> true >IIf the truth of a proposition can be established in more than one way, the corresponding connective has multipleintroduction rules.A true

A_B true _I1 B trueA_B true _I2Note that in the nullary case, i.e., for falsehood, there are no introduction rules. Thus one can never infer falsehoodfrom simpler judgments.

18 CHAPTER 4. NATURAL DEDUCTION

Dual to introduction rules are elimination rules to describe how to de-construct information about a compound propo-sition into information about its constituents. Thus, from "A B true, we can conclude "A true and "B true":A^B trueA true ^E1 A^B trueB true ^E2

As an example of the use of inference rules, consider commutativity of conjunction. If A B is true, then B A istrue; This derivation can be drawn by composing inference rules in such a fashion that premises of a lower inferencematch the conclusion of the next higher inference.A ^B trueB true ^E2

A ^B trueA true ^E1

B ^A true ^IThe inference gures we have seen so far are not sucient to state the rules of implication introduction or disjunctionelimination; for these, we need a more general notion of hypothetical derivation.

4.4 Hypothetical derivations

A pervasive operation in mathematical logic is reasoning from assumptions. For example, consider the followingderivation:A ^ (B ^ C) trueB ^ C trueB true

^E1^E2

This derivation does not establish the truth of B as such; rather, it establishes the following fact:

If A (B C) is true then B is true.

In logic, one says "assuming A (B C) is true, we show that B is true"; in other words, the judgment "B true" dependson the assumed judgment "A (B C) true". This is a hypothetical derivation, which we write as follows:A ^ (B ^ C) true

...B true

The interpretation is: "B true is derivable from A (B C) true". Of course, in this specic example we actuallyknow the derivation of "B true" from "A (B C) true", but in general we may not a-priori know the derivation. Thegeneral form of a hypothetical derivation is:D1 D2 Dn

...J

Each hypothetical derivation has a collection of antecedent derivations (theDi) written on the top line, and a succedentjudgment (J) written on the bottom line. Each of the premises may itself be a hypothetical derivation. (For simplicity,we treat a judgment as a premise-less derivation.)The notion of hypothetical judgment is internalised as the connective of implication. The introduction and eliminationrules are as follows.

A trueu

...B true

A B true IuA B true A true

B trueE

In the introduction rule, the antecedent named u is discharged in the conclusion. This is a mechanism for delimitingthe scope of the hypothesis: its sole reason for existence is to establish "B true"; it cannot be used for any otherpurpose, and in particular, it cannot be used below the introduction. As an example, consider the derivation of "A (B (A B)) true":

4.5. CONSISTENCY, COMPLETENESS, AND NORMAL FORMS 19

A trueu

B truew

A ^B true ^IB (A ^B) true

A (B (A ^B)) true IuIw

This full derivation has no unsatised premises; however, sub-derivations are hypothetical. For instance, the derivationof "B (A B) true" is hypothetical with antecedent "A true" (named u).With hypothetical derivations, we can now write the elimination rule for disjunction:

A _B true A trueu

...C true

B truew

...C true

C true_Eu;w

In words, if A B is true, and we can derive C true both from A true and from B true, then C is indeed true. Note thatthis rule does not commit to either A true or B true. In the zero-ary case, i.e. for falsehood, we obtain the followingelimination rule:?trueC true ?EThis is read as: if falsehood is true, then any proposition C is true.Negation is similar to implication.

A trueu

...p true

:A true:Iu;p:A true A true

C true:E

The introduction rule discharges both the name of the hypothesis u, and the succedent p, i.e., the proposition p mustnot occur in the conclusion A. Since these rules are schematic, the interpretation of the introduction rule is: if from"A true" we can derive for every proposition p that "p true", thenAmust be false, i.e., "not A true". For the elimination,if both A and not A are shown to be true, then there is a contradiction, in which case every proposition C is true.Because the rules for implication and negation are so similar, it should be fairly easy to see that not A and A areequivalent, i.e., each is derivable from the other.

4.5 Consistency, completeness, and normal forms

A theory is said to be consistent if falsehood is not provable (from no assumptions) and is complete if every theoremis provable using the inference rules of the logic. These are statements about the entire logic, and are usually tied tosome notion of a model. However, there are local notions of consistency and completeness that are purely syntacticchecks on the inference rules, and require no appeals to models. The rst of these is local consistency, also knownas local reducibility, which says that any derivation containing an introduction of a connective followed immediatelyby its elimination can be turned into an equivalent derivation without this detour. It is a check on the strength ofelimination rules: they must not be so strong that they include knowledge not already contained in its premises. Asan example, consider conjunctions.Dually, local completeness says that the elimination rules are strong enough to decompose a connective into the formssuitable for its introduction rule. Again for conjunctions:These notions correspond exactly to -reduction (beta reduction) and -conversion (eta conversion) in the lambdacalculus, using the CurryHoward isomorphism. By local completeness, we see that every derivation can be convertedto an equivalent derivation where the principal connective is introduced. In fact, if the entire derivation obeys thisordering of eliminations followed by introductions, then it is said to be normal. In a normal derivation all eliminationshappen above introductions. In most logics, every derivation has an equivalent normal derivation, called a normalform. The existence of normal forms is generally hard to prove using natural deduction alone, though such accountsdo exist in the literature, most notably by Dag Prawitz in 1961.[7] It is much easier to show this indirectly by meansof a cut-free sequent calculus presentation.

20 CHAPTER 4. NATURAL DEDUCTION

4.6 First and higher-order extensions

Summary of rst-order system

The logic of the earlier section is an example of a single-sorted logic, i.e., a logic with a single kind of object: propo-sitions. Many extensions of this simple framework have been proposed; in this section we will extend it with a secondsort of individuals or terms. More precisely, we will add a new kind of judgment, "t is a term" (or "t term") where tis schematic. We shall x a countable set V of variables, another countable set F of function symbols, and constructterms as follows:For propositions, we consider a third countable set P of predicates, and dene atomic predicates over terms with the

4.7. DIFFERENT PRESENTATIONS OF NATURAL DEDUCTION 21

following formation rule:In addition, we add a pair of quantied propositions: universal () and existential ():These quantied propositions have the following introduction and elimination rules.In these rules, the notation [t/x]A stands for the substitution of t for every (visible) instance of x inA, avoiding capture;see the article on lambda calculus for more detail about this standard operation. As before the superscripts on thename stand for the components that are discharged: the term a cannot occur in the conclusion of I (such terms areknown as eigenvariables or parameters), and the hypotheses named u and v in E are localised to the second premisein a hypothetical derivation. Although the propositional logic of earlier sections was decidable, adding the quantiersmakes the logic undecidable.So far the quantied extensions are rst-order: they distinguish propositions from the kinds of objects quantied over.Higher-order logic takes a dierent approach and has only a single sort of propositions. The quantiers have as thedomain of quantication the very same sort of propositions, as reected in the formation rules:A discussion of the introduction and elimination forms for higher-order logic is beyond the scope of this article. Itis possible to be in between rst-order and higher-order logics. For example, second-order logic has two kinds ofpropositions, one kind quantifying over terms, and the second kind quantifying over propositions of the rst kind.

4.7 Dierent presentations of natural deduction

4.7.1 Tree-like presentationsGentzens discharging annotations used to internalise hypothetical judgments can be avoided by representing proofsas a tree of sequents A instead of a tree of A true judgments.

4.7.2 Sequential presentationsJakowskis representations of natural deduction led to dierent notations such as Fitch-style calculus (or Fitchsdiagrams) or Suppes' method, of which Lemmon gave a variant called system L. Such presentation systems, whichare more accurately described as tabular, include the following.

1940: In a textbook, Quine[8] indicated antecedent dependencies by line numbers in square brackets, antici-pating Suppes 1957 line-number notation.

1950: In a textbook, Quine (1982, pp. 241255) demonstrated a method of using one or more asterisks to theleft of each line of proof to indicate dependencies. This is equivalent to Kleenes vertical bars. (It is not totallyclear if Quines asterisk notation appeared in the original 1950 edition or was added in a later edition.)

1957: An introduction to practical logic theorem proving in a textbook by Suppes (1999, pp. 25150). Thisindicated dependencies (i.e. antecedent propositions) by line numbers at the left of each line.

1963: Stoll (1979, pp. 183190, 215219) uses sets of line numbers to indicate antecedent dependencies ofthe lines of sequential logical arguments based on natural deduction inference rules.

1965: The entire textbook by Lemmon (1965) is an introduction to logic proofs using a method based on thatof Suppes.

1967: In a textbook, Kleene (2002, pp. 5058, 128130) briey demonstrated two kinds of practical logicproofs, one system using explicit quotations of antecedent propositions on the left of each line, the other systemusing vertical bar-lines on the left to indicate dependencies.[9]

4.8 Proofs and type-theoryThe presentation of natural deduction so far has concentrated on the nature of propositions without giving a formaldenition of a proof. To formalise the notion of proof, we alter the presentation of hypothetical derivations slightly.We label the antecedents with proof variables (from some countable set V of variables), and decorate the succedent

22 CHAPTER 4. NATURAL DEDUCTION

with the actual proof. The antecedents or hypotheses are separated from the succedent by means of a turnstile ().This modication sometimes goes under the name of localised hypotheses. The following diagram summarises thechange.The collection of hypotheses will be written as when their exact composition is not relevant. To make proofsexplicit, we move from the proof-less judgment "A true" to a judgment: " is a proof of (A true)", which is writtensymbolically as " : A true". Following the standard approach, proofs are specied with their own formation rulesfor the judgment " proof". The simplest possible proof is the use of a labelled hypothesis; in this case the evidenceis the label itself.For brevity, we shall leave o the judgmental label true in the rest of this article, i.e., write " : A". Let usre-examine some of the connectives with explicit proofs. For conjunction, we look at the introduction rule I todiscover the form of proofs of conjunction: they must be a pair of proofs of the two conjuncts. Thus:The elimination rules E1 and E2 select either the left or the right conjunct; thus the proofs are a pair of projections rst (fst) and second (snd).For implication, the introduction form localises or binds the hypothesis, written using a ; this corresponds to thedischarged label. In the rule, ", u:A" stands for the collection of hypotheses , together with the additional hypothesisu.With proofs available explicitly, one can manipulate and reason about proofs. The key operation on proofs is thesubstitution of one proof for an assumption used in another proof. This is commonly known as a substitution theorem,and can be proved by induction on the depth (or structure) of the second judgment.

Substitution theorem If 1 : A and , u:A 2 : B, then [1/u] 2 : B.

So far the judgment " : A" has had a purely logical interpretation. In type theory, the logical view is exchangedfor a more computational view of objects. Propositions in the logical interpretation are now viewed as types, andproofs as programs in the lambda calculus. Thus the interpretation of " : A" is "the program has type A". Thelogical connectives are also given a dierent reading: conjunction is viewed as product (), implication as the functionarrow (), etc. The dierences are only cosmetic, however. Type theory has a natural deduction presentation in termsof formation, introduction and elimination rules; in fact, the reader can easily reconstruct what is known as simpletype theory from the previous sections.The dierence between logic and type theory is primarily a shift of focus from the types (propositions) to the programs(proofs). Type theory is chiey interested in the convertibility or reducibility of programs. For every type, there arecanonical programs of that type which are irreducible; these are known as canonical forms or values. If every programcan be reduced to a canonical form, then the type theory is said to be normalising (or weakly normalising). If thecanonical form is unique, then the theory is said to be strongly normalising. Normalisability is a rare feature of mostnon-trivial type theories, which is a big departure from the logical world. (Recall that almost every logical derivationhas an equivalent normal derivation.) To sketch the reason: in type theories that admit recursive denitions, it ispossible to write programs that never reduce to a value; such looping programs can generally be given any type.In particular, the looping program has type , although there is no logical proof of " true". For this reason, thepropositions as types; proofs as programs paradigm only works in one direction, if at all: interpreting a type theory asa logic generally gives an inconsistent logic.Like logic, type theory has many extensions and variants, including rst-order and higher-order versions. An inter-esting branch of type theory, known as dependent type theory, allows quantiers to range over programs themselves.These quantied types are written as and instead of and , and have the following formation rules:These types are generalisations of the arrow and product types, respectively, as witnessed by their introduction andelimination rules.Dependent type theory in full generality is very powerful: it is able to express almost any conceivable propertyof programs directly in the types of the program. This generality comes at a steep price either typecheckingis undecidable (extensional type theory), or extensional reasoning is more dicult (intensional type theory). Forthis reason, some dependent type theories do not allow quantication over arbitrary programs, but rather restrict toprograms of a given decidable index domain, for example integers, strings, or linear programs.Since dependent type theories allow types to depend on programs, a natural question to ask is whether it is possible forprograms to depend on types, or any other combination. There aremany kinds of answers to such questions. A popularapproach in type theory is to allow programs to be quantied over types, also known as parametric polymorphism;of this there are two main kinds: if types and programs are kept separate, then one obtains a somewhat more well-

4.9. CLASSICAL AND MODAL LOGICS 23

behaved system called predicative polymorphism; if the distinction between program and type is blurred, one obtainsthe type-theoretic analogue of higher-order logic, also known as impredicative polymorphism. Various combinationsof dependency and polymorphism have been considered in the literature, the most famous being the lambda cube ofHenk Barendregt.The intersection of logic and type theory is a vast and active research area. New logics are usually formalised ina general type theoretic setting, known as a logical framework. Popular modern logical frameworks such as thecalculus of constructions and LF are based on higher-order dependent type theory, with various trade-os in termsof decidability and expressive power. These logical frameworks are themselves always specied as natural deductionsystems, which is a testament to the versatility of the natural deduction approach.

4.9 Classical and modal logicsFor simplicity, the logics presented so far have been intuitionistic. Classical logic extends intuitionistic logic with anadditional axiom or principle of excluded middle:

For any proposition p, the proposition p p is true.

This statement is not obviously either an introduction or an elimination; indeed, it involves two distinct connectives.Gentzens original treatment of excluded middle prescribed one of the following three (equivalent) formulations,which were already present in analogous forms in the systems of Hilbert and Heyting:(XM3 is merely XM2 expressed in terms of E.) This treatment of excluded middle, in addition to being objectionablefrom a purists standpoint, introduces additional complications in the denition of normal forms.A comparatively more satisfactory treatment of classical natural deduction in terms of introduction and eliminationrules alone was rst proposed by Parigot in 1992 in the form of a classical lambda calculus called . The key insightof his approach was to replace a truth-centric judgmentA truewith a more classical notion, reminiscent of the sequentcalculus: in localised form, instead of A, he used , with a collection of propositions similar to . wastreated as a conjunction, and as a disjunction. This structure is essentially lifted directly from classical sequentcalculi, but the innovation in was to give a computational meaning to classical natural deduction proofs in termsof a callcc or a throw/catch mechanism seen in LISP and its descendants. (See also: rst class control.)Another important extension was for modal and other logics that need more than just the basic judgment of truth.These were rst described, for the alethic modal logics S4 and S5, in a natural deduction style by Prawitz in 1965,[4]and have since accumulated a large body of related work. To give a simple example, the modal logic S4 requires onenew judgment, "A valid", that is categorical with respect to truth:

If A true under no assumptions of the form B true, then A valid.

This categorical judgment is internalised as a unary connective A (read "necessarily A") with the following intro-duction and elimination rules:Note that the premise "A valid" has no dening rules; instead, the categorical denition of validity is used in its place.This mode becomes clearer in the localised form when the hypotheses are explicit. We write "; A true" where contains the true hypotheses as before, and contains valid hypotheses. On the right there is just a single judgment"A true"; validity is not needed here since " A valid" is by denition the same as "; A true". The introductionand elimination forms are then:The modal hypotheses have their own version of the hypothesis rule and substitution theorem.

Modal substitution theorem If ; 1 : A true and , u: (A valid) ; 2 : C true, then ; [1/u] 2 : Ctrue.

This framework of separating judgments into distinct collections of hypotheses, also known asmulti-zoned or polyadiccontexts, is very powerful and extensible; it has been applied for many dierent modal logics, and also for linear andother substructural logics, to give a few examples. However, relatively few systems of modal logic can be formaliseddirectly in natural deduction. To give proof-theoretic characterisations of these systems, extensions such as labellingor systems of deep inference.

24 CHAPTER 4. NATURAL DEDUCTION

The addition of labels to formulae permits much ner control of the conditions under which rules apply, allowingthe more exible techniques of analytic tableaux to be applied, as has been done in the case of labelled deduction.Labels also allow the naming of worlds in Kripke semantics; Simpson (1993) presents an inuential technique forconverting frame conditions of modal logics in Kripke semantics into inference rules in a natural deduction formali-sation of hybrid logic. Stouppa (2004) surveys the application of many proof theories, such as Avron and Pottingershypersequents and Belnaps display logic to such modal logics as S5 and B.

4.10 Comparison with other foundational approaches

4.10.1 Sequent calculusMain article: Sequent calculus

The sequent calculus is the chief alternative to natural deduction as a foundation of mathematical logic. In naturaldeduction the ow of information is bi-directional: elimination rules ow information downwards by deconstruction,and introduction rules ow information upwards by assembly. Thus, a natural deduction proof does not have a purelybottom-up or top-down reading, making it unsuitable for automation in proof search. To address this fact, Gentzen in1935 proposed his sequent calculus, though he initially intended it as a technical device for clarifying the consistencyof predicate logic. Kleene, in his seminal 1952 book Introduction to Metamathematics, gave the rst formulation ofthe sequent calculus in the modern style.[10]

In the sequent calculus all inference rules have a purely bottom-up reading. Inference rules can apply to elementson both sides of the turnstile. (To dierentiate from natural deduction, this article uses a double arrow instead ofthe right tack for sequents.) The introduction rules of natural deduction are viewed as right rules in the sequentcalculus, and are structurally very similar. The elimination rules on the other hand turn into left rules in the sequentcalculus. To give an example, consider disjunction; the right rules are familiar:On the left:Recall the E rule of natural deduction in localised form:The proposition A B, which is the succedent of a premise in E, turns into a hypothesis of the conclusion in theleft rule L. Thus, left rules can be seen as a sort of inverted elimination rule. This observation can be illustrated asfollows:In the sequent calculus, the left and right rules are performed in lock-step until one reaches the initial sequent, whichcorresponds to the meeting point of elimination and introduction rules in natural deduction. These initial rules aresupercially similar to the hypothesis rule of natural deduction, but in the sequent calculus they describe a transpositionor a handshake of a left and a right proposition:The correspondence between the sequent calculus and natural deduction is a pair of soundness and completenesstheorems, which are both provable by means of an inductive argument.

Soundness of wrt. If A, then A.Completeness of wrt. If A, then A.

It is clear by these theorems that the sequent calculus does not change the notion of truth, because the same collectionof propositions remain true. Thus, one can use the same proof objects as before in sequent calculus derivations. Asan example, consider the conjunctions. The right rule is virtually identical to the introduction ruleThe left rule, however, performs some additional substitutions that are not performed in the corresponding eliminationrules.The kinds of proofs generated in the sequent calculus are therefore rather dierent from those of natural deduction.The sequent calculus produces proofs in what is known as the -normal -long form, which corresponds to a canonicalrepresentation of the normal form of the natural deduction proof. If one attempts to describe these proofs using naturaldeduction itself, one obtains what is called the intercalation calculus (rst described by John Byrnes), which can beused to formally dene the notion of a normal form for natural deduction.The substitution theorem of natural deduction takes the form of a structural rule or structural theorem known as cutin the sequent calculus.

4.11. SEE ALSO 25

Cut (substitution) If 1 : A and , u:A 2 : C, then [1/u] 2 : C.

In most well behaved logics, cut is unnecessary as an inference rule, though it remains provable as a meta-theorem;the superuousness of the cut rule is usually presented as a computational process, known as cut elimination. This hasan interesting application for natural deduction; usually it is extremely tedious to prove certain properties directly innatural deduction because of an unbounded number of cases. For example, consider showing that a given propositionis not provable in natural deduction. A simple inductive argument fails because of rules like E or E which canintroduce arbitrary propositions. However, we know that the sequent calculus is complete with respect to naturaldeduction, so it is enough to show this unprovability in the sequent calculus. Now, if cut is not available as aninference rule, then all sequent rules either introduce a connective on the right or the left, so the depth of a sequentderivation is fully bounded by the connectives in the nal conclusion. Thus, showing unprovability is much easier,because there are only a nite number of cases to consider, and each case is composed entirely of sub-propositionsof the conclusion. A simple instance of this is the global consistency theorem: " true" is not provable. In thesequent calculus version, this is manifestly true because there is no rule that can have " " as a conclusion! Prooftheorists often prefer to work on cut-free sequent calculus formulations because of such properties.

4.11 See also Mathematical logic Sequent calculus Gerhard Gentzen System L (tabular natural deduction)