Natural Deduction with Alternatives Greg Restall applied proof - PowerPoint PPT Presentation

Structural Rules X Π 1 X � A Y, A � B Y A Π 2 X, Y � B B Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules X Π 1 X � A Y, A � B Y A Π 2 X, Y � B B Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules X Π 1 X � A Y, A � B Y A Π 2 X, Y � B B Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules X [ A ] i [ A ] i Π X, A, A � B B → I i X, A � B A → B A → E B Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules X [ A ] i [ A ] i Π X, A, A � B B → I i X, A � B A → B A → E B Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules X Π X � B B X, A � B → I A → B A → E B Greg Restall Natural Deduction, with Alternatives 11 of 72

Structural Rules X Π X � B B X, A � B → I A → B A → E B Greg Restall Natural Deduction, with Alternatives 11 of 72

Discharge Policies duplicates no duplicates Standard Affine vacuous Relevant Linear no vacuous Greg Restall Natural Deduction, with Alternatives 12 of 72

Natural deduction is opinionated �⊢ p ∨ ¬ p ¬¬ p �⊢ p Greg Restall Natural Deduction, with Alternatives 13 of 72

Natural deduction is opinionated �⊢ p ∨ ¬ p ¬¬ p �⊢ p �⊢ ( p → q ) ∨ ( q → r ) �⊢ ((( p → q )) → p ) → p Greg Restall Natural Deduction, with Alternatives 13 of 72

‘Textbook’ natural deduction plugs the gap, but it has no taste . [ ¬ A ] i [ A ] i [ ¬ A ] j Π Π Π Π ¬¬ A ⊥ C C DNE A Cases i,j ⊥ E c A C Greg Restall Natural Deduction, with Alternatives 14 of 72

We get classicallogic , but the rules are no longer separated [ ¬ p ] 2 [ p ] 1 ¬ E ⊥ ⊥ E q → I 1 [( p → q ) → p ] 3 p → q → E [ ¬ p ] 2 p ¬ E ⊥ ¬ I 2 ¬¬ p DNE p → I 3 (( p → q ) → p ) → p Greg Restall Natural Deduction, with Alternatives 15 of 72

classical sequent calculus

Gentzen’s Sequent Calculus p � q, p → R p � p � p → q, p → L ( p → q ) → p � p, p W ( p → q ) → p � p → R � (( p → q ) → p ) → p Greg Restall Natural Deduction, with Alternatives 17 of 72

Gentzen’s Sequent Calculus p � q, p → R p � p � p → q, p p � p p � p → L ¬ R ¬ L ( p → q ) → p � p, p � p, ¬ p p, ¬ p � W ∨ R ∧ L ( p → q ) → p � p � p ∨ ¬ p p ∧ ¬ p � → R � (( p → q ) → p ) → p Greg Restall Natural Deduction, with Alternatives 17 of 72

Gentzen’s Sequent Calculus p � q, p → R p � p � p → q, p p � p p � p → L ¬ R ¬ L ( p → q ) → p � p, p � p, ¬ p p, ¬ p � W ∨ R ∧ L ( p → q ) → p � p � p ∨ ¬ p p ∧ ¬ p � → R � (( p → q ) → p ) → p Classical • Separated Rules • Normalising • Analytic Greg Restall Natural Deduction, with Alternatives 17 of 72

Gentzen’s Sequent Calculus p � q, p → R p � p � p → q, p p � p p � p → L ¬ R ¬ L ( p → q ) → p � p, p � p, ¬ p p, ¬ p � W ∨ R ∧ L ( p → q ) → p � p � p ∨ ¬ p p ∧ ¬ p � → R � (( p → q ) → p ) → p Classical • Separated Rules • Normalising • Analytic ... but what kind of proof does X � Y score ? Greg Restall Natural Deduction, with Alternatives 17 of 72

Me, in 2005: Not a proof, but . . . https://consequently.org/writing/multipleconclusions/ . . . deriving X � Y does tell you that it’s out of bounds to assert each member of X and deny each member of Y , and that’s something ! Greg Restall Natural Deduction, with Alternatives 18 of 72

Steinberger on the Principle of Answerability Florian Steinberger, “Why Conclusions Should Remain Single” JPL (2011) 40:333–355 https://dx.doi.org/10.1007/s10992-010-9153-3 Greg Restall Natural Deduction, with Alternatives 19 of 72

This is not just conservatism W hat is a proof of p ? Greg Restall Natural Deduction, with Alternatives 20 of 72

This is not just conservatism W hat is a proof of p ? A proof of p meets a justification request for the assertion of p . (Not every way to meet a justification request is a proof , but proofs meet justification requests in a very stringent way.) Greg Restall Natural Deduction, with Alternatives 20 of 72

Slogan A proof of A (in a context) meets a justification request for A on the basis of the claims we take for granted. Greg Restall Natural Deduction, with Alternatives 21 of 72

Slogan A proof of A (in a context) meets a justification request for A on the basis of the claims we take for granted. A sequent calculus derivation doesn’t do that , at least, not without quite a bit of work . Greg Restall Natural Deduction, with Alternatives 21 of 72

Slogan A proof of A (in a context) meets a justification request for A on the basis of the claims we take for granted. A sequent calculus derivation doesn’t do that , at least, not without quite a bit of work . Is there a way to read a classical sequent derivation as constructing this kind of proof? Greg Restall Natural Deduction, with Alternatives 21 of 72

Signed Natural Deduction [− p ∨ ¬ p ] 1 − ∨ E − p + ¬ I + ¬ p + ∨ I [− p ∨ ¬ p ] 2 + p ∨ ¬ p RAA 1,2 + p ∨ ¬ p Decorate your proof with signs . Greg Restall Natural Deduction, with Alternatives 22 of 72

Double up your Rules [+ A ] j [+ B ] k Π Π Π ′ Π ′′ Π + A + B + A ∨ B φ φ + ∨ I + ∨ I ∨ E j,k + A ∨ B + A ∨ B φ Π Π Π Π ′ − A ∨ B − ∨ E − A ∨ B − ∧ E − A − B − ∨ E − A − B − A ∨ B Π Π Π Π − A + ¬ A + ¬ E + A − ¬ A − ¬ E + ¬ I − ¬ I + ¬ A − A − ¬ A + A Greg Restall Natural Deduction, with Alternatives 23 of 72

Add some ‘Structural’ Rules [ α ] j [ α ] k [ α ] i Π ′ Π Π ′ Π ′′ Π α α ∗ β β ∗ ⊥ I ⊥ Reductio i ⊥ SR j,k α ∗ α ∗ α and β are signed formulas. (− A ) ∗ = + A and (+ A ) ∗ = − A . Greg Restall Natural Deduction, with Alternatives 24 of 72

This is very complex Te duality of assertion and denial are important to the defender of classical logic, but doubling up every connective rule is like cracking a small nut with a sledgehammer . Greg Restall Natural Deduction, with Alternatives 25 of 72

So far ... ◮ A nswerability to our practice is a constraint worth meeting. Greg Restall Natural Deduction, with Alternatives 26 of 72

So far ... ◮ A nswerability to our practice is a constraint worth meeting. ◮ Sequents help keep score in a proof. Greg Restall Natural Deduction, with Alternatives 26 of 72

So far ... ◮ A nswerability to our practice is a constraint worth meeting. ◮ Sequents help keep score in a proof. ◮ Bilateralism (paying attention to assertion and denial ) is important to the defender of classical logic. Greg Restall Natural Deduction, with Alternatives 26 of 72

So far ... ◮ A nswerability to our practice is a constraint worth meeting. ◮ Sequents help keep score in a proof. ◮ Bilateralism (paying attention to assertion and denial ) is important to the defender of classical logic. ◮ Sequent calculus and signed natural deduction do not approach the simplicity of standard natural deduction as an account of proof . Greg Restall Natural Deduction, with Alternatives 26 of 72

assertion, denial, negation and contradiction

Are these rules truly separated ? [ A ] i Π Π ′ Π Π ¬ A A ¬ E ⊥ ⊥ E ⊥ ¬ I i A ⊥ ¬ A Greg Restall Natural Deduction, with Alternatives 28 of 72

Are these rules truly separated ? [ A ] i Π Π ′ Π Π ¬ A A ¬ E ⊥ ⊥ E ⊥ ¬ I i A ⊥ ¬ A If ⊥ is a formula we do not have the subformula property for normal proofs. ¬ p p ¬ E ⊥ ⊥ E q Greg Restall Natural Deduction, with Alternatives 28 of 72

In the sequent calculus, it’s structure , not a formula ¬ p p p � p ¬ E ¬ L ¬ p, p � ⊥ ⊥ E K q ¬ p, p � q Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure , not a formula ¬ p p p � p ¬ E ¬ L ¬ p, p � ⊥ ⊥ E K q ¬ p, p � q Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure , not a formula ¬ p p p � p ¬ E ¬ L ¬ p, p � ⊥ ⊥ E K q ¬ p, p � q Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure , not a formula ¬ p p p � p ¬ E ¬ L ¬ p, p � ⊥ ⊥ E K q ¬ p, p � q Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure , not a formula ¬ p p p � p ¬ E ¬ L ¬ p, p � ♯ K ♯ E q ¬ p, p � q F ollowing Tennant ( Natural Logic , 1978), I’ll use “ ♯ ” as a contradiction marker. It’s not a formula. We regain the subformula property for normal proofs. Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure , not a formula ¬ p p p � p ¬ E ¬ L ¬ p, p � ♯ K ♯ E q ¬ p, p � q F ollowing Tennant ( Natural Logic , 1978), I’ll use “ ♯ ” as a contradiction marker. It’s not a formula. We regain the subformula property for normal proofs. [ A ] i Π ′ Π Π Π ¬ A A ¬ E ♯ ♯ ♯ E ¬ I i ♯ A ¬ A Greg Restall Natural Deduction, with Alternatives 29 of 72

In the sequent calculus, it’s structure , not a formula ¬ p p p � p ¬ E ¬ L ¬ p, p � ♯ K ♯ E q ¬ p, p � q F ollowing Tennant ( Natural Logic , 1978), I’ll use “ ♯ ” as a contradiction marker. It’s not a formula. We regain the subformula property for normal proofs. [ A ] i Π ′ Π Π Π Π Π ¬ A A ¬ E ♯ ♯ f f E ♯ ♯ E f I ♯ ¬ I i ♯ A f ¬ A Greg Restall Natural Deduction, with Alternatives 29 of 72

Relevance, Vacuous Discharge, and ♯ E ¬ p p p � p ¬ E ¬ L ¬ p, p � ♯ K ♯ E q ¬ p, p � q Greg Restall Natural Deduction, with Alternatives 30 of 72

Relevance, Vacuous Discharge, and ♯ E ¬ p p p � p ¬ E ¬ L ¬ p, p � ♯ K ♯ E q ¬ p, p � q p � p p K p, q � p → I q → p → I p � q → p Greg Restall Natural Deduction, with Alternatives 30 of 72

Relevance, Vacuous Discharge, and ♯ E ¬ p p p � p ¬ E ¬ L ¬ p, p � ♯ K ♯ E q ¬ p, p � q p � p p K p, q � p → I q → p → I p � q → p Greg Restall Natural Deduction, with Alternatives 30 of 72

Relevance, Vacuous Discharge, and ♯ E ¬ p p p � p ¬ E ¬ L ¬ p, p � ♯ K ♯ E q ¬ p, p � q p � p p K p, q � p → I q → p → I p � q → p Greg Restall Natural Deduction, with Alternatives 30 of 72

Relevance, Vacuous Discharge, and ♯ E ¬ p p p � p ¬ E ¬ L ¬ p, p � ♯ K ♯ E q ¬ p, p � q p � p p K p, q � p → I q → p → I p � q → p W hat connects vacuous discharge and ♯ E ? In the sequent calculus, they are both weakening . But in natural deduction? Greg Restall Natural Deduction, with Alternatives 30 of 72

Is ♯ genuinely structural ? If ♯ is a genuine structural feature of proofs, why does it feature only in the f and ¬ rules? Greg Restall Natural Deduction, with Alternatives 31 of 72

Is ♯ genuinely structural ? If ♯ is a genuine structural feature of proofs, why does it feature only in the f and ¬ rules? For bilateralists , the notion of a contradiction is more fundamental than any particular connective . Asserting and denying the one thing can also lead to a dead end ... Greg Restall Natural Deduction, with Alternatives 31 of 72

Is ♯ genuinely structural ? If ♯ is a genuine structural feature of proofs, why does it feature only in the f and ¬ rules? For bilateralists , the notion of a contradiction is more fundamental than any particular connective . Asserting and denying the one thing can also lead to a dead end ... and so can setting aside the current conclusion, to look for an alternative . Greg Restall Natural Deduction, with Alternatives 31 of 72

alternatives

Adding alternatives Π A ↑ A ↑ ♯ Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives [ A ↑ ] i Π Π A ↑ ♯ A ↓ i ↑ ♯ A Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives [ A ↑ ] i Π Π A ↑ ♯ A ↓ i ↑ ♯ A [ X : Y ] � A ↑ [ X : A, Y ] � ♯ Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives [ A ↑ ] i Π Π A ↑ ♯ A ↓ i ↑ ♯ A [ X : Y ] � A ↑ [ X : A, Y ] � ♯ Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives [ A ↑ ] i Π Π A ↑ ♯ A ↓ i ↑ ♯ A [ X : Y ] � A [ X : A, Y ] � ♯ ↑ ↓ [ X : A, Y ] � ♯ [ X : Y ] � A Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives [ A ↑ ] i Π Π A ↑ ♯ A ↓ i ↑ ♯ A [ X : Y ] � A [ X : A, Y ] � ♯ ↑ ↓ [ X : A, Y ] � ♯ [ X : Y ] � A Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives [ A ↑ ] i Π Π A ↑ ♯ A ↓ i ↑ ♯ A X � A ; Y X � ♯ ; A, Y ↑ ↓ X � ♯ ; A, Y X � A ; Y Greg Restall Natural Deduction, with Alternatives 33 of 72

Adding alternatives [ A ↑ ] i Π Π A ↑ ♯ A ↓ i ↑ ♯ A X � A ; Y X � ♯ ; A, Y ↑ ↓ X � ♯ ; A, Y X � A ; Y W e add the store and retrieve rules and keep the other rules fixed . Te store and retrieve rules are the only rules that manipulate alternatives. Greg Restall Natural Deduction, with Alternatives 33 of 72

A minimal set of rules [ A ↑ ] i Π Π ♯ A ↑ A ↓ i ↑ A ♯ A [ A ] i Π Π Π Π ′ Π ♯ B A → B A → E f f E → I i f I A → B B f ♯ Greg Restall Natural Deduction, with Alternatives 34 of 72

Example Proof: Contraposition [( A → f ) → B : ] � ( B → f ) → A [ A ] 1 [ A ↑ ] 2 ↓ ♯ f I f → I 1 ( A → f ) → B A → f → E [ B → f ] 3 B → E f f E ♯ ↓ 2 A → I 3 ( B → f ) → A Greg Restall Natural Deduction, with Alternatives 35 of 72

Example Proof: Contraposition [( A → f ) → B : ] � ( B → f ) → A [ A ] 1 [ A ↑ ] 2 ↓ ♯ f I f → I 1 ( A → f ) → B A → f → E [ B → f ] 3 B → E f f E ♯ ↓ 2 A → I 3 ( B → f ) → A [ A : ] � A Greg Restall Natural Deduction, with Alternatives 35 of 72