Speech Acts & the Quest for a Natural Account of Classical Proof - PowerPoint PPT Presentation

We get intuitionistic logic �⊢ p ∨ ¬ p ¬¬ p �⊢ p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 13 of 51

We get intuitionistic logic �⊢ p ∨ ¬ p ¬¬ p �⊢ p �⊢ ( p → q ) ∨ ( q → r ) Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 13 of 51

We get intuitionistic logic �⊢ p ∨ ¬ p ¬¬ p �⊢ p �⊢ ( p → q ) ∨ ( q → r ) �⊢ ((( p → q )) → p ) → p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 13 of 51

‘Textbook’ natural deduction plugs the gap, but it has no taste . Π ¬¬ A DNE A Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 14 of 51

‘Textbook’ natural deduction plugs the gap, but it has no taste . [ ¬ A ] i Π Π ¬¬ A ⊥ DNE A ⊥ E c A Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 14 of 51

‘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 Speech Acts & the Quest for, a Natural Account of Classical Proof 14 of 51

We get classicallogic , but at some cost [ ¬ 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 Speech Acts & the Quest for, a Natural Account of Classical Proof 15 of 51

other frameworks

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 Speech Acts & the Quest for, a Natural Account of Classical Proof 17 of 51

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 Speech Acts & the Quest for, a Natural Account of Classical Proof 17 of 51

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 Speech Acts & the Quest for, a Natural Account of Classical Proof 17 of 51

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 does deriving X � Y have to do with proof ? Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 17 of 51

Me, in 2005: Nothing much . . . https://consequently.org/writing/multipleconclusions/ . . . but 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 Speech Acts & the Quest for, a Natural Account of Classical Proof 18 of 51

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 Speech Acts & the Quest for, a Natural Account of Classical Proof 19 of 51

This is not just conservatism W hat is a proof of p ? Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 20 of 51

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 . Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 20 of 51

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 Speech Acts & the Quest for, a Natural Account of Classical Proof 20 of 51

An Example [ q ] 1 q → s [ p ] 3 → E [ r ] 2 p → ( q ∨ r ) s → E ∨ I ∨ I q ∨ r r ∨ s r ∨ s ∨ E 1,2 [ ¬ ( r ∨ s )] 4 r ∨ s ¬ E ⊥ ¬ I 3 ¬ p → I 4 ¬ ( r ∨ s ) → ¬ p We’vegranted p → ( q ∨ r ) and q → s . Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51

An Example [ q ] 1 q → s [ p ] 3 → E [ r ] 2 p → ( q ∨ r ) s → E ∨ I ∨ I q ∨ r r ∨ s r ∨ s ∨ E 1,2 [ ¬ ( r ∨ s )] 4 r ∨ s ¬ E ⊥ ¬ I 3 ¬ p → I 4 ¬ ( r ∨ s ) → ¬ p We’vegranted p → ( q ∨ r ) and q → s . Iassert ¬ ( r ∨ s ) → ¬ p ,andyouchallengeme. Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51

An Example [ q ] 1 q → s [ p ] 3 → E [ r ] 2 p → ( q ∨ r ) s → E ∨ I ∨ I q ∨ r r ∨ s r ∨ s ∨ E 1,2 [ ¬ ( r ∨ s )] 4 r ∨ s ¬ E ⊥ ¬ I 3 ¬ p → I 4 ¬ ( r ∨ s ) → ¬ p We’vegranted p → ( q ∨ r ) and q → s . Iassert ¬ ( r ∨ s ) → ¬ p ,andyouchallengeme. Isay, suppose ¬ ( r ∨ s ) . We’ve got ¬ p . You challenge me again, Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51

An Example [ q ] 1 q → s [ p ] 3 → E [ r ] 2 p → ( q ∨ r ) s → E ∨ I ∨ I q ∨ r r ∨ s r ∨ s ∨ E 1,2 [ ¬ ( r ∨ s )] 4 r ∨ s ¬ E ⊥ ¬ I 3 ¬ p → I 4 ¬ ( r ∨ s ) → ¬ p We’vegranted p → ( q ∨ r ) and q → s . Iassert ¬ ( r ∨ s ) → ¬ p ,andyouchallengeme. Isay, suppose ¬ ( r ∨ s ) . We’ve got ¬ p . You challenge me again, so I say, suppose p , and I’ll show that thisisinconsistent. Youaskmetodothat, Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51

An Example [ q ] 1 q → s [ p ] 3 → E [ r ] 2 p → ( q ∨ r ) s → E ∨ I ∨ I q ∨ r r ∨ s r ∨ s ∨ E 1,2 [ ¬ ( r ∨ s )] 4 r ∨ s ¬ E ⊥ ¬ I 3 ¬ p → I 4 ¬ ( r ∨ s ) → ¬ p We’vegranted p → ( q ∨ r ) and q → s . Iassert ¬ ( r ∨ s ) → ¬ p ,andyouchallengeme. Isay, suppose ¬ ( r ∨ s ) . We’ve got ¬ p . You challenge me again, so I say, suppose p , and I’ll show that thisisinconsistent. Youaskmetodothat,soI’llsayweget r ∨ s ,whichclasheswiththe ¬ ( r ∨ s ) weassumed. Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51

An Example [ q ] 1 q → s [ p ] 3 → E [ r ] 2 p → ( q ∨ r ) s → E ∨ I ∨ I q ∨ r r ∨ s r ∨ s ∨ E 1,2 [ ¬ ( r ∨ s )] 4 r ∨ s ¬ E ⊥ ¬ I 3 ¬ p → I 4 ¬ ( r ∨ s ) → ¬ p We’vegranted p → ( q ∨ r ) and q → s . Iassert ¬ ( r ∨ s ) → ¬ p ,andyouchallengeme. Isay, suppose ¬ ( r ∨ s ) . We’ve got ¬ p . You challenge me again, so I say, suppose p , and I’ll show that thisisinconsistent. Youaskmetodothat,soI’llsayweget r ∨ s ,whichclasheswiththe ¬ ( r ∨ s ) weassumed. Youaskmehowdoyoudothat? Isay,we’ll,we’vegot q ∨ r ,fromour p → ( q ∨ r ) and p . So, let’ssplitintotwocases. Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51

An Example [ q ] 1 q → s [ p ] 3 → E [ r ] 2 p → ( q ∨ r ) s → E ∨ I ∨ I q ∨ r r ∨ s r ∨ s ∨ E 1,2 [ ¬ ( r ∨ s )] 4 r ∨ s ¬ E ⊥ ¬ I 3 ¬ p → I 4 ¬ ( r ∨ s ) → ¬ p We’vegranted p → ( q ∨ r ) and q → s . Iassert ¬ ( r ∨ s ) → ¬ p ,andyouchallengeme. Isay, suppose ¬ ( r ∨ s ) . We’ve got ¬ p . You challenge me again, so I say, suppose p , and I’ll show that thisisinconsistent. Youaskmetodothat,soI’llsayweget r ∨ s ,whichclasheswiththe ¬ ( r ∨ s ) weassumed. Youaskmehowdoyoudothat? Isay,we’ll,we’vegot q ∨ r ,fromour p → ( q ∨ r ) and p . So, let’ssplitintotwocases. Inthe q case, we’vegot r ∨ s , Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51

An Example [ q ] 1 q → s [ p ] 3 → E [ r ] 2 p → ( q ∨ r ) s → E ∨ I ∨ I q ∨ r r ∨ s r ∨ s ∨ E 1,2 [ ¬ ( r ∨ s )] 4 r ∨ s ¬ E ⊥ ¬ I 3 ¬ p → I 4 ¬ ( r ∨ s ) → ¬ p We’vegranted p → ( q ∨ r ) and q → s . Iassert ¬ ( r ∨ s ) → ¬ p ,andyouchallengeme. Isay, suppose ¬ ( r ∨ s ) . We’ve got ¬ p . You challenge me again, so I say, suppose p , and I’ll show that thisisinconsistent. Youaskmetodothat,soI’llsayweget r ∨ s ,whichclasheswiththe ¬ ( r ∨ s ) weassumed. Youaskmehowdoyoudothat? Isay,we’ll,we’vegot q ∨ r ,fromour p → ( q ∨ r ) and p . So, let’ssplitintotwocases. Inthe q case, we’vegot r ∨ s ,andwehave itinthe r case, too. Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51

An Example [ q ] 1 q → s [ p ] 3 → E [ r ] 2 p → ( q ∨ r ) s → E ∨ I ∨ I q ∨ r r ∨ s r ∨ s ∨ E 1,2 [ ¬ ( r ∨ s )] 4 r ∨ s ¬ E ⊥ ¬ I 3 ¬ p → I 4 ¬ ( r ∨ s ) → ¬ p We’vegranted p → ( q ∨ r ) and q → s . Iassert ¬ ( r ∨ s ) → ¬ p ,andyouchallengeme. Isay, suppose ¬ ( r ∨ s ) . We’ve got ¬ p . You challenge me again, so I say, suppose p , and I’ll show that thisisinconsistent. Youaskmetodothat,soI’llsayweget r ∨ s ,whichclasheswiththe ¬ ( r ∨ s ) weassumed. Youaskmehowdoyoudothat? Isay,we’ll,we’vegot q ∨ r ,fromour p → ( q ∨ r ) and p . So, let’ssplitintotwocases. Inthe q case, we’vegot r ∨ s ,andwehave itinthe r case, too. Soineithercase,we’vegot r ∨ s . Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51

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 Speech Acts & the Quest for, a Natural Account of Classical Proof 22 of 51

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 Speech Acts & the Quest for, a Natural Account of Classical Proof 22 of 51

Signed Natural Deduction [− p ∨ ¬ p ] 1 − ∨ E − p + ¬ I + ¬ p + ∨ I [− p ∨ ¬ p ] 2 + p ∨ ¬ p RAA 1,2 + p ∨ ¬ p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 23 of 51

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 Speech Acts & the Quest for, a Natural Account of Classical Proof 23 of 51

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 Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 24 of 51

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 Speech Acts & the Quest for, a Natural Account of Classical Proof 24 of 51

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 Speech Acts & the Quest for, a Natural Account of Classical Proof 25 of 51

An Example [− p ] 2 [+ p ] 1 ⊥ I ⊥ ⊥ E + q → I 1 [+ ( p → q ) → p ] 3 + p → q → E [− p ] 2 + p ⊥ I ⊥ Reductio 2 + p → I 3 + (( p → q ) → p ) → p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 26 of 51

An Example [− p ] 2 [+ p ] 1 ⊥ I ⊥ ⊥ E + q → I 1 [+ ( p → q ) → p ] 3 + p → q → E [− p ] 2 + p ⊥ I ⊥ Reductio 2 + p → I 3 + (( p → q ) → p ) → p Classical • Separated Rules • Normalising • Analytic • Single Conclusion Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 26 of 51

An Example [− p ] 2 [+ p ] 1 ⊥ I ⊥ ⊥ E + q → I 1 [+ ( p → q ) → p ] 3 + p → q → E [− p ] 2 + p ⊥ I ⊥ Reductio 2 + p → I 3 + (( p → q ) → p ) → p Classical • Separated Rules • Normalising • Analytic • Single Conclusion ... but what are ‘ + ’ and ‘ − ’ really doing ? Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 26 of 51

What are these ‘ + ’ and ‘ − ’ doing anyway? the official line: + A is an assertion of A − A is a denial or rejection of A . Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 27 of 51

What are these ‘ + ’ and ‘ − ’ doing anyway? the official line: + A is an assertion of A − A is a denial or rejection of A . − A � = + ¬ A , since denial is a speech act that cannot be embedded in other contexts, while negation modifies content, and can embed. Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 27 of 51

What are these ‘ + ’ and ‘ − ’ doing anyway? the official line: + A is an assertion of A − A is a denial or rejection of A . − A � = + ¬ A , since denial is a speech act that cannot be embedded in other contexts, while negation modifies content, and can embed. Proofs contain speech acts , not contents . Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 27 of 51

A Problem: Supposition � = Assertion N atural deduction proofs already contain different speech acts. At the leaves we can suppose A to later discharge it. Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 28 of 51

A Problem: Supposition � = Assertion N atural deduction proofs already contain different speech acts. At the leaves we can suppose A to later discharge it. Supposing − A is . . . what , exactly? Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 28 of 51

The Lessons ◮ A nswerability to our practice is a constraint worth meeting. Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 29 of 51

The Lessons ◮ A nswerability to our practice is a constraint worth meeting. ◮ Bilateralism (paying attention to assertion and denial ) is important to the defender of classical logic. Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 29 of 51

The Lessons ◮ A nswerability to our practice is a constraint worth meeting. ◮ 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 Speech Acts & the Quest for, a Natural Account of Classical Proof 29 of 51

natural deduction with alternatives

Parigot’s λµ -Calculus M ichel Parigot “ λµ -Calculus: an algorithmic interpretation of classical natural deduction” International Conference on Logic for Programming Artificial Intelligence and Reasoning , 1992 Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 31 of 51

I’ll translate this for an audience of non-specialists, showing how it meets the answerability criterion much better than previous efforts, staying close to our practice of giving a proof , without decorating formulas with signs, while retaining the good properties of intuitionistic natural deduction. Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 32 of 51

The Rules [ A ] i Π Π ′ Π A A → B A → E B → I i B A → B Π Π ′ Π Π A ∧ B ∧ E A ∧ B ∧ E A B ∧ I A B A ∧ B [ A ] j [ B ] k Π Π Π ′ Π ′′ Π A B ∨ I ∨ I A ∨ B C C ∨ E A ∨ B A ∨ B C [ A ] i Π ′ Π Π Π Π ¬ A A ¬ E ⊥ ⊥ E A Alt, ↓ A ⊥ ¬ I i ⊥ A B ¬ A Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 33 of 51

The Rules [ A ] i Π Π ′ Π A A → B A → E B → I i B A → B Π Π ′ Π Π A ∧ B ∧ E A ∧ B ∧ E A B ∧ I A B A ∧ B [ A ] j [ B ] k Π Π Π ′ Π ′′ Π A B ∨ I ∨ I A ∨ B C C ∨ E A ∨ B A ∨ B C [ A ] i Π ′ Π Π Π Π ¬ A A ¬ E ⊥ ⊥ E A Alt, ↓ A ⊥ ¬ I i ⊥ A B ¬ A Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 33 of 51

The Rules [ A ] i Π Π ′ Π A A → B A → E, ↑ B B → I i , ↑ A → B B A → B Π Π ′ Π Π A ∧ B ∧ E, ↑ A A ∧ B ∧ E, ↑ B A B ∧ I, ↑ A ∧ B A B A ∧ B [ A ] j [ B ] k Π Π Π ′ Π ′′ Π A B ∨ I, ↑ A ∨ B ∨ I, ↑ A ∨ B A ∨ B C C ∨ E, ↑ C A ∨ B A ∨ B C [ A ] i Π ′ Π Π Π Π ¬ A A ¬ E ⊥ ⊥ E, ↑ A A Alt, ↓ A , ↑ B ⊥ ¬ I i , ↑ ¬ A ⊥ A B ¬ A Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 33 of 51

Add just one rule: the Alternative Rule Π A Alt, ↓ A B X � A ; Y X � B ; A, Y Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 34 of 51

Add just one rule: the Alternative Rule Π A Alt, ↓ A , ↑ B B X � A ; B, Y X � B ; A, Y Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 34 of 51

Add just one rule: the Alternative Rule Π A Alt, ↓ A B [ X : Y ] � A [ X : A, Y ] � B Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 34 of 51

Add just one rule: the Alternative Rule Π A Alt, ↓ A , ↑ B B [ X : B, Y ] � A [ X : A, Y ] � B Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 34 of 51

Example Proof (Peirce’s Law) [ p ] 1 Alt, ↓ p 2 q → I 1 [( p → q ) → p ] 3 p → q → E, ↑ p 2 p → I 3 (( p → q ) → p ) → p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 35 of 51

Example Proof (Peirce’s Law) [ p ] 1 Alt, ↓ p 2 q → I 1 [( p → q ) → p ] 3 p → q → E, ↑ p 2 p → I 3 (( p → q ) → p ) → p [ p : ] � p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 35 of 51

Example Proof (Peirce’s Law) [ p ] 1 Alt, ↓ p 2 q → I 1 [( p → q ) → p ] 3 p → q → E, ↑ p 2 p → I 3 (( p → q ) → p ) → p [ p : p ] � q Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 35 of 51

Example Proof (Peirce’s Law) [ p ] 1 Alt, ↓ p 2 q → I 1 [( p → q ) → p ] 3 p → q → E, ↑ p 2 p → I 3 (( p → q ) → p ) → p [ : p ] � p → q Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 35 of 51

Example Proof (Peirce’s Law) [ p ] 1 Alt, ↓ p 2 q → I 1 [( p → q ) → p ] 3 p → q → E, ↑ p 2 p → I 3 (( p → q ) → p ) → p [( p → q ) → p : p ] � p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 35 of 51

Example Proof (Peirce’s Law) [ p ] 1 Alt, ↓ p 2 q → I 1 [( p → q ) → p ] 3 p → q → E, ↑ p 2 p → I 3 (( p → q ) → p ) → p [( p → q ) → p : ] � p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 35 of 51

Example Proof (Peirce’s Law) [ p ] 1 Alt, ↓ p 2 q → I 1 [( p → q ) → p ] 3 p → q → E, ↑ p 2 p → I 3 (( p → q ) → p ) → p [ : ] � (( p → q ) → p ) → p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 35 of 51

Another Proof [ p ] 1 Alt, ↓ p 2 ⊥ ¬ I 2 ¬ p ∨ I p ∨ ¬ p Alt, ↓ p ∨ ¬ p 3 , ↑ p 2 p ∨ I , ↑ p ∨ ¬ p 3 p ∨ ¬ p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 36 of 51

Another Proof [ p ] 1 Alt, ↓ p 2 ⊥ ¬ I 2 [ p : ] � p ¬ p ∨ I p ∨ ¬ p Alt, ↓ p ∨ ¬ p 3 , ↑ p 2 p ∨ I , ↑ p ∨ ¬ p 3 p ∨ ¬ p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 36 of 51

Another Proof [ p ] 1 Alt, ↓ p 2 ⊥ ¬ I 2 ¬ p [ p : p ] � ⊥ ∨ I p ∨ ¬ p Alt, ↓ p ∨ ¬ p 3 , ↑ p 2 p ∨ I , ↑ p ∨ ¬ p 3 p ∨ ¬ p Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 36 of 51