The Meaning of Proofs in Different Proof Systems Sara Ayhan (Ruhr - - PowerPoint PPT Presentation

SLIDE 1

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

SLIDE 2

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

SLIDE 3

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

SLIDE 4

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

SLIDE 5

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

SLIDE 6

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

SLIDE 7

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

SLIDE 8

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

SLIDE 9

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

SLIDE 10

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

SLIDE 11

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

SLIDE 12

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

SLIDE 13

Introduction

Different senses: normal vs. non-normal terms

NDp ⊃ p

[x : p]

⊃I

λx.x : p ⊃ p

NDnon-normalp ⊃ p

[x : p]

⊃I

λx.x : p ⊃ p [y : q]

⊃I

λy.y : q ⊃ q

∧I

λx.x, λy.y : (p ⊃ p) ∧ (q ⊃ q)

∧E

fst(λx.x, λy.y) : p ⊃ p

fst(λx.x, λy.y) λx.x

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 10 / 30

SLIDE 14

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 ⊢ inlfst(y) : p ∨ p

⊃R

⊢ λy.inlfst(y) : (p ∧ p) ⊃ (p ∨ p)

SCcut⊢ (p ∧ p) ⊃ (p ∨ p)

z : p ⊢ z : p

∧L

y : p ∧ p ⊢ fst(y) : p z : p ⊢ z : p

∧L

y : p ∧ p ⊢ snd(y) : p

∧R

y : p ∧ p, y : p ∧ p ⊢ fst(y), snd(y) : p ∧ p

C

y : p ∧ p ⊢ fst(y), snd(y) : p ∧ p z : p ⊢ z : p

∧L

y : p ∧ p ⊢ fst(y) : p

cut

y : p ∧ p ⊢ fst fst(y), snd(y) : p

∨R

y : p ∧ p ⊢ inlfst fst(y), snd(y) : p ∨ p

⊃R

⊢ λy.inlfst fst(y), snd(y) : (p ∧ p) ⊃ (p ∨ p)

λy.inlfst fst(y), snd(y) λy.inlfst(y)

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 11 / 30

SLIDE 15

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

SLIDE 16

Distinguishing sense and denotation of proofs

Are these different proofs?

SC1⊢ ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

Rf

q ⊢ q

∨R

q ⊢ p ∨ q

∧L

q ∧ r ⊢ p ∨ q

Rf

r ⊢ r

∨R

r ⊢ p ∨ r

∧L

q ∧ r ⊢ p ∨ r

∧R

q ∧ r, q ∧ r ⊢ (p ∨ q) ∧ (p ∨ r)

C

q ∧ r ⊢ (p ∨ q) ∧ (p ∨ r)

Rf

p ⊢ p

∨R

p ⊢ p ∨ q

Rf

p ⊢ p

∨R

p ⊢ p ∨ r

∧R

p, p ⊢ (p ∨ q) ∧ (p ∨ r)

C

p ⊢ (p ∨ q) ∧ (p ∨ r)

∨L

(q ∧ r) ∨ p ⊢ (p ∨ q) ∧ (p ∨ r)

⊃R

⊢ ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

SC2⊢ ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

Rf

q ⊢ q

∧L

q ∧ r ⊢ q

∨R

q ∧ r ⊢ p ∨ q

Rf

r ⊢ r

∧L

q ∧ r ⊢ r

∨R

q ∧ r ⊢ p ∨ r

∧R

q ∧ r, q ∧ r ⊢ (p ∨ q) ∧ (p ∨ r)

C

q ∧ r ⊢ (p ∨ q) ∧ (p ∨ r)

Rf

p ⊢ p

∨R

p ⊢ p ∨ q

Rf

p ⊢ p

∨R

p ⊢ p ∨ r

∧R

p, p ⊢ (p ∨ q) ∧ (p ∨ r)

C

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

SLIDE 17

Distinguishing sense and denotation of proofs

Are these different proofs?

SC1⊢ ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

Rf

q ⊢ q

∨R

q ⊢ p ∨ q

∧L

q ∧ r ⊢ p ∨ q

Rf

r ⊢ r

∨R

r ⊢ p ∨ r

∧L

q ∧ r ⊢ p ∨ r

∧R

q ∧ r, q ∧ r ⊢ (p ∨ q) ∧ (p ∨ r)

C

q ∧ r ⊢ (p ∨ q) ∧ (p ∨ r)

Rf

p ⊢ p

∨R

p ⊢ p ∨ q

Rf

p ⊢ p

∨R

p ⊢ p ∨ r

∧R

p, p ⊢ (p ∨ q) ∧ (p ∨ r)

C

p ⊢ (p ∨ q) ∧ (p ∨ r)

∨L

(q ∧ r) ∨ p ⊢ (p ∨ q) ∧ (p ∨ r)

⊃R

⊢ ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

SC3⊢ ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

Rf

q ⊢ q

∨R

q ⊢ p ∨ q

∧L

q ∧ r ⊢ p ∨ q

Rf

p ⊢ p

∨R

p ⊢ p ∨ q

∨L

(q ∧ r) ∨ p ⊢ p ∨ q

Rf

r ⊢ r

∨R

r ⊢ p ∨ r

∧L

q ∧ r ⊢ p ∨ r

Rf

p ⊢ p

∨R

p ⊢ p ∨ r

∨L

(q ∧ r) ∨ p ⊢ p ∨ r

∧R

(q ∧ r) ∨ p, (q ∧ r) ∨ p ⊢ (p ∨ q) ∧ (p ∨ r)

C

(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 14 / 30

SLIDE 18

Distinguishing sense and denotation of proofs

Change of order does not lead to different proofs

SC1⊢ ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

Rf

y : q ⊢ y : q

∨R

y : q ⊢ inry : p ∨ q

∧L

v : q ∧ r ⊢ inrfst(v) : p ∨ q

Rf

z : r ⊢ z : r

∨R

z : r ⊢ inrz : p ∨ r

∧L

v : q ∧ r ⊢ inrsnd(v) : p ∨ r

∧R

v : q ∧ r, v : q ∧ r ⊢ inrfst(v), inrsnd(v) : (p ∨ q) ∧ (p ∨ r)

C

v : q ∧ r ⊢ inrfst(v), inrsnd(v) : (p ∨ q) ∧ (p ∨ r)

Rf

x : p ⊢ x : p

∨R

x : p ⊢ inlx : p ∨ q

Rf

x : p ⊢ x : p

∨R

x : p ⊢ inlx : p ∨ r

∧R

x : p, x : p ⊢ inlx, inlx : (p ∨ q) ∧ (p ∨ r)

C

x : p ⊢ inlx, inlx : (p ∨ q) ∧ (p ∨ r)

∨L

u : (q ∧ r) ∨ p ⊢ case u {v. inrfst(v), inrsnd(v) | x. inlx, inlx} : (p ∨ q) ∧ (p ∨ r)

⊃R

⊢ λu.case u {v. inrfst(v), inrsnd(v) | x. inlx, inlx} : ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

SC2⊢ ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

Rf

y : q ⊢ y : q

∧L

v : q ∧ r ⊢ fst(v) : q

∨R

v : q ∧ r ⊢ inrfst(v) : p ∨ q

Rf

z : r ⊢ z : r

∧L

v : q ∧ r ⊢ snd(v) : r

∨R

v : q ∧ r ⊢ inrsnd(v) : p ∨ r

∧R

v : q ∧ r, v : q ∧ r ⊢ inrfst(v), inrsnd(v) : (p ∨ q) ∧ (p ∨ r)

C

v : q ∧ r ⊢ inrfst(v), inrsnd(v) : (p ∨ q) ∧ (p ∨ r)

Rf

x : p ⊢ x : p

∨R

x : p ⊢ inlx : p ∨ q

Rf

x : p ⊢ x : p

∨R

x : p ⊢ inlx : p ∨ r

∧R

x : p, x : p ⊢ inlx, inlx : (p ∨ q) ∧ (p ∨ r)

C

x : p ⊢ inlx, inlx : (p ∨ q) ∧ (p ∨ r)

∨L

u : (q ∧ r) ∨ p ⊢ case u {v. inrfst(v), inrsnd(v) | x. inlx, inlx} : (p ∨ q) ∧ (p ∨ r)

⊃R

⊢ λu.case u {v. inrfst(v), inrsnd(v) | x. inlx, inlx} : ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r)) Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 15 / 30

SLIDE 19

Distinguishing sense and denotation of proofs

Change of order does lead to different proofs

SC1⊢ ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

Rf

y : q ⊢ y : q

∨R

y : q ⊢ inry : p ∨ q

∧L

v : q ∧ r ⊢ inrfst(v) : p ∨ q

Rf

z : r ⊢ z : r

∨R

z : r ⊢ inrz : p ∨ r

∧L

v : q ∧ r ⊢ inrsnd(v) : p ∨ r

∧R

v : q ∧ r, v : q ∧ r ⊢ inrfst(v), inrsnd(v) : (p ∨ q) ∧ (p ∨ r)

C

v : q ∧ r ⊢ inrfst(v), inrsnd(v) : (p ∨ q) ∧ (p ∨ r)

Rf

x : p ⊢ x : p

∨R

x : p ⊢ inlx : p ∨ q

Rf

x : p ⊢ x : p

∨R

x : p ⊢ inlx : p ∨ r

∧R

x : p, x : p ⊢ inlx, inlx : (p ∨ q) ∧ (p ∨ r)

C

x : p ⊢ inlx, inlx : (p ∨ q) ∧ (p ∨ r)

∨L

u : (q ∧ r) ∨ p ⊢ case u {v. inrfst(v), inrsnd(v) | x. inlx, inlx} : (p ∨ q) ∧ (p ∨ r)

⊃R

⊢ λu.case u {v. inrfst(v), inrsnd(v) | x. inlx, inlx} : ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

SC3⊢ ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r))

Rf

y : q ⊢ y : q

∨R

y : q ⊢ inry : p ∨ q

∧L

v : q ∧ r ⊢ inrfst(v) : p ∨ q

Rf

x : p ⊢ x : p

∨R

x : p ⊢ inlx : p ∨ q

∨L

u : (q ∧ r) ∨ p ⊢ case u {v.inrfst(v) | x.inlx} : p ∨ q

Rf

z : r ⊢ z : r

∨R

z : r ⊢ inrz : p ∨ r

∧L

v : q ∧ r ⊢ inrsnd(v) : p ∨ r

Rf

x : p ⊢ x : p

∨R

x : p ⊢ inlx : p ∨ r

∨L

u : (q ∧ r) ∨ p ⊢ case u {v.inrsnd(v) | x.inlx} : p ∨ r

∧R

u : (q ∧ r) ∨ p, u : (q ∧ r) ∨ p ⊢ case u {v.inrfst(v) | x.inlx}, case u {v.inrsnd(v) | x.inlx} : (p ∨ q) ∧ (p ∨ r)

C

u : (q ∧ r) ∨ p ⊢ case u {v.inrfst(v) | x.inlx}, case u {v.inrsnd(v) | x.inlx} : (p ∨ q) ∧ (p ∨ r)

⊃R

⊢ λu. case u {v.inrfst(v) | x.inlx}, case u {v.inrsnd(v) | x.inlx} : ((q ∧ r) ∨ p) ⊃ ((p ∨ q) ∧ (p ∨ r)) Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 16 / 30

SLIDE 20

Distinguishing sense and denotation of proofs

Distinguishing sense and denotation in proofs

Encoding the rules with λ-terms makes transparent why some changes in the order of rule applications leads to different terms, while others do not: In SC the ∧L rule as well as the ⊃L rule are substitution operations → they can change their place in the order because only the inner structure of the term is affected (for ⊃L only if the right premise is not an axiom) → no completely new term is created In ND there are no substitution operations as rules, i.e. there’s always a new kind of term produced

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 17 / 30

SLIDE 21

Distinguishing sense and denotation of proofs Sense of derivations

Distinguishing sense and denotation in proofs

As can be seen in the case above a change of order of rule applications can lead to different proofs (SC1 and SC3) In the other case (SC1 and SC2) there is only a difference in sense, but not in denotation

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 18 / 30

SLIDE 22

Distinguishing sense and denotation of proofs Sense of derivations

Distinguishing sense and denotation in proofs

As can be seen in the case above a change of order of rule applications can lead to different proofs (SC1 and SC3) In the other case (SC1 and SC2) there is only a difference in sense, but not in denotation Sense of a derivation Consists of the set of terms that occur within the derivation If a sense-denoting set can be obtained from another by replacing any

• ccurrence of a variable (bound or free) by another variable of the

same type, they express the same sense

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 18 / 30

SLIDE 23

Distinguishing sense and denotation of proofs Sense of derivations

Philosophical motivation

Fregean sense is a procedure to determine its denotation (Dummett 1973) a sequence of instructions; terms represent programs; purpose of a program is to calculate its denotation (Girard 1990)

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 19 / 30

SLIDE 24

Distinguishing sense and denotation of proofs Sense of derivations

Philosophical motivation

Fregean sense is a procedure to determine its denotation (Dummett 1973) a sequence of instructions; terms represent programs; purpose of a program is to calculate its denotation (Girard 1990) → There can be the same underlying proof in different proof systems, like in ND and SC but also within the same proof system: there can be difference in sense but not in denotation → The proof is essentially the same but the way it is given to us, the way of inference, differs, i.e. the sense, differs → This can be read off the set of terms occurring in a derivation: they end up building the same term but the way it is built differs

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 19 / 30

SLIDE 25

Distinguishing sense and denotation of proofs Sense of derivations

Example for different senses

Example: SC1 vs. SC2 (same denotation)

Sense of SC1: {x, y, z, u, v, inlx, inry, inrz, inlx, inlx , inrfst(v), inrsnd(v), inrfst(v), inrsnd(v) , case u {v. inrfst(v), inrsnd(v) | x. inlx, inlx}, λu.case u {v. inrfst(v), inrsnd(v) | x. inlx, inlx}} Sense of SC2: {x, y, z, u, v, inlx, inlx, inlx , fst(v), snd(v), inrfst(v), inrsnd(v), inrfst(v), inrsnd(v) , case u {v. inrfst(v), inrsnd(v) | x. inlx, inlx}, λu.case u {v. inrfst(v), inrsnd(v) | x. inlx, inlx}}

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 20 / 30

SLIDE 26

Analogy between Frege’s cases and the proof cases

Frege’s cases

1 Different signs, same sense, same denotation; examples for:

Singular Terms: “Gottlob’s brother”, “the brother of Gottlob” Sentences: “M gave document A to N”, “Document A was given to N by M” (Frege 1979: 141)

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 21 / 30

SLIDE 27

Analogy between Frege’s cases and the proof cases

Frege’s cases

1 Different signs, same sense, same denotation; examples for:

Singular Terms: “Gottlob’s brother”, “the brother of Gottlob” Sentences: “M gave document A to N”, “Document A was given to N by M” (Frege 1979: 141)

2 Different signs, difference in sense, same denotation; examples for:

Singular Terms: “morning star”, “evening star” Sentences: “The morning star is the planet Venus”, “The evening star is the planet Venus”

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 21 / 30

SLIDE 28

Analogy between Frege’s cases and the proof cases

Frege’s cases

1 Different signs, same sense, same denotation; examples for:

Singular Terms: “Gottlob’s brother”, “the brother of Gottlob” Sentences: “M gave document A to N”, “Document A was given to N by M” (Frege 1979: 141)

2 Different signs, difference in sense, same denotation; examples for:

Singular Terms: “morning star”, “evening star” Sentences: “The morning star is the planet Venus”, “The evening star is the planet Venus”

The following cases should NOT happen according to Frege:

One sign, different senses: ambiguous terms (happen in natural languages but shouldn’t occur in formal languages) One sense, different denotations: the sense should determine the reference

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 21 / 30

SLIDE 29

Analogy between Frege’s cases and the proof cases

Transferring Frege’s cases onto the context of proofs

1

Different signs, same sense (same set of terms within derivation), same denotation (same term for the endsequent): SC⊢ (p ∧ p) ⊃ (p ∨ p) z : p ⊢ z : p

∧L

y : p ∧ p ⊢ fst(y) : p

∨R

y : p ∧ p ⊢ inlfst(y) : p ∨ p

⊃R

⊢ λy.inlfst(y) : (p ∧ p) ⊃ (p ∨ p) SC⊢ (p ∧ q) ⊃ (p ∨ r) z : p ⊢ z : p

∧L

y : p ∧ q ⊢ fst(y) : p

∨R

y : p ∧ q ⊢ inlfst(y) : p ∨ r

⊃R

⊢ λy.inlfst(y) : (p ∧ q) ⊃ (p ∨ r)

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 22 / 30

SLIDE 30

Analogy between Frege’s cases and the proof cases

Transferring Frege’s cases onto the context of proofs

1

Different signs, same sense (same set of terms within derivation), same denotation (same term for the endsequent), this case is also possible between the two proof systems: ND⊢ (p ∨ p) ⊃ (p ∧ p)

[y : p ∨ p] [x : p] [x : p]

∨E

case y {x.x | x.x} : p [y : p ∨ p] [x : p] [x : p]

∨E

case y {x.x | x.x} : p

∧I

case y {x.x | x.x}, case y {x.x | x.x} : p ∧ p

⊃I

λy. case y {x.x | x.x}, case y {x.x | x.x} : (p ∨ p) ⊃ (p ∧ p)

SC⊢ (p ∨ p) ⊃ (p ∧ p)

x : p ⊢ x : p x : p ⊢ x : p

∨L

y : p ∨ p ⊢ case y {x.x | x.x} : p x : p ⊢ x : p x : p ⊢ x : p

∨L

y : p ∨ p ⊢ case y {x.x | x.x} : p

∧R

y : p ∨ p, y : p ∨ p ⊢ case y {x.x | x.x}, case y {x.x | x.x} : p ∧ p

C

y : p ∨ p ⊢ case y {x.x | x.x}, case y {x.x | x.x} : p ∧ p

⊃R

⊢ λy. case y {x.x | x.x}, case y {x.x | x.x} : (p ∨ p) ⊃ (p ∧ p)

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 23 / 30

SLIDE 31

Analogy between Frege’s cases and the proof cases

Transferring Frege’s cases onto the context of proofs

2

Different signs, difference in sense, same denotation: see example above between SC1 and SC2 but also consistent with distinguishing between SC-derivations containing cut vs. cut-free ones:

SC⊢ (p ∧ p) ⊃ (p ∨ p), Sense: {z, y, fst(y), inlfst(y), λy.inlfst(y)}

z : p ⊢ z : p

∧L

y : p ∧ p ⊢ fst(y) : p

∨R

y : p ∧ p ⊢ inlfst(y) : p ∨ p

⊃R

⊢ λy.inlfst(y) : (p ∧ p) ⊃ (p ∨ p)

SC-cut⊢ (p ∧ p) ⊃ (p ∨ p), Sense: {z, y, fst(y), inlz, inlfst(y), λy.inlfst(y)}

z : p ⊢ z : p

∧L

y : p ∧ p ⊢ fst(y) : p z : p ⊢ z : p

∨R

z : p ⊢ inlz : p ∨ p

cut

y : p ∧ p ⊢ inlfst(y) : p ∨ p

⊃R

⊢ λy.inlfst(y) : (p ∧ p) ⊃ (p ∨ p)

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 24 / 30

SLIDE 32

Conclusion

Conclusion

Frege was interested in identity: we can also use these results to say something about proof identity Identity over different calculi or within the same calculus consists in having end-terms reducible to the same normal form:

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 25 / 30

SLIDE 33

Conclusion

Conclusion

Frege was interested in identity: we can also use these results to say something about proof identity Identity over different calculi or within the same calculus consists in having end-terms reducible to the same normal form:

With syntactically different derivations in ND this can only happen when we have one derivation in normal and the other in non-normal form (reducible to the former) In SC this can happen when

1

we have one derivation in cut-free form and one corresponding derivation containing cut, or

2

when there’s a change in the order of rule applications, either of the: ∧L rule, or of the ⊃L rule, though iff its right premise is not an axiom

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 25 / 30

SLIDE 34

Conclusion

Conclusion

Frege was interested in identity: we can also use these results to say something about proof identity Identity over different calculi or within the same calculus consists in having end-terms reducible to the same normal form:

With syntactically different derivations in ND this can only happen when we have one derivation in normal and the other in non-normal form (reducible to the former) In SC this can happen when

1

we have one derivation in cut-free form and one corresponding derivation containing cut, or

2

when there’s a change in the order of rule applications, either of the: ∧L rule, or of the ⊃L rule, though iff its right premise is not an axiom

If they are supposed to be identical in meaning then this means that the way the inference is given is essentially the same, so the set of terms building up the end-term must be the same

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 25 / 30

SLIDE 35

Conclusion

Thanks for your attention!

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 26 / 30

SLIDE 36

Conclusion

Literature

Dummett, Michael (1973): Frege. Philosophy of Language, New York: Harper & Row. Frege, Gottlob (1948): “Sense and Reference”, The Philosophical Review 57(3), 209-230. Frege, Gottlob (1979): Posthumous Writings, Oxford: Basil Blackwell. Girard, Jean-Yves (1990): Proofs and Types, Cambridge: Cambridge University Press. Restall, Greg (2017): “Proof Terms for Classical Derivations”, article in progress. Tranchini, Luca (2016): “Proof-theoretic semantics, paradoxes and the distinction between sense and denotation”, Journal of Logic and Computation 26(2), 495-512.

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 27 / 30

SLIDE 37

Appendix

Term-annotated natural deduction

Γ · · · s : A ∆ · · · t : B

∧I

s, t : A ∧ B Γ · · · t : A ∧ B

∧E1

fst(t) : A Γ · · · t : A ∧ B

∧E2

snd(t) : B Γ · · · s : A

∨I1

inls : A ∨ B Γ · · · t : B

∨I2

inrt : A ∨ B Γ · · · r : A ∨ B ∆, [x : A] · · · s : C Θ, [y : B] · · · t : C

∨E

case r {x.s | y.t} : C Γ, [x : A] · · · t : B

⊃I

λx.t : A ⊃ B ∆ · · · s : A ⊃ B Γ · · · t : A

⊃E

App(s, t) : B Γ · · · t : ⊥

⊥E

abort(t) : A Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 28 / 30

SLIDE 38

Appendix

Term-annotated G0ip

Logical axiom:

Rf

x : A ⊢ x : A Logical rules: Γ ⊢ s : A ∆ ⊢ t : B

∧R

Γ, ∆ ⊢ s, t : A ∧ B Γ, x : A ⊢ s : C

∧L1

Γ, y : A ∧ B ⊢ s[fst(y)/x] : C Γ, x : B ⊢ s : C

∧L2

Γ, y : A ∧ B ⊢ s[snd(y)/x] : C Γ ⊢ s : A

∨R1

Γ ⊢ inls : A ∨ B Γ ⊢ t : B

∨R2

Γ ⊢ inrt : A ∨ B Γ, x : A ⊢ s : C ∆, y : B ⊢ t : C

∨L

Γ, z : A ∨ B ⊢ case z {x.s | y.t} : C Γ, x : A ⊢ t : B

⊃R

Γ ⊢ λx.t : A ⊃ B Γ ⊢ t : A ∆, y : B ⊢ s : C

⊃L

Γ, ∆, x : A ⊃ B ⊢ s[App(x, t)/y] : C

⊥L

x : ⊥ ⊢ abort(x) : C Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 29 / 30

SLIDE 39

Appendix

Term-annotated G0ip

Structural rules: Weakening: Γ ⊢ t : C

W

Γ, x : A ⊢ t : C Contraction: Γ, x : A, y : A ⊢ t : C Γ, x : A ⊢ t[x/y] : C Cut: Γ ⊢ t : D ∆, x : D ⊢ s : C Γ, ∆ ⊢ s[t/x] : C

Sara Ayhan (Ruhr University Bochum) The Meaning of Proofs in Different Proof Systems 26 April, 2019 30 / 30