Proof-theoretic semantics, self-contradiction and the format of deductive reasoning

Transcription

1 St. Andrews, p. 1 To appear as an article in: L. Tranchini (ed.), Anti-Realistic Notions of Truth, Special issue of Topoi vol. 31 no. 1, 2012 Proof-theoretic semantics, self-contradiction and the format of deductive reasoning Peter Schroeder-Heister In Honour of Roy Dyckhoff

2 St. Andrews, p. 2 Subtitle: In defence of definitional freedom If anything needs to be changed in view of paradoxes, it is proofs, not definitions. Parallel: Partial recursive functions Non-terminating Turing-machines are perfectly well defined [Another approach of this kind is that of Curry and Fitch ( contraction)]

3 St. Andrews, p. 3 Paradoxes and self-contradiction We define a proposition R as its own negation: R := R or, in the intuitionistic spirit: R := R Russell s paradox is a sophisticated way of generating such a definition We avoid set-theoretic terminology after all, the problem lies with reasoning with respect to self-contradiction, and only indirectly with set-theoretic concepts.

4 St. Andrews, p. 4 Thesis: The sequent calculus, and not natural deduction is the appropriate formal model of deductive reasoning Characteristic feature: Specific introduction of assumtions according to their meaning Example: A, C A B, C Bidirectionality The philosophical significance of the sequent calculus has not been properly acknowledged. [Contraction-free approaches also speak in favour of the sequent calculus]

5 St. Andrews, p. 5 Background: Proof-Theoretic Semantics Not happy about theory of meaning Semantics should not be left to the denotationalists or truth-conditionalists alone There is no opposition between semantics and proof-theory There are many issues that proof-theoretic semantics shares with truth-condition semantics, much beyond the broad interest in meaning

6 St. Andrews, p. 6 Consequence (simpliciter) vs. logical consequence I am dealing with consequence simpliciter Logical consequence is a special case of consequence logical constants domain independence The traditional preoccupation with logical consequence obstructs the view on many phenomena Material consequence is not logical consequence with respect to certain assumptions (axioms) Basic prejudice since Aristotle

7 St. Andrews, p. 7 Definitional reasoning Consequence is relativized to a definition, which represents the material base Definitions are understood as consisting of clauses of the form A if B 1,B 2,... I prefer the paradigm of logic programming to that of functional programming, as it can better deal with non-well-founded phenomena Definitional freedom: A if A A if not A is both possible

8 St. Andrews, p. 8 Definitional reasoning Traditional criteria such as conservativeness (non-creativity) and eliminability may be (and in regular cases are) particular features of the definitional system considered, but they are not requirements for it s being admissible The traditional philosophical preoccupation in philosophy with explicit definitions is ill-guided Inductive definitions are the standard case Logic programming is a computational treatment of inductive definitions

9 St. Andrews, p. 9 Dogmas of standard semantics The priority of the categorical over the hypothetical The transmission view of consequence The view of hypotheses as placeholders The priority of closed over open The reducibility of abstract objects to concrete ones (well-foundedness) The view that valid consequence guarantees correct inference

10 St. Andrews, p. 10 The formal model of reasoning The natural-deduction model of consequence is strongly tied to the dogmas: Criticizing the latter leads to criticizing the former. A less internal argument: Reasoning with self-contradiction suggests an alternative model. We want to be able to deal with self-contradiction not just avoid it.

11 St. Andrews, p. 11 Contradiction and absurdity in natural deduction Inference rules: R R R R Derivation of absurdity: [R] (1) R [R] (1) (1) R [R] (2) R [R] (2) (2) R R Observation (Prawitz): This proof does not normalize.

12 St. Andrews, p. 12 Non-termination of reduction [R] R [R] R [R] R [R] R R [R] R [R] R R R [R] R [R] R R [R] R [R] R [R] R [R] R R

13 Contradiction and absurdity with terms { R := R t : R rt : R t : R r t : R r rt t gives non-normalizable terms: [x : R] (1) r x : R [x : R] (1) r xx : (1) λx.r xx : R (λx.r xx)rλx.r xx : [x : R] (2) r x : R [x : R] (2) r xx : (2) λx.r xx : R rλx.r xx : R r (rλx.r xx)(rλx.r xx) (λx.r xx)(rλx.r xx) r (rλx.r xx)(rλx.r xx) St. Andrews, p. 13

14 St. Andrews, p. 14 The meaning of reduction in standard proof-theoretic semantics (Dummett-Prawitz): There is direct (canonical) and indirect (non-canonical) knowledge. Indirect knowledge reduces to direct knowledge. Second-class knowledge can always be upgraded to first-class. This is crucial for the solution of the paradox of inference.

15 St. Andrews, p. 15 The interpretation of non-termination There is indirect knowledge, which cannot be directified. There is irreducibly indirect knowledge. Self-contradiction yield second-class knowledge of absurdity, which cannot be upgraded to first-class knowledge. Cp. discussion of theoretical terms in philosophy of science: They are only indirectly linked with observation terms. Quinean perspective

16 St. Andrews, p. 16 Counterargument There should be no knowledge of absurdity whatsoever. Absurdity is not on par with theoretical terms. A non-normalizable proof is no proof at all.

17 St. Andrews, p. 17 Way out: Side condition on modus ponens s : A B t : A st : B st! st! means: st is normalizable [x : R] (1) r x : R [x : R] (1) r xx : (1) λx.r xx : R (λx.r xx)rλx.r xx : (λx.r xx)rλx.r xx! is not satisfied. [x : R] (2) r x : R [x : R] (2) r xx : (2) λx.r xx : R rλx.r xx : R (λx.r xx)rλx.r xx!

18 St. Andrews, p. 18 Way out: Side condition on modus ponens s : A B t : A st : B st! Problems: Proviso not necessarily decidable need a (metalinguistic) proof system for is normalizable Proviso not closed under substitution All provisos have to be checked again when proofs are composed Result: High degree on non-locality, way beyond standard non-locality in natural deduction

19 St. Andrews, p. 19 Self-contradiction in the sequent calculus Prima facie same situation: Γ R Γ R Γ,R C Γ,R C Derivation of absurdity: R R R,R R,R R R R R,R R R,R R R Now cut rather than modus ponens. Cut elimination loops.

20 St. Andrews, p. 20 Cut vs. modus ponens Modus ponens is a meaning-giving rule. We cannot just dispense with it. Cut is a structural rule that comes in addition to the semantical rules. In principle, we can give up cut. This should be done in the case of self-contradiction.

21 St. Andrews, p. 21 Overall picture We reason with respect to a definition. Normally, if the definition is well-behaved (especially well-founded), cut is admissible. In other cases such as self-contradiction it is not admissible. Cut is not a primitive rule. But something that holds depending on the definitions presupposed. Admissibility of cut corresponds to termination.

22 St. Andrews, p. 22 Cut and substitution Γ A A, C Γ, C In natural deduction, this corresponds to combining proofs, i.e. substitution a proof for an open assumption. Γ. A, A,. C Γ. A,. C

23 St. Andrews, p. 23 Cut and substitution For terms, this is ordinary substitution: Γ s : A x : A, t : C Γ, t[x/s] : C This substitution feature can be blamed for paradoxes

24 Formal representation of contradiction with terms Γ t : R Γ rt : R Γ,x : R t : C Γ,y : R t[x/r y] : C r rt t Note that this is not a Dyckhoff-style representation, which would instead be Γ,x : R t : C Γ,y : R F (y,x.t) : C for some selector F, whose natural deduction translation would be: φ(f (y,x.t) = t[x/r y]) So we are using natural deduction terms in the style of Barendregt and Ghilezan. Reason: Terms should represent knowledge and not just codify proofs. St. Andrews, p. 24

25 St. Andrews, p. 25 Derivation of absurdity x : R x : R x : R,y : R yx : x : R,z : R r zx : x : R r xx : λx.r xx : R rλx.r xx : R x : R x : R x : R,y : R yx : x : R,z : R r zx : x : R r xx : r (rλx.r xx)(rλx.r xx) : r (rλx.r xx)(rλx.r xx) (λx.r xx)(rλx.r xx) r (rλx.r xx)(rλx.r xx)

26 St. Andrews, p. 26 Termination proviso in the sequent calculus Γ t : A x : A, s : C Γ, s[x/t] : C s[x/t]!! : normalizes Again we need a proof system for normalization. But: The side-condition is purely local. From the Dyckhoff-translation follows: s[x/t]! implies that this cut is admissible.

27 St. Andrews, p. 27 Restricted modus ponens vs. restricted cut Restricted modus ponens: s : A B t : A App(s,t) : B App(s,t)! Restricted cut: t : A,x : A s : C s[x/t] : C s[x/t]!

28 St. Andrews, p. 28 Under assumptions Restricted modus ponens: y : D. s : A B t : A App(s,t)! App(s,t) : B Restricted cut: t : A,y : D,x : A s : C,y : D s[x/t] : C s[x/t]!

29 St. Andrews, p. 29 Performing substitution In natural deduction by combining proofs:. t : D. s : A B t[y/t ] : A App(s,t[y/t ])! App(s,t[y/t ]) : B In the sequent calculus by an additional cut: t : D,y : D t : A,x : A s : C s[x/t]!,y : D s[x/t] : C s[x/t[y/t ]]! s[x/t[y/t ]] : C

30 St. Andrews, p. 30 Results The sequent calculus is the preferable system due to the locality of its rules. Highly nonlocal features of natural deduction such as the combination of proofs are handled in the sequent calculus by the local rule of cut. In principle we can do without cut. However, cut with side conditions stays local (if the side conditions do). The fundamental feature of paradoxes is that substitution of a denoting term into a denoting term does not need to result in a denoting term: This can be handled by the sequent calculus. Substitution is perhaps the most fundamental notion in logic.

31 Appendix: Free type theory We may consider turning the side condition in t : A,x : A s : C s[x/t] : C s[x/t]! into an actual premiss: t : A,x : A s : C s[x/t]! s[x/t] : C Pro: Gain expressive power Contra: The formal and ontological framework of type theory has to be re-worked This is not against the spirit of type theory: Formation rules for terms rather than only for types. St. Andrews, p. 31