Proof-theoretic semantics, self-contradiction and the format of - PowerPoint PPT Presentation

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 Dyckho ff St. Andrews, 19.11.2011 – p. 1

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)] St. Andrews, 19.11.2011 – p. 2

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. St. Andrews, 19.11.2011 – p. 3

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 A, ∆ ⊢ C Example: “Bidirectionality” A ∧ B, ∆ ⊢ C The philosophical significance of the sequent calculus has not been properly acknowledged. [Contraction-free approaches also speak in favour of the sequent calculus] St. Andrews, 19.11.2011 – p. 4

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 St. Andrews, 19.11.2011 – p. 5

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 St. Andrews, 19.11.2011 – p. 6

Definitional reasoning • Consequence is relativized to a definition, which represents the material base • Definitions are understood as consisting of clauses of A if B 1 , B 2 ,... the form • 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 St. Andrews, 19.11.2011 – p. 7

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 St. Andrews, 19.11.2011 – p. 8

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 St. Andrews, 19.11.2011 – p. 9

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. St. Andrews, 19.11.2011 – p. 10

Contradiction and absurdity in natural deduction Inference rules: R → ⊥ R R R → ⊥ Derivation of absurdity: [ R ] (2) [ R ] (1) [ R ] (2) R → ⊥ [ R ] (1) R → ⊥ ⊥ (2) R → ⊥ ⊥ (1) R → ⊥ R ⊥ Observation (Prawitz): This proof does not normalize. St. Andrews, 19.11.2011 – p. 11

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 ⊥ St. Andrews, 19.11.2011 – p. 12

Contradiction and absurdity with terms � R R → ⊥ : = t : R → ⊥ t : R r ′ rt ⊲ t r ′ t : R → ⊥ rt : R gives non-normalizable terms: [ x : R ] (2) [ x : R ] (1) [ x : R ] (2) r ′ x : R → ⊥ [ x : R ] (1) r ′ x : R → ⊥ r ′ xx : ⊥ (2) λx.r ′ xx : R → ⊥ r ′ xx : ⊥ (1) λx.r ′ xx : R → ⊥ rλx.r ′ xx : R ( λx.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 ) St. Andrews, 19.11.2011 – p. 13

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”. St. Andrews, 19.11.2011 – p. 14

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 − St. Andrews, 19.11.2011 – p. 15

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. St. Andrews, 19.11.2011 – p. 16

Way out: Side condition on modus ponens s : A → B t : A st! st : B st ! means: st is normalizable [ x : R ] (2) [ x : R ] (1) [ x : R ] (2) r ′ x : R → ⊥ [ x : R ] (1) r ′ x : R → ⊥ r ′ xx : ⊥ (2) λx.r ′ xx : R → ⊥ r ′ xx : ⊥ (1) λx.r ′ xx : R → ⊥ rλx.r ′ xx : R ( λx.r ′ xx ) rλx.r ′ xx ! ( λx.r ′ xx ) rλx.r ′ xx : ⊥ ( λx.r ′ xx ) rλx.r ′ xx ! is not satisfied. St. Andrews, 19.11.2011 – p. 17

Way out: Side condition on modus ponens s : A → B t : A st! st : B 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 St. Andrews, 19.11.2011 – p. 18

Self-contradiction in the sequent calculus Prima facie same situation: Γ ⊢ R → ⊥ Γ , R → ⊥ ⊢ C Γ ⊢ R 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. St. Andrews, 19.11.2011 – p. 19

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. St. Andrews, 19.11.2011 – p. 20

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. St. Andrews, 19.11.2011 – p. 21

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, ∆ Γ . . . A, ∆ , . . � . . . A C . . C St. Andrews, 19.11.2011 – p. 22

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 St. Andrews, 19.11.2011 – p. 23

Formal representation of contradiction with terms Γ ⊢ t : R → ⊥ Γ , x : R → ⊥ ⊢ t : C r ′ rt ⊲ t Γ , y : R ⊢ t [ x/r ′ y ] : C Γ ⊢ rt : R Note that this is not a Dyckho ff -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, 19.11.2011 – p. 24