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
To appear as an article in: L. Tranchini (ed.), Anti-Realistic Notions of Truth, Special issue of Topoi vol. 31 no. 1, 2012
Peter Schroeder-Heister In Honour of Roy Dyckhoff
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)]
We define a proposition R as its own negation: R := ¬R
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
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]
proof-theory
shares with truth-condition semantics, much beyond the broad interest in meaning
consequence
consequence obstructs the view on many phenomena
consequence with respect to certain assumptions (axioms)
represents the material base
the form A if B1, B2,...
with non-well-founded phenomena
is both possible
(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
philosophy with explicit definitions is ill-guided Inductive definitions are the standard case
inductive definitions
(well-foundedness)
inference
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.
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.
[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 ⊥
:= 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 → ⊥ [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 : ⊥ r′(rλx.r′xx)(rλx.r′xx) ⊲ (λx.r′xx)(rλx.r′xx) ⊲ r′(rλx.r′xx)(rλx.r′xx)
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”.
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.
They are only indirectly linked with observation terms. − → Quinean perspective
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.
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 → ⊥ [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 ! (λx.r′xx)rλx.r′xx : ⊥ (λx.r′xx)rλx.r′xx ! is not satisfied.
s : A → B t : A st : B st! Problems:
(metalinguistic) proof system for ‘is normalizable’
have to be checked again when proofs are composed Result: High degree on non-locality, way beyond standard non-locality in natural deduction
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.
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.
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.
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
. . . A, ∆ . . . C
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
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.
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)
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.
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] !
Restricted modus ponens: s : A → B y : D . . . 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] !
In natural deduction by combining proofs: s : A → B . . . t′ : D . . . 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
the locality of its rules.
the combination of proofs are handled in the sequent calculus by the local rule of cut.
side conditions do).
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.
logic.
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.