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Conditionals Revision theory More conditionals Discussion Thanks References Truth and conditionals Shawn Standefer University of Pittsburgh CAPE Seminar University of Kyoto September 17, 2012 Conditionals Revision theory More

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Conditionals Revision theory More conditionals Discussion Thanks References

Truth and conditionals

Shawn Standefer University of Pittsburgh CAPE Seminar University of Kyoto September 17, 2012

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Conditionals Revision theory More conditionals Discussion Thanks References

Conditionals

→

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Conditionals Revision theory More conditionals Discussion Thanks References

T-sentences

T(A) ↔ A

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Conditionals Revision theory More conditionals Discussion Thanks References

Reasoning

A ∴ B

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Conditionals Revision theory More conditionals Discussion Thanks References

Reasoning

A ∴ B A, A → B ⊢ B

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Conditionals Revision theory More conditionals Discussion Thanks References

Rejecting

⊣ λ

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Conditionals Revision theory More conditionals Discussion Thanks References

Weakening - I

| = T(λ) ↔ λ

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Conditionals Revision theory More conditionals Discussion Thanks References

Weakening - I

| = T(λ) ↔ λ T(A) | = A

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Conditionals Revision theory More conditionals Discussion Thanks References

Weakening - II

A, A → B | = B

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Conditionals Revision theory More conditionals Discussion Thanks References

Weakening

| = T(λ) ↔ λ A, A → B | = B

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Conditionals Revision theory More conditionals Discussion Thanks References

Feferman objection

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Conditionals Revision theory More conditionals Discussion Thanks References

Feferman objection

A . . . B A ⊃ B ????

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Conditionals Revision theory More conditionals Discussion Thanks References

Conditionals

⊃

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Conditionals Revision theory More conditionals Discussion Thanks References

Conditionals

⊃ ⊃ = →

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Field

A → B | = C → A → .C → B

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Beall

A, A → B | = B

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Several people

T(A) ↔ A

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Conditionals Revision theory More conditionals Discussion Thanks References

Field again, sort of

DA =Df A& ∼ (A → ∼A) ⊣ A = ∼D∗A

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Conditionals Revision theory More conditionals Discussion Thanks References

Roles

Reasoning Truth-theoretic features

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Revision theory

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Circular definitions

Gx =Df A(x, G)

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Conditionals Revision theory More conditionals Discussion Thanks References

Example

Gx =Df (x = a & ∼ Gx) ∨ (x = b & Gx)

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Conditionals Revision theory More conditionals Discussion Thanks References

Hypotheses

h ⊆ D

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Conditionals Revision theory More conditionals Discussion Thanks References

Revising

Gx =Df A(x, G) → δ

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Conditionals Revision theory More conditionals Discussion Thanks References

Revising

Gx =Df A(x, G) → δ h, δ(h), δ(δ(h)), δ3(h), . . . , δω(h), . . .

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Conditionals Revision theory More conditionals Discussion Thanks References

Example

Gx =Df (x = a & ∼ Gx) ∨ (x = b & Gx) 1 2 . . . ∅ ∅ {a} ∅ {a} {a} ∅ {a} {b} {b} {a, b} {b} {a, b} {a, b} {b} {a, b}

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Conditionals Revision theory More conditionals Discussion Thanks References

Revision theory of truth

T(A1) =Df A1 T(A2) =Df A2 . . . T(An) =Df An . . .

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Conditionals Revision theory More conditionals Discussion Thanks References

Classical logic and T-sentences

a = ∼Ta

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Conditionals Revision theory More conditionals Discussion Thanks References

Classical logic and T-sentences

a = ∼Ta ⊢K ∼ (A ≡ ∼A)

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Conditionals Revision theory More conditionals Discussion Thanks References

Classical logic and T-sentences

a = ∼Ta ⊢K ∼ (A ≡ ∼A) ⊢RTTa ≡ ∼Ta

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It gets worse

⊢RT ∼ (Ta ≡ ∼Ta)

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Definitional equivalence =Df = ≡

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Conditionals for revision theory

A → B, A ← B

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Conditionals for revision theory

A → B, A ← B A ↔ B := (A → B)&(A ← B)

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New hypotheses

h ⊆ F × V

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Conditionals Revision theory More conditionals Discussion Thanks References

Semantics

M, v, h | = A → B ⇔ M, v, h | = A or B, v ∈M h M, v, h | = B ← A ⇔ A, v ∈M h or M, v, h | = B

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Conditionals Revision theory More conditionals Discussion Thanks References

Rules

Ak+1 . . . Bk A → Bk+1 →I Ak+1 A → Bk+1 Bk →E Ak . . . Bk+1 B ← Ak+1 ←I Ak B ← Ak+1 Bk+1 ←E

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Features

| =RT + T(A) ↔ A

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Features

A =Df B = A ↔ B

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Features

Gx =Df A(x, C(Gx ↔ B))

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Conditionals Revision theory More conditionals Discussion Thanks References

Logic - Sameness

(A → C) ⊃ (A&B → C) (A → B)&(A → C) ⊃ .A → (B&C) A ∨ B → C ⊃ .A → C (A → C)&(B → C) ⊃ .A ∨ B → C (∼A → B)&(∼A → ∼B) ⊃ .A

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Logic - Difference

| = ((C ← B) ← A) ≡ (C ← A&B) | = (A → (B → C)) ⊃ A&B → C | = (A&B → C) ⊃ (A → (B → C)) A → (A → B) | = A → B (B ← A) ← A | = B ← A

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Logic - Interaction

(A → B) ≡ (∼A ← ∼B)

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Flaws

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Flaws . . .

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Flaws . . . ??

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Flaws . . . ??

| = A → A

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Flaws . . . ??

| = A → A A → B, B → C | = A → C

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Flaws . . . ??

| = A → A A → B, B → C | = A → C A ↔ B | = B ↔ A

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Seriously?

→ , ← = ⇒

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Roles revisited

Reasoning Truth →F →F →BX →BX ⊃ →, ←

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Too complicated

M, v, h | = A → B ⇔ M, v, h | = A or B, v ∈ h

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Completeness

| =D

RT + A ⇔ ⊢D RT +A

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Naturally fits into the revision theory

Gx =Df A(x, G)

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Naturally fits into the revision theory

=Df = ≡

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Conclusions

Distinguish roles conditionals play in our theories These roles can be used to motivate the addition of conditionals to logics Adding conditionals to revision theory fixes one of its problems These conditionals fill out the formal and philosophical picture

  • f the revision theory

Our earlier distinction can be used to defend these conditionals against objections

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Thank you

. . . to you, the audience. . . . to Shunsuke Yatabe for inviting me. . . . to James Shaw and the Pittsburgh philosophy department dissertation seminar for discussion. . . . to Anil Gupta for the support, discussion, and many ideas and insights.

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Field quote

Field says the following of the strong Kleene material conditional. But while [the material conditional] does a passable job as a conditional in the presence of excluded middle, it is totally inadequate as a conditional without excluded middle: with ⊃ as one’s candidate for →, one wouldn’t even get such elementary laws of the conditional as A → A, A → (A ∨ B), or the inference from A → B to (C → A) → (C → B). . . . The lack of a conditional (and also of a biconditional) cripples ordinary reasoning.Field (2008, 73) Field says that his conditional “enables us to come much closer to carrying out ordinary reasoning” than the strong Kleene material conditional does.Field (2008, 276)

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Feferman quote

“[N]othing like sustained ordinary reasoning can be carried on in [strong Kleene logic].”Feferman (1984, 95) The whole quotation is emphasized in the original.

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Beall quote

“The question is whether we have detachable Tr-biconditionals (i.e., ttruth-biconditionals). If we do, then such biconditionals are not our usual material biconditionals, as noted above. I think that we do enjoy detachable Tr-biconditionals. . . .” (Beall, 2009, 26)

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Bibliography

Beall, J. (2009). Spandrels of Truth. Oxford University Press. Feferman, S. (1984). Toward useful type-free theories. I. Journal

  • f Symbolic Logic, 49(1):75–111.

Field, H. (2008). Saving Truth from Paradox. Oxford.