The Meaning of Proofs in Different Proof Systems Sara Ayhan (Ruhr University Bochum) PhDs in Logic XI Bern, April 24-26, 2019 Institut f¨ ur Exakte Wissenschaften, University of Bern 26 April, 2019 Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 1 / 30
Introduction Questions of this talk What is the meaning of proofs in a proof-theoretic semantics account? Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 2 / 30
Introduction Excursus I: Main ideas of PTS Semantical approach to the meaning of logical expressions based on notion of proof Opposed to standard semantics, i.e. model theory, based on notion of truth Rules of inference give the meaning of logical constants, not models, truth tables, etc. → Proofs considered to be more than just technical devices but important from a semantical point of view Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 3 / 30
Introduction Main ideas of PTS Gentzen’s remarks on his calculus of natural deduction: The introductions represent, as it were, the ‘definitions’ of the symbols concerned, and the eliminations are no more, in the final analysis, than the consequences of these definitions. (Gentzen 1934/5) Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 4 / 30
Introduction Excursus II: Fregean ideas on meaning Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 5 / 30
Introduction Excursus II: Fregean ideas on meaning What does it mean to say a = b ? If it were just about what a and b refer to (their reference or denotation ), then there wouldn’t be any difference in cognitive value between a = b and a = a What is interesting is if the difference in the signs a and b corresponds to a difference in the “mode of presentation” (difference in sense ) → The morning star is the evening star Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 5 / 30
Introduction Questions of this talk What is the meaning of proofs in a proof-theoretic semantics account? How do we get from meaning of logical constants to meaning of proofs as a whole? → Meaning of proof must be in some way based on rules of inference it contains Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 6 / 30
Introduction Questions of this talk What is the meaning of proofs in a proof-theoretic semantics account? How do we get from meaning of logical constants to meaning of proofs as a whole? → Meaning of proof must be in some way based on rules of inference it contains → Approach proposed by Tranchini: distinguishes for a derivation to have a denotation (a proof object it refers to) and to have sense: “being constituted of applications of correct inferences rules” ( Tranchini 2016: 508) Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 6 / 30
Introduction Questions of this talk What exactly does the sense of a derivation consist of? → Can we make a distinction between sense and denotation of proofs analogous to Frege’s distinction for singular terms or sentences? Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 7 / 30
Introduction Questions of this talk What exactly does the sense of a derivation consist of? → Can we make a distinction between sense and denotation of proofs analogous to Frege’s distinction for singular terms or sentences? What is the relation of different kinds of proof systems with respect to such a distinction? → Do two derivations with the same denotation in different proof systems always differ in sense or can sense be shared over two proof systems? Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 7 / 30
Introduction The starting point There can be different ways to go from the same premises to the same conclusion, not only in different (kinds of) proof systems but also within one proof system (cf. Restall 2017 for an approach in classical logic) Focus so far: Normal vs. non-normal proofs in Natural Deduction (ND) Proofs containing an application of the cut rule vs. cut-free proofs in Sequent Calculus (SC) However, this can also happen due to changing the order of rule applications → Does this change the denotation of the derivation, i.e. the proof it refers to, or only the sense, i.e. the way the inference is built up? Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 8 / 30
Introduction The basic approach Encoding the proof systems with λ -terms → makes sense and denotation transparent: Denotation is referred to by the normal form of the term that denotes the sequent or formula to be proven → henceforth: the ‘end-term’ → Two derivations with β -equivalent end-terms denote the same proof Concerning sense , usually the difference between normal and non-normal terms is mentioned ( Girard 1990, Tranchini 2016, Restall 2017) Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 9 / 30
Introduction Different senses: normal vs. non-normal terms ND p ⊃ p [ x : p ] ⊃ I λ x . x : p ⊃ p ND non-normal p ⊃ p [ x : p ] [ y : q ] ⊃ I ⊃ I λ x . x : p ⊃ p λ y . y : q ⊃ q ∧ I � λ x . x , λ y . y � : ( p ⊃ p ) ∧ ( q ⊃ q ) ∧ E fst ( � λ x . x , λ y . y � ) : p ⊃ p fst ( � λ x . x , λ y . y � ) � λ x . x Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 10 / 30
Introduction Different senses: normal vs. non-normal terms SC ⊢ ( p ∧ p ) ⊃ ( p ∨ p ) z : p ⊢ z : p ∧ L y : p ∧ p ⊢ fst ( y ) : p ∨ R y : p ∧ p ⊢ inl fst ( y ) : p ∨ p ⊃ R ⊢ λ y . inl fst ( y ) : ( p ∧ p ) ⊃ ( p ∨ p ) SC cut ⊢ ( p ∧ p ) ⊃ ( p ∨ p ) z : p ⊢ z : p z : p ⊢ z : p ∧ L ∧ L y : p ∧ p ⊢ fst ( y ) : p y : p ∧ p ⊢ snd ( y ) : p ∧ R y : p ∧ p , y : p ∧ p ⊢ � fst ( y ) , snd ( y ) � : p ∧ p z : p ⊢ z : p C ∧ L y : p ∧ p ⊢ � fst ( y ) , snd ( y ) � : p ∧ p y : p ∧ p ⊢ fst ( y ) : p cut y : p ∧ p ⊢ fst � fst ( y ) , snd ( y ) � : p ∨ R y : p ∧ p ⊢ inl fst � fst ( y ) , snd ( y ) � : p ∨ p ⊃ R ⊢ λ y . inl fst � fst ( y ) , snd ( y ) � : ( p ∧ p ) ⊃ ( p ∨ p ) λ y . inl fst � fst ( y ) , snd ( y ) � � λ y . inl fst ( y ) Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 11 / 30
Introduction The basic approach Encoding the proof systems with λ -terms → makes connection between change of order of rule applications and sense-denotation-distinction transparent: In SC it is often possible to change the order of rule application However, this does not necessarily lead to a different proof (in ND it does) Sometimes it does, sometimes it doesn’t: we need terms to distinguish the cases Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 12 / 30
Distinguishing sense and denotation of proofs Are these different proofs? SC 1 ⊢ (( q ∧ r ) ∨ p ) ⊃ (( p ∨ q ) ∧ ( p ∨ r )) Rf Rf q ⊢ q r ⊢ r ∨ R ∨ R Rf Rf q ⊢ p ∨ q r ⊢ p ∨ r p ⊢ p p ⊢ p ∧ L ∧ L ∨ R ∨ R q ∧ r ⊢ p ∨ q q ∧ r ⊢ p ∨ r p ⊢ p ∨ q p ⊢ p ∨ r ∧ R ∧ R q ∧ r , q ∧ r ⊢ ( p ∨ q ) ∧ ( p ∨ r ) p , p ⊢ ( p ∨ q ) ∧ ( p ∨ r ) C C q ∧ r ⊢ ( p ∨ q ) ∧ ( p ∨ r ) p ⊢ ( p ∨ q ) ∧ ( p ∨ r ) ∨ L ( q ∧ r ) ∨ p ⊢ ( p ∨ q ) ∧ ( p ∨ r ) ⊃ R ⊢ (( q ∧ r ) ∨ p ) ⊃ (( p ∨ q ) ∧ ( p ∨ r )) SC 2 ⊢ (( q ∧ r ) ∨ p ) ⊃ (( p ∨ q ) ∧ ( p ∨ r )) Rf Rf q ⊢ q r ⊢ r ∧ L ∧ L Rf Rf q ∧ r ⊢ q q ∧ r ⊢ r p ⊢ p p ⊢ p ∨ R ∨ R ∨ R ∨ R q ∧ r ⊢ p ∨ q q ∧ r ⊢ p ∨ r p ⊢ p ∨ q p ⊢ p ∨ r ∧ R ∧ R q ∧ r , q ∧ r ⊢ ( p ∨ q ) ∧ ( p ∨ r ) p , p ⊢ ( p ∨ q ) ∧ ( p ∨ r ) C C q ∧ r ⊢ ( p ∨ q ) ∧ ( p ∨ r ) p ⊢ ( p ∨ q ) ∧ ( p ∨ r ) ∨ L ( q ∧ r ) ∨ p ⊢ ( p ∨ q ) ∧ ( p ∨ r ) ⊃ R ⊢ (( q ∧ r ) ∨ p ) ⊃ (( p ∨ q ) ∧ ( p ∨ r )) Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 13 / 30
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