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Speech Acts & the Quest for a Natural Account of Classical Proof Greg Restall berkeley logic colloquium 18 september 2020 https://consequently.org/presentation/2020/ speech-acts-for-classical-natural-deduction-berkeley My Aim T o
Speech Acts & the Quest for a Natural Account of Classical Proof
Greg Restall
berkeley logic colloquium · 18 september 2020
https://consequently.org/presentation/2020/ speech-acts-for-classical-natural-deduction-berkeley
My Aim To introduce and defend Michel Parigot’s λµ-calculus as an appropriate framework for inferentialists to study classical logical concepts.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 2 of 51
My Plan
Inferentialism & Natural Deduction Natural Deduction is Opinionated Other Frameworks Natural Deduction with Alternatives Meeting Objections Going Beyond
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 3 of 51
Natural Deduction is Beautiful!
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 5 of 51
Natural Deduction is Beautiful!
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 5 of 51
Natural Deduction is Beautiful!
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 5 of 51
Natural Deduction is Beautiful!
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 5 of 51
Natural Deduction is Beautiful!
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 5 of 51
Natural Deduction is Beautiful!
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 5 of 51
Natural Deduction is Beautiful!
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 5 of 51
Natural Deduction is Beautiful!
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 5 of 51
Natural Deduction is Beautiful!
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 5 of 51
The Rules
A
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 6 of 51
The Rules
A [A]i Π B
→Ii
A → B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 6 of 51
The Rules
A [A]i Π B
→Ii
A → B Π A → B Π′ A →E B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 6 of 51
The Rules
A [A]i Π B
→Ii
A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 6 of 51
The Rules
A [A]i Π B
→Ii
A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 6 of 51
The Rules
A [A]i Π B
→Ii
A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A
∨I
A ∨ B Π B
∨I
A ∨ B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 6 of 51
The Rules
A [A]i Π B
→Ii
A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A
∨I
A ∨ B Π B
∨I
A ∨ B Π A ∨ B [A]j Π′ C [B]k Π′′ C ∨E C
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 6 of 51
The Rules
A [A]i Π B
→Ii
A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A
∨I
A ∨ B Π B
∨I
A ∨ B Π A ∨ B [A]j Π′ C [B]k Π′′ C ∨E C [A]i Π ⊥
¬Ii
¬A
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 6 of 51
The Rules
A [A]i Π B
→Ii
A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A
∨I
A ∨ B Π B
∨I
A ∨ B Π A ∨ B [A]j Π′ C [B]k Π′′ C ∨E C [A]i Π ⊥
¬Ii
¬A Π ¬A Π′ A ¬E ⊥
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 6 of 51
The Rules
A [A]i Π B
→Ii
A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A
∨I
A ∨ B Π B
∨I
A ∨ B Π A ∨ B [A]j Π′ C [B]k Π′′ C ∨E C [A]i Π ⊥
¬Ii
¬A Π ¬A Π′ A ¬E ⊥ Π ⊥ ⊥E A
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 6 of 51
Inferentialists like Natural Deduction ◮ Inference is something we can do, and can learn.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 7 of 51
Inferentialists like Natural Deduction ◮ Inference is something we can do, and can learn. ◮ Te rules are separated.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 7 of 51
Inferentialists like Natural Deduction ◮ Inference is something we can do, and can learn. ◮ Te rules are separated. ◮ I/E rules play a similar role to ‘truth conditions.’
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 7 of 51
Inferentialists like Natural Deduction ◮ Inference is something we can do, and can learn. ◮ Te rules are separated. ◮ I/E rules play a similar role to ‘truth conditions.’ ◮ Proofs normalise. (We can straighten out detours.)
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 7 of 51
Inferentialists like Natural Deduction ◮ Inference is something we can do, and can learn. ◮ Te rules are separated. ◮ I/E rules play a similar role to ‘truth conditions.’ ◮ Proofs normalise. (We can straighten out detours.) ◮ Normal proofs are analytic.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 7 of 51
Normalisation
[A]i Π1 B
→Ii
A → B Π2 A
→E
B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 8 of 51
Normalisation
[A]i Π1 B
→Ii
A → B Π2 A
→E
B
A Π1 B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 8 of 51
. . . and it’s type theory and the λ-calculus under the hood.
[x : A]i Π1 t(x) : B
→Ii
λx.t(x) : A → B Π2 s : A
→E
(λx.t(x))s : B
s : A Π1 t(s) : B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 9 of 51
What’s not to love?
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 10 of 51
Soundness and Completeness I try to be a philosophical logician, with equal emphasis on ‘philosophical’ and ‘logician,’ and I try to take both proof theory and model theory equally seriously for foundational purposes.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 11 of 51
Soundness and Completeness I try to be a philosophical logician, with equal emphasis on ‘philosophical’ and ‘logician,’ and I try to take both proof theory and model theory equally seriously for foundational purposes. Soundness and completeness help me explore the relationship between inferentialism and representationalism.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 11 of 51
We get intuitionistic logic ⊢ 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
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. Π ¬¬A
DNE
A [¬A]i Π ⊥
⊥Ec
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. Π ¬¬A
DNE
A [¬A]i Π ⊥
⊥Ec
A [A]i Π C [¬A]j Π C
Casesi,j
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 → q) → p]3 [¬p]2 [p]1
¬E
⊥ ⊥E q
→I1
p → q
→E
p
¬E
⊥
¬I2
¬¬p
DNE
p
→I3
((p → q) → p) → p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 15 of 51
Gentzen’s Sequent Calculus
p p p q, p
→R
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 p p q, p
→R
p → q, p
→L
(p → q) → p p, p
W
(p → q) → p p
→R
((p → q) → p) → p p p
¬R
p, ¬p
∨R
p ∨ ¬p p p
¬L
p, ¬p
∧L
p ∧ ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 17 of 51
Gentzen’s Sequent Calculus
p p p q, p
→R
p → q, p
→L
(p → q) → p p, p
W
(p → q) → p p
→R
((p → q) → p) → p p p
¬R
p, ¬p
∨R
p ∨ ¬p p p
¬L
p, ¬p
∧L
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 p p q, p
→R
p → q, p
→L
(p → q) → p p, p
W
(p → q) → p p
→R
((p → q) → p) → p p p
¬R
p, ¬p
∨R
p ∨ ¬p p p
¬L
p, ¬p
∧L
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 What 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 What 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 What 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
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
We’vegrantedp → (q ∨ r)andq → s.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51
An Example
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
We’vegrantedp → (q ∨ r)andq → s. Iassert¬(r ∨ s) → ¬p,andyouchallengeme.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51
An Example
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
We’vegrantedp → (q ∨ r)andq → 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
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
We’vegrantedp → (q ∨ r)andq → 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
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51
An Example
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
We’vegrantedp → (q ∨ r)andq → 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
weassumed.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51
An Example
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
We’vegrantedp → (q ∨ r)andq → 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
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51
An Example
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
We’vegrantedp → (q ∨ r)andq → 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
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51
An Example
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
We’vegrantedp → (q ∨ r)andq → 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
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 21 of 51
An Example
[¬(r ∨ s)]4 p → (q ∨ r) [p]3
→E
q ∨ r q → s [q]1
→E
s
∨I
r ∨ s [r]2
∨I
r ∨ s
∨E1,2
r ∨ s
¬E
⊥
¬I3
¬p
→I4
¬(r ∨ s) → ¬p
We’vegrantedp → (q ∨ r)andq → 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
Soineithercase,we’vegotr ∨ 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
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
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 [− p ∨ ¬p]2
RAA1,2
+ 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 [− p ∨ ¬p]2
RAA1,2
+ 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
Π + A
+∨I
+ A ∨ B Π + B
+∨I
+ A ∨ B Π + A ∨ B [+ A]j Π′ φ [+ B]k Π′′ φ
∨Ej,k
φ Π − A ∨ B −∨E − A Π − A ∨ B −∧E − B Π − A Π′ − B −∨E − A ∨ B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 24 of 51
Double up your Rules
Π + A
+∨I
+ A ∨ B Π + B
+∨I
+ A ∨ B Π + A ∨ B [+ A]j Π′ φ [+ B]k Π′′ φ
∨Ej,k
φ Π − A ∨ B −∨E − A Π − A ∨ B −∧E − B Π − A Π′ − B −∨E − A ∨ B Π − A
+¬I
+ ¬A Π + ¬A +¬E − A Π + A
−¬I
− ¬A Π − ¬A −¬E + A
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 24 of 51
Add some ‘Structural’ Rules
Π α Π′ α∗
⊥I
⊥ [α]i Π ⊥ Reductioi α∗ [α]j Π′ β [α]k Π′′ β∗
SRj,k
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 → q) → p]3 [− p]2 [+ p]1
⊥I
⊥
⊥E
+ q
→I1
+ p → q
→E
+ p
⊥I
⊥
Reductio2
+ p
→I3
+ ((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 → q) → p]3 [− p]2 [+ p]1
⊥I
⊥
⊥E
+ q
→I1
+ p → q
→E
+ p
⊥I
⊥
Reductio2
+ p
→I3
+ ((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 → q) → p]3 [− p]2 [+ p]1
⊥I
⊥
⊥E
+ q
→I1
+ p → q
→E
+ p
⊥I
⊥
Reductio2
+ p
→I3
+ ((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 Natural 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 Natural 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
◮ Answerability 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
◮ Answerability 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
◮ Answerability 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
Parigot’s λµ-Calculus
Michel Parigot “λµ-Calculus: an algorithmic interpretation of classical natural deduction” International Conference on Logic for Programming Artificial Intelligence and Reasoning, 1992
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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
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The Rules
A [A]i Π B
→Ii
A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A
∨I
A ∨ B Π B
∨I
A ∨ B Π A ∨ B [A]j Π′ C [B]k Π′′ C ∨E C [A]i Π ⊥
¬Ii
¬A Π ¬A Π′ A ¬E ⊥ Π ⊥ ⊥E A Π A Alt, ↓A B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 33 of 51
The Rules
A [A]i Π B
→Ii
A → B Π A → B Π′ A →E B Π A Π′ B ∧I A ∧ B Π A ∧ B ∧E A Π A ∧ B ∧E B Π A
∨I
A ∨ B Π B
∨I
A ∨ B Π A ∨ B [A]j Π′ C [B]k Π′′ C ∨E C [A]i Π ⊥
¬Ii
¬A Π ¬A Π′ A ¬E ⊥ Π ⊥ ⊥E A Π A Alt, ↓A B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 33 of 51
The Rules
A [A]i Π B
→Ii,↑A → B
A → B Π A → B Π′ A →E,↑B B Π A Π′ B ∧I,↑A ∧ B A ∧ B Π A ∧ B ∧E,↑A A Π A ∧ B ∧E,↑B B Π A
∨I,↑A ∨ B
A ∨ B Π B
∨I,↑A ∨ B
A ∨ B Π A ∨ B [A]j Π′ C [B]k Π′′ C ∨E,↑C C [A]i Π ⊥
¬Ii,↑¬A
¬A Π ¬A Π′ A ¬E ⊥ Π ⊥ ⊥E,↑A A Π A Alt, ↓A,↑B B
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 → q) → p]3 [p]1
Alt, ↓p2
q
→I1
p → q
→E, ↑p2
p
→I3
((p → q) → p) → p
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Example Proof (Peirce’s Law) [(p → q) → p]3 [p]1
Alt, ↓p2
q
→I1
p → q
→E, ↑p2
p
→I3
((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 → q) → p]3 [p]1
Alt, ↓p2
q
→I1
p → q
→E, ↑p2
p
→I3
((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 → q) → p]3 [p]1
Alt, ↓p2
q
→I1
p → q
→E, ↑p2
p
→I3
((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 → q) → p]3 [p]1
Alt, ↓p2
q
→I1
p → q
→E, ↑p2
p
→I3
((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 → q) → p]3 [p]1
Alt, ↓p2
q
→I1
p → q
→E, ↑p2
p
→I3
((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 → q) → p]3 [p]1
Alt, ↓p2
q
→I1
p → q
→E, ↑p2
p
→I3
((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,↓p2
⊥
¬I2
¬p
∨I
p ∨ ¬p
Alt,↓p ∨ ¬p3, ↑p2
p
∨I, ↑p ∨ ¬p3
p ∨ ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 36 of 51
Another Proof [p]1
Alt,↓p2
⊥
¬I2
¬p
∨I
p ∨ ¬p
Alt,↓p ∨ ¬p3, ↑p2
p
∨I, ↑p ∨ ¬p3
p ∨ ¬p [p : ] p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 36 of 51
Another Proof [p]1
Alt,↓p2
⊥
¬I2
¬p
∨I
p ∨ ¬p
Alt,↓p ∨ ¬p3, ↑p2
p
∨I, ↑p ∨ ¬p3
p ∨ ¬p [p : p] ⊥
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 36 of 51
Another Proof [p]1
Alt,↓p2
⊥
¬I2
¬p
∨I
p ∨ ¬p
Alt,↓p ∨ ¬p3, ↑p2
p
∨I, ↑p ∨ ¬p3
p ∨ ¬p [ : p] ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 36 of 51
Another Proof [p]1
Alt,↓p2
⊥
¬I2
¬p
∨I
p ∨ ¬p
Alt,↓p ∨ ¬p3, ↑p2
p
∨I, ↑p ∨ ¬p3
p ∨ ¬p [ : p] p ∨ ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 36 of 51
Another Proof [p]1
Alt,↓p2
⊥
¬I2
¬p
∨I
p ∨ ¬p
Alt,↓p ∨ ¬p3, ↑p2
p
∨I, ↑p ∨ ¬p3
p ∨ ¬p [ : p ∨ ¬p] p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 36 of 51
Another Proof [p]1
Alt,↓p2
⊥
¬I2
¬p
∨I
p ∨ ¬p
Alt,↓p ∨ ¬p3, ↑p2
p
∨I, ↑p ∨ ¬p3
p ∨ ¬p [ : ] p ∨ ¬p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 36 of 51
Alternative Formulations of the Rules: Negation
[A]1
Alt,↓A
⊥
¬I1
¬A
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 37 of 51
Alternative Formulations of the Rules: Negation
[A]1
Alt,↓A
⊥
¬I1
¬A [A]1
Alt,↓A2
⊥
¬I1
¬A Alt,↓¬A, ↑A2 A
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 37 of 51
Alternative Formulations of the Rules: Negation
¬I′,↓A
¬A
¬I′,↓¬A
A
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 37 of 51
Alternative Formulations of the Rules: Negation
¬I′,↓A
¬A
¬I′,↓¬A
A [A]i Π ⊥
¬Ii
¬A becomes
¬I′,↓¬Ai
A Π ⊥
⊥E,↑¬Ai
¬A
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 37 of 51
Alternative Formulations of the Rules: Disjunction
Π A ∨ B [A]1 [B]2
Alt,↓B
A
∨E1,2
A Π A ∨ B [A]1
Alt, ↓A
B [B]2
∨E1,2
B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 38 of 51
Alternative Formulations of the Rules: Disjunction
Π A ∨ B ∨E′, ↓B A Π A ∨ B ∨E′, ↓A B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 38 of 51
Alternative Formulations of the Rules: Disjunction
Π A ∨ B ∨E′, ↓B A Π A ∨ B ∨E′, ↓A B Π A ∨ B [A]i Π′ C [B]j Π′′ C ∨E C becomes Π A ∨ B ∨E′, ↓Bi A Π′ C Alt, ↓Cj↑Bi B Π′′
↑Cj
C
Π A ∨ B ∨E′, ↓Ai B Π′′ C Alt, ↓Cj↑Ai A Π′
↑Cj
C
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 38 of 51
Completeness and Soundness
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 39 of 51
Completeness and Soundness
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 39 of 51
Completeness and Soundness
in any Boolean valuation v where v(X) = 1 and v(Y) = 0 then v(A) = 1.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 39 of 51
Answerability Tis is so much closer to our everyday proof practice than either the sequent calculus or a signed system.
(In the paper I show top-down or bottom-up readings
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 41 of 51
Is this really a single conclusion system? Tere are multiple conclusion sequents X A; Y just lurking under the surface, after all.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 42 of 51
Is this really a single conclusion system? Tere are multiple conclusion sequents X A; Y just lurking under the surface, after all. Of course, but in the sequent [X : Y] A, the X and Y (the assumptions and the alternatives) are the background context and the A is what we have proved against that background.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 42 of 51
Bilateralism does some work for us When I put a current conclusion aside as an alternative, I temporarily (for the sake of the argument) deny it, to consider a different option in its place. Tis is very mildly bilateral, but not so much that it litters every formula in a proof with a sign.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 43 of 51
Benefits Classical • Separated Rules • Normalising Analytic • Single Conclusion • Answerable
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 44 of 51
Grounds We can put the λµ terms back into our proofs, and explore what this means for grounds.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 46 of 51
Grounds We can put the λµ terms back into our proofs, and explore what this means for grounds. A proof of A from [X : Y] constructs a ground for A from grounds for each member of X and grounds against each member of Y.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 46 of 51
Grounds We can put the λµ terms back into our proofs, and explore what this means for grounds. A proof of A from [X : Y] constructs a ground for A from grounds for each member of X and grounds against each member of Y.
Te flourishing tradition of “classical computation” using λµ terms, constructions and closures is worth exploring by those philosophical logicians interested in the epistemic power of proof.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 46 of 51
Substructural Logics
Π B
→I
A → B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 47 of 51
Substructural Logics
Π B
→I
A → B [A]i Π B
∧I
A ∧ B ∧E B
→Ii
A → B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 47 of 51
Substructural Logics
Π B
→I
A → B [A]i Π B
∧I
A ∧ B ∧E B
→Ii
A → B A B ⊗I A ⊗ B A ⊗ B [A]i [B]j Π C ⊗Ei,j C
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 47 of 51
Substructural Logics A Alt,↓A B looks fishy
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 48 of 51
Substructural Logics A Alt,↓A B looks fishy
So, let’s split it up into more basic parts. A Alt,↓A ⊥ ⊥ ⊥E,↑B B ⊥ ⊥E, vacuous B
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 48 of 51
Vacuous absorption is justlike vacuous discharge
p
Alt,↓p1
⊥
⊥E, vacuous
¬q [q]2
¬E
⊥ ⊥E,↑p1 p
→I2
q → p
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 49 of 51
Vacuous absorption is justlike vacuous discharge
p
Alt,↓p1
⊥
⊥E, vacuous
¬q [q]2
¬E
⊥ ⊥E,↑p1 p
→I2
q → p
You get well-behaved proof systems for ‘classical’ relevant, affine and linear logics by restricting discharge and absorption in the natural ways, and it ‘just works.’
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 49 of 51
Vacuous absorption is justlike vacuous discharge
p
Alt,↓p1
⊥
⊥E, vacuous
¬q [q]2
¬E
⊥ ⊥E,↑p1 p
→I2
q → p
You get well-behaved proof systems for ‘classical’ relevant, affine and linear logics by restricting discharge and absorption in the natural ways, and it ‘just works.’ Tis seems like good evidence that this technique is worth exploring, and isn’t just a ‘hack’ cooked up to solve just one single problem.
Greg Restall Speech Acts & the Quest for, a Natural Account of Classical Proof 49 of 51
Thank you!
slides: https://consequently.org/presentation/2020/
speech-acts-for-classical-natural-deduction-berkeley
feedback: @consequently on Twitter,
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