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 conditionals Discussion Thanks References Conditionals →
Conditionals Revision theory More conditionals Discussion Thanks References T-sentences T ( � A � ) ↔ A
Conditionals Revision theory More conditionals Discussion Thanks References Reasoning A ∴ B
Conditionals Revision theory More conditionals Discussion Thanks References Reasoning A ∴ B A , A → B ⊢ B
Conditionals Revision theory More conditionals Discussion Thanks References Rejecting ⊣ λ
Conditionals Revision theory More conditionals Discussion Thanks References Weakening - I �| = T ( � λ � ) ↔ λ
Conditionals Revision theory More conditionals Discussion Thanks References Weakening - I �| = T ( � λ � ) ↔ λ T ( � A � ) | = A
Conditionals Revision theory More conditionals Discussion Thanks References Weakening - II A , A → B �| = B
Conditionals Revision theory More conditionals Discussion Thanks References Weakening �| = T ( � λ � ) ↔ λ A , A → B �| = B �
Conditionals Revision theory More conditionals Discussion Thanks References Feferman objection
Conditionals Revision theory More conditionals Discussion Thanks References Feferman objection A . . . B A ⊃ B ????
Conditionals Revision theory More conditionals Discussion Thanks References Conditionals ⊃
Conditionals Revision theory More conditionals Discussion Thanks References Conditionals ⊃ ⊃ � = →
Conditionals Revision theory More conditionals Discussion Thanks References Field A → B | = C → A → . C → B
Conditionals Revision theory More conditionals Discussion Thanks References Beall A , A → B | = B
Conditionals Revision theory More conditionals Discussion Thanks References Several people T ( � A � ) ↔ A
Conditionals Revision theory More conditionals Discussion Thanks References Field again, sort of DA = Df A & ∼ ( A → ∼ A ) ⊣ A = ∼ D ∗ A
Conditionals Revision theory More conditionals Discussion Thanks References Roles Reasoning Truth-theoretic features
Conditionals Revision theory More conditionals Discussion Thanks References Revision theory
Conditionals Revision theory More conditionals Discussion Thanks References Circular definitions Gx = Df A ( x , G )
Conditionals Revision theory More conditionals Discussion Thanks References Example Gx = Df ( x = a & ∼ Gx ) ∨ ( x = b & Gx )
Conditionals Revision theory More conditionals Discussion Thanks References Hypotheses h ⊆ D
Conditionals Revision theory More conditionals Discussion Thanks References Revising Gx = Df A ( x , G ) �→ δ
Conditionals Revision theory More conditionals Discussion Thanks References Revising Gx = Df A ( x , G ) �→ δ h , δ ( h ) , δ ( δ ( h )) , δ 3 ( h ) , . . . , δ ω ( h ) , . . .
Conditionals Revision theory More conditionals Discussion Thanks References Example Gx = Df ( x = a & ∼ Gx ) ∨ ( x = b & Gx ) 0 1 2 . . . ∅ ∅ { a } ∅ { a } { a } ∅ { a } { b } { b } { a , b } { b } { a , b } { a , b } { b } { a , b }
Conditionals Revision theory More conditionals Discussion Thanks References Revision theory of truth T ( � A 1 � ) = Df A 1 T ( � A 2 � ) = Df A 2 . . . T ( � A n � ) = Df A n . . .
Conditionals Revision theory More conditionals Discussion Thanks References Classical logic and T-sentences a = � ∼ Ta �
Conditionals Revision theory More conditionals Discussion Thanks References Classical logic and T-sentences a = � ∼ Ta � ⊢ K ∼ ( A ≡ ∼ A )
Conditionals Revision theory More conditionals Discussion Thanks References Classical logic and T-sentences a = � ∼ Ta � ⊢ K ∼ ( A ≡ ∼ A ) � ⊢ RT Ta ≡ ∼ Ta
Conditionals Revision theory More conditionals Discussion Thanks References It gets worse ⊢ RT ∼ ( Ta ≡ ∼ Ta )
Conditionals Revision theory More conditionals Discussion Thanks References Definitional equivalence ≡ = Df � =
Conditionals Revision theory More conditionals Discussion Thanks References Conditionals for revision theory A → B , A ← B
Conditionals Revision theory More conditionals Discussion Thanks References Conditionals for revision theory A → B , A ← B A ↔ B := ( A → B )&( A ← B )
Conditionals Revision theory More conditionals Discussion Thanks References New hypotheses h ⊆ F × V
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
Conditionals Revision theory More conditionals Discussion Thanks References Rules A k +1 A k +1 . . . A → B k +1 B k B k → E A → B k +1 → I A k A k . . . B ← A k +1 B k +1 B k +1 ← E B ← A k +1 ← I
Conditionals Revision theory More conditionals Discussion Thanks References Features | = RT + T ( � A � ) ↔ A
Conditionals Revision theory More conditionals Discussion Thanks References Features A = Df B = A ↔ B
Conditionals Revision theory More conditionals Discussion Thanks References Features Gx = Df A ( x , C ( Gx ↔ B ))
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
Conditionals Revision theory More conditionals Discussion Thanks References 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
Conditionals Revision theory More conditionals Discussion Thanks References Logic - Interaction ( A → B ) ≡ ( ∼ A ← ∼ B )
Conditionals Revision theory More conditionals Discussion Thanks References Flaws
Conditionals Revision theory More conditionals Discussion Thanks References Flaws . . .
Conditionals Revision theory More conditionals Discussion Thanks References Flaws . . . ??
Conditionals Revision theory More conditionals Discussion Thanks References Flaws . . . ?? �| = A → A
Conditionals Revision theory More conditionals Discussion Thanks References Flaws . . . ?? �| = A → A A → B , B → C �| = A → C
Conditionals Revision theory More conditionals Discussion Thanks References Flaws . . . ?? �| = A → A A → B , B → C �| = A → C A ↔ B �| = B ↔ A
Conditionals Revision theory More conditionals Discussion Thanks References Seriously? → , ← � = ⇒
Conditionals Revision theory More conditionals Discussion Thanks References Roles revisited Reasoning Truth → F → F → BX → BX ⊃ → , ←
Conditionals Revision theory More conditionals Discussion Thanks References Too complicated M , v , h | = A → B ⇔ M , v , h | = A or � B , v � ∈ h
Conditionals Revision theory More conditionals Discussion Thanks References Completeness = D RT + A ⇔ ⊢ D | RT + A
Conditionals Revision theory More conditionals Discussion Thanks References Naturally fits into the revision theory Gx = Df A ( x , G )
Conditionals Revision theory More conditionals Discussion Thanks References Naturally fits into the revision theory = Df � = ≡
Conditionals Revision theory More conditionals Discussion Thanks References 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 of the revision theory Our earlier distinction can be used to defend these conditionals against objections
Conditionals Revision theory More conditionals Discussion Thanks References 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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