ProofTheory: Logicaland Philosophical Aspects Class 4: - - PowerPoint PPT Presentation
ProofTheory: Logicaland Philosophical Aspects Class 4: Hypersequents forModal Logics Greg Restall and Shawn Standefer nasslli july 2016 rutgers Our Aim To introduce proof theory , with a focus in its applications in philosophy,
ProofTheory: Logicaland Philosophical Aspects
Class 4: Hypersequents forModal Logics Greg Restall and Shawn Standefer
nasslli · july 2016 · rutgers
Our Aim
To introduce proof theory, with a focus in its applications in philosophy, linguistics and computer science.
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 2 of 47
Our Aim for Today
Explore the behaviour of hypersequent systems for modal logics, including two dimensional modal logic with more than one modal operator.
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 3 of 47
Today's Plan
Flat Hypersequents Two Dimensional Modal Logic
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 4 of 47
The Modal Logic s5
The modal logic of equivalence relations. Equivalently, it is the modal logic of universal relations. A model is a pair . iff for every , iff for some ,
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 5 of 47
The Modal Logic s5
The modal logic of equivalence relations. Equivalently, it is the modal logic of universal relations. A model is a pair . iff for every , iff for some ,
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 5 of 47
The Modal Logic s5
The modal logic of equivalence relations. Equivalently, it is the modal logic of universal relations. A model is a pair ⟨W, v⟩. vw(□A) = 1 iff for every u, vu(A) = 1 vw(♢A) = 1 iff for some u, vu(A) = 1
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 5 of 47
How can we simplify hypersequents for s5?
H[X ⊢ Y X′, A ⊢ Y ′]
[□L]
H[X, □A ⊢ Y X′ ⊢ Y ′] H[X ⊢ Y ⊢ A]
[□R]
H[X ⊢ □A, Y] H[X ⊢ Y A ⊢ ]
[♢L]
H[♢A, X ⊢ Y] H[X ⊢ Y X′ ⊢ A, Y ′]
[♢R]
H[X ⊢ ♢A, Y X′ ⊢ Y ′]
Eliminate the arrows!
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 6 of 47
How can we simplify hypersequents for s5?
H[X ⊢ Y X′, A ⊢ Y ′]
[□L]
H[X, □A ⊢ Y X′ ⊢ Y ′] H[X ⊢ Y ⊢ A]
[□R]
H[X ⊢ □A, Y] H[X ⊢ Y A ⊢ ]
[♢L]
H[♢A, X ⊢ Y] H[X ⊢ Y X′ ⊢ A, Y ′]
[♢R]
H[X ⊢ ♢A, Y X′ ⊢ Y ′]
Eliminate the arrows!
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 6 of 47
flat hypersequents
A flat hypersequent is a non-empty multiset of sequents. X1 ⊢ Y1 | X2 ⊢ Y2 | · · · | Xn ⊢ Yn
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 7 of 47
Modal Rules
H[X ⊢ Y X′, A ⊢ Y ′]
[□L]
H[X, □A ⊢ Y X′ ⊢ Y ′] H[X ⊢ Y ⊢ A]
[□R]
H[X ⊢ □A, Y] H[X ⊢ Y A ⊢ ]
[♢L]
H[♢A, X ⊢ Y] H[X ⊢ Y X′ ⊢ A, Y ′]
[♢R]
H[X ⊢ ♢A, Y X′ ⊢ Y ′] There is subtlety here—concerning reflexivity. In the and can be the same.
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 9 of 47
Modal Rules
H[X ⊢ Y | X′, A ⊢ Y ′]
[□L]
H[X, □A ⊢ Y | X′ ⊢ Y ′] H[X ⊢ Y | ⊢ A]
[□R]
H[X ⊢ □A, Y] H[X ⊢ Y | A ⊢ ]
[♢L]
H[♢A, X ⊢ Y] H[X ⊢ Y | X′ ⊢ A, Y ′]
[♢R]
H[X ⊢ ♢A, Y | X′ ⊢ Y ′] There is subtlety here—concerning reflexivity. In the and can be the same.
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 9 of 47
Modal Rules
H[X ⊢ Y | X′, A ⊢ Y ′]
[□L]
H[X, □A ⊢ Y | X′ ⊢ Y ′] H[X ⊢ Y | ⊢ A]
[□R]
H[X ⊢ □A, Y] H[X ⊢ Y | A ⊢ ]
[♢L]
H[♢A, X ⊢ Y] H[X ⊢ Y | X′ ⊢ A, Y ′]
[♢R]
H[X ⊢ ♢A, Y | X′ ⊢ Y ′] There is subtlety here—concerning reflexivity. In H[X ⊢ Y | X′ ⊢ Y ′] the X ⊢ Y and X′ ⊢ Y ′ can be the same.
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 9 of 47
Modal Rules
H[X ⊢ Y | X′, A ⊢ Y ′]
[□L]
H[X, □A ⊢ Y | X′ ⊢ Y ′] H[X′, A ⊢ Y ′]
[□L]
H[X′, □A ⊢ Y ′] H[X ⊢ Y | X′ ⊢ A, Y ′]
[♢R]
H[X ⊢ ♢A, Y | X′ ⊢ Y ′] H[X′ ⊢ A, Y ′]
[♢R]
H[X′ ⊢ ♢A, Y ′] is a hypersequent in which and are components.
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 10 of 47
Modal Rules
H[X ⊢ Y | X′, A ⊢ Y ′]
[□L]
H[X, □A ⊢ Y | X′ ⊢ Y ′] H[X′, A ⊢ Y ′]
[□L]
H[X′, □A ⊢ Y ′] H[X ⊢ Y | X′ ⊢ A, Y ′]
[♢R]
H[X ⊢ ♢A, Y | X′ ⊢ Y ′] H[X′ ⊢ A, Y ′]
[♢R]
H[X′ ⊢ ♢A, Y ′] H[X ⊢ Y | X′ ⊢ Y ′] is a hypersequent in which X ⊢ Y and X′ ⊢ Y ′ are components.
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 10 of 47
Forms of Weakening
H[X ⊢ Y]
[iKL]
H[X, A ⊢ Y] H[X ⊢ Y]
[iKR]
H[X ⊢ A, Y]
[eK]
axK
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 11 of 47
Forms of Weakening
H[X ⊢ Y]
[iKL]
H[X, A ⊢ Y] H[X ⊢ Y]
[iKR]
H[X ⊢ A, Y] H[X ⊢ Y]
[eK]
H[X ⊢ Y | X′ ⊢ Y ′]
axK
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 11 of 47
Forms of Weakening
H[X ⊢ Y]
[iKL]
H[X, A ⊢ Y] H[X ⊢ Y]
[iKR]
H[X ⊢ A, Y] H[X ⊢ Y]
[eK]
H[X ⊢ Y | X′ ⊢ Y ′] H[X, A ⊢ A, Y]
[axK]
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 11 of 47
Forms of Contraction
H[X, A, A ⊢ Y]
[iWL]
H[X, A ⊢ Y] H[X ⊢ A, A, Y]
[iWR]
H[X ⊢ A, Y]
[eWo]
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 12 of 47
Forms of Contraction
H[X, A, A ⊢ Y]
[iWL]
H[X, A ⊢ Y] H[X ⊢ A, A, Y]
[iWR]
H[X ⊢ A, Y] H[X ⊢ Y | X′ ⊢ Y ′]
[eWo]
H[X, X′ ⊢ Y, Y ′]
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 12 of 47
Forms of Cut
X ⊢ A, Y | H X, A ⊢ Y | H
[aCut]
X ⊢ Y | H X ⊢ A, Y | H X′, A ⊢ Y ′ | H′
[mCut]
X, X′ ⊢ Y, Y ′ | H | H′
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 13 of 47
Example Derivation
A ⊢ A
[□L]
□A ⊢ | ⊢ A
[K]
□A, □B ⊢ | ⊢ A B ⊢ B
[□L]
□B ⊢ | ⊢ B
[K]
□A, □B ⊢ | ⊢ B
[∧R]
□A, □B ⊢ | ⊢ A ∧ B
[□R]
□A, □B ⊢ □(A ∧ B)
[∧R]
□A ∧ □B ⊢ □(A ∧ B)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 14 of 47
More Example Derivations
A ⊢ A
[□L]
□A ⊢ | ⊢ A
[□R]
□A ⊢ | ⊢ □A
[□R]
□A ⊢ □□A A ⊢ A
[¬L]
¬A, A ⊢
[□L]
□¬A ⊢ | A ⊢
[¬R]
⊢ ¬□¬A | A ⊢
[sym]
⊢ ¬□¬A | A ⊢
[□R]
A ⊢ □¬□¬A
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 15 of 47
Modifying the Hypersequent Rules for s5
H[X, □A ⊢ Y | X′, A ⊢ Y ′]
[□L]
H[X, □A ⊢ Y | X′ ⊢ Y ′] H[X ⊢ □A, Y | ⊢ A]
[□R]
H[X ⊢ □A, Y] H[X ⊢ Y | A ⊢]
[♢L]
H[X, ♢A ⊢ Y] ‘ H[X ⊢ ♢A, Y | X′ ⊢ A, Y ′]
[♢R]
H[X ⊢ ♢A, Y | X′ ⊢ Y ′]
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 16 of 47
Height Preserving Admissibility
With these modified rules, internal and external weakening, and internal and external contraction, are height-preserving admissible. The von Plato–Negri cut elimination argument works straightforwardly for this system. (See Poggiolesi 2008.)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 17 of 47
Height Preserving Admissibility
With these modified rules, internal and external weakening, and internal and external contraction, are height-preserving admissible. The von Plato–Negri cut elimination argument works straightforwardly for this system. (See Poggiolesi 2008.)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 17 of 47
(m)Cut Elimination: the □ Case
δl X ⊢ Y | ⊢ A | H
[□R]
X ⊢ □A, Y | H δl X′ ⊢ Y ′ | X′′, A ⊢ Y ′′ | H′
[□L]
X′, □A ⊢ Y ′ | X′′ ⊢ Y ′′ | H′
[mCut]
X, X′ ⊢ Y, Y ′ | X′′ ⊢ Y ′′ | H | H′ simplifies to
[mCut] [eW]
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 18 of 47
(m)Cut Elimination: the □ Case
δl X ⊢ Y | ⊢ A | H
[□R]
X ⊢ □A, Y | H δl X′ ⊢ Y ′ | X′′, A ⊢ Y ′′ | H′
[□L]
X′, □A ⊢ Y ′ | X′′ ⊢ Y ′′ | H′
[mCut]
X, X′ ⊢ Y, Y ′ | X′′ ⊢ Y ′′ | H | H′ simplifies to δl X ⊢ Y | ⊢ A | H δr X′ ⊢ Y ′ | X′′, A ⊢ Y ′′ | H′
[mCut]
X ⊢ Y | X′ ⊢ Y ′ | X′′ ⊢ Y ′′ | H | H′
[eW]
X, X′ ⊢ Y, Y ′ | X′′ ⊢ Y ′′ | H | H′
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 18 of 47
Hypersequent Validity
X1 ⊢ Y1 | · · · | Xn ⊢ Yn holds in M iff there are no worlds wi where each element of Xi is true at wi and each element of Yi is false at wi. Equivalent formula:
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 19 of 47
Hypersequent Validity
X1 ⊢ Y1 | · · · | Xn ⊢ Yn holds in M iff there are no worlds wi where each element of Xi is true at wi and each element of Yi is false at wi. Equivalent formula: ¬(♢( ∧ X1 ∧ ¬ ∨ Y1) ∧ · · · ∧ ♢( ∧ Xn ∧ ¬ ∨ Yn))
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 19 of 47
Hypersequent Validity
X1 ⊢ Y1 | · · · | Xn ⊢ Yn holds in M iff there are no worlds wi where each element of Xi is true at wi and each element of Yi is false at wi. Equivalent formula: ¬(♢( ∧ X1 ∧ ¬ ∨ Y1) ∧ · · · ∧ ♢( ∧ Xn ∧ ¬ ∨ Yn)) □( ∧ X1 ⊃ ∨ Y1) ∨ · · · ∨ □( ∧ Xn ⊃ ∨ Yn)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 19 of 47
Features of this Proof System
Soundness and Completeness Separation Decision Procedure Easy Extension
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 20 of 47
The Modal Logic s5@
The modal logic of universal relations with a distinguished world w@. A model is a pair
@ .
iff for every , iff for some , @ iff
@ Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 22 of 47
The Modal Logic s5@
The modal logic of universal relations with a distinguished world w@. A model is a pair ⟨W, v, w@⟩. vw(□A) = 1 iff for every u, vu(A) = 1 vw(♢A) = 1 iff for some u, vu(A) = 1 vw(@A) = 1 iff vw@(A) = 1
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 22 of 47
Hypersequents with @
X1 ⊢ Y1 | · · · | Xn ⊢ Yn X1 ⊢@ Y1 | · · · | Xn ⊢ Yn Multisets of sequents where one (at most) is tagged with the label ‘@’. When you take the union of two hypersequents with @, the @-sequents in the parent hypersequents are merged.
@ @ @
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 23 of 47
Hypersequents with @
X1 ⊢ Y1 | · · · | Xn ⊢ Yn X1 ⊢@ Y1 | · · · | Xn ⊢ Yn Multisets of sequents where one (at most) is tagged with the label ‘@’. When you take the union of two hypersequents with @, the @-sequents in the parent hypersequents are merged. (X1 ⊢@ Y1 | X2 ⊢ Y2) | (X′
1 ⊢@ Y ′ 1 | X′ 2 ⊢ Y ′ 2) =
X1, X′
1 ⊢@ Y1, Y ′ 1 | X2 ⊢ Y2 | X′ 2 ⊢ Y ′ 2
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 23 of 47
Rules for the @ operator
H[X ⊢ Y | X′, A ⊢@ Y ′]
[@L]
H[X, @A ⊢ Y | X′ ⊢@ Y ′] H[X ⊢ Y | X′ ⊢@ A, Y ′]
[@R]
H[X ⊢ @A, Y | X′ ⊢@ Y ′]
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 24 of 47
@-Hypersequent Notation
H[X ⊢ Y | X′ ⊢ Y ′] — a hypersequent with components X ⊢ Y and X′ ⊢ Y ′, which may or may not be identical. H[X ⊢ Y] — a hypersequent with a component X ⊢ Y, which may or may not be tagged with ‘@’. H[X ⊢! Y] — a hypersequent with a component X ⊢ Y, which is not tagged with ‘@.’ H[X ⊢@ Y] — a hypersequent with a component X ⊢@ Y, if X or Y are non-empty.
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 25 of 47
Modal Rules
H[X ⊢ Y | X′, A ⊢ Y ′]
[□L]
H[X, □A ⊢ Y | X′ ⊢ Y ′] H[X ⊢ Y | ⊢! A]
[□R]
H[X ⊢ □A, Y] H[X ⊢ Y | A ⊢! ]
[♢L]
H[♢A, X ⊢ Y] H[X ⊢ Y | X′ ⊢ A, Y ′]
[♢R]
H[X ⊢ ♢A, Y | X′ ⊢ Y ′] Here, can’t tag the A ⊢ component of [♢L] and the ⊢ A component of [□R] with @. (If we tag it, the premise is not general enough.) We have ⊢@ p ⊃ @p, but not ⊢@ □(p ⊃ @p).
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 26 of 47
The proviso on X ⊢@ Y …
… means that the inference step ⊢@ A
[@L]
⊢ A is indeed an instance of [@L] as it is specified. H[X ⊢ Y | X′, A ⊢@ Y ′]
[@L]
H[X, @A ⊢ Y | X′ ⊢@ Y ′]
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 27 of 47
Example Derivations
p ⊢@ p | ⊢
[@R]
p ⊢@ | ⊢ @p
[□R]
p ⊢@ □@p
[⊃R]
⊢@ p ⊃ □@p
@ [@R] @
@
[ R] @
@
[ R] @
@
[@R]
@ @
[ R]
@ @
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 28 of 47
Example Derivations
p ⊢@ p | ⊢
[@R]
p ⊢@ | ⊢ @p
[□R]
p ⊢@ □@p
[⊃R]
⊢@ p ⊃ □@p p ⊢@ p | ⊢
[@R]
p ⊢@ | ⊢ @p
[□R]
p ⊢@ □@p
[⊃R]
⊢@ p ⊃ □@p
[@R]
⊢ @(p ⊃ □@p)
[□R]
⊢ □@(p ⊃ □@p)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 28 of 47
(m)Cut Elimination is unscathed
δl X ⊢ Y | X′ ⊢@ A, Y ′ | H
[@R]
X ⊢ @A, Y | X′ ⊢@ Y ′ | H δr X′′ ⊢ Y ′′ | X′′′, A ⊢@ Y ′′′ | H′
[@L]
X′′, @A ⊢ Y ′′ | X′′′ ⊢@ Y ′′′ | H′
[mCut]
X, X′′ ⊢ Y, Y ′′ | X′, X′′′ ⊢@ Y ′, Y ′′′ | H | H′ simplifies to
@ @ [mCut] @ [eW] @
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 29 of 47
(m)Cut Elimination is unscathed
δl X ⊢ Y | X′ ⊢@ A, Y ′ | H
[@R]
X ⊢ @A, Y | X′ ⊢@ Y ′ | H δr X′′ ⊢ Y ′′ | X′′′, A ⊢@ Y ′′′ | H′
[@L]
X′′, @A ⊢ Y ′′ | X′′′ ⊢@ Y ′′′ | H′
[mCut]
X, X′′ ⊢ Y, Y ′′ | X′, X′′′ ⊢@ Y ′, Y ′′′ | H | H′ simplifies to δl X ⊢ Y | X′ ⊢@ A, Y ′ | H δr X′′ ⊢ Y ′′ | X′′′, A ⊢@ Y ′′′ | H′
[mCut]
X ⊢ Y | X′′ ⊢ Y ′′ | X′, X′′′ ⊢@ Y ′, Y ′′′ | H | H′
[eW]
X, X′′ ⊢ Y, Y ′′ | X′, X′′′ ⊢@ Y ′, Y ′′′ | H | H′
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 29 of 47
Two Dimensional Modal Logic: Relativising the Actual
A 2d model is a pair ⟨W, v⟩. vw,w′(□A) = 1 iff for every u; vu,w′(A) = 1 vw,w′(♢A) = 1 iff for some u; vu,w′(A) = 1 vw,w′(@A) = 1 iff vw′,w′(A) = 1 iff for every ,
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 30 of 47
Two Dimensional Modal Logic: Relativising the Actual
A 2d model is a pair ⟨W, v⟩. vw,w′(□A) = 1 iff for every u; vu,w′(A) = 1 vw,w′(♢A) = 1 iff for some u; vu,w′(A) = 1 vw,w′(@A) = 1 iff vw′,w′(A) = 1 vw,w′(FA) = 1 iff for every u, vw,u(A) = 1
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 30 of 47
Martin Davies & LloydHumberstone
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 w2 @ w3 @ . . . . . . . . . wn @ . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A w2 @ w3 @ . . . . . . . . . wn @ . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A · · · A · · · w2 @ w3 @ . . . . . . . . . wn @ . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A A · · · A · · · w2 @ w3 @ . . . . . . . . . wn @ . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A A · · · A · · · F@B w2 @ w3 @ . . . . . . . . . wn @ . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A A · · · A · · · F@B w2 @B w3 @B . . . . . . . . . wn @B . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A A · · · A · · · F@B @B w2 @B w3 @B . . . . . . . . . wn @B . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A A · · · A · · · F@B @B, B w2 @B w3 @B . . . . . . . . . wn @B . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A A · · · A · · · F@B @B, B w2 @B B w3 @B B . . . . . . . . . wn @B B . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A A · · · A · · · [K]B B w2 @ B w3 @ B . . . . . . . . . wn @ B . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A A · · · A · · · [K]B B w2 @ B w3 @ B . . . . . . . . . wn @ B . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A A · · · A · · · [K]B B w2 @ B w3 @ B . . . . . . . . . wn @ B . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A A · · · A · · · [K]B B w2 @ B w3 @ B . . . . . . . . . wn @ B . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
□ and F@ — the necessary and the fixedlyactual
w1 w2 w3 · · · wn · · · w1 □A A A A · · · A · · · [K]B B w2 @ B w3 @ B . . . . . . . . . wn @ B . . . . . . . . .
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 32 of 47
Different Alternatives □p ⊢ | ⊢ p [K]p ⊢ ∥ ⊢@ p
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 33 of 47
An example derivation…
In fact, we will have the following sort of derivation: p ⊢@ p [K]p ⊢ ∥ ⊢@ p
[[K]R]
[K]p ⊢ | ⊢ [K]p
[□R]
[K]p ⊢ □[K]p
[⊃R]
⊢ [K]p ⊃ □[K]p
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 34 of 47
2d Hypersequents
X1
1 ⊢@ Y1 1
| X1
2 ⊢ Y1 2 |
· · · | X1
m1 ⊢ Y1 m1
∥ X2
1 ⊢@ Y2 1
| X2
2 ⊢ Y2 2 |
· · · | X2
m2 ⊢ Y2 m2
∥ . . . . . . . . . Xn
1 ⊢@ Yn 1
| Xn
2 ⊢ Yn 2 |
· · · | Xn
mn ⊢ Yn mn
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 35 of 47
2d Hypersequent Notation
H[X ⊢ Y | X′ ⊢ Y ′] H[X ⊢ Y ∥ X′ ⊢ Y ′]
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 36 of 47
2d Hypersequent Rules
H[X ⊢ Y ∥ X′, A ⊢@ Y ′]
[APK L]
H[X, [K]A ⊢ Y ∥ X′ ⊢@ Y ′] H[ ⊢@ A ∥ X ⊢ Y]
[APK R]
H[X ⊢ [K]A, Y]
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 37 of 47
Example Derivation
p ⊢@ p
[@R]
p ⊢@ @p
[⊃R]
⊢@ p ⊃ @p
[[K]R]
⊢ [K](p ⊃ @p)
[□R]
⊢ □[K](p ⊃ @p)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 38 of 47
Cut Elimination is standard
δ1 H[ ⊢@ A ∥ X ⊢ Y ∥ X′ ⊢@ Y ′]
[APK R]
H[X ⊢ [K]A, Y ∥ X′ ⊢@ Y ′] δ2 H[X ⊢ Y ∥ X′, A ⊢@ Y ′]
[APK L]
H[X, [K]A ⊢ Y ∥ X′ ⊢@ Y ′]
[aCut]
H[X ⊢ Y ∥ X′ ⊢@ Y ′] δ1 H[ ⊢@ A ∥ X ⊢ Y ∥ X′ ⊢@ Y ′] δ2 H[X ⊢ Y ∥ X′, A ⊢@ Y ′]
[aCut]
H[X ⊢ Y ∥ X′ ⊢@ Y ′ ∥ X′ ⊢@ Y ′]
[eW]
H[X ⊢ Y ∥ X′ ⊢@ Y ′]
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 39 of 47
Proof Search for invalid sequents generates models
̸⊢@ □([K]p ⊃ p)
@ @ @ @
(For more details on this construction, see tomorrow.)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 40 of 47
Proof Search for invalid sequents generates models
̸⊢@ □([K]p ⊃ p) [ : □([K]p ⊃ p) ]@
@ @ @
(For more details on this construction, see tomorrow.)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 40 of 47
Proof Search for invalid sequents generates models
̸⊢@ □([K]p ⊃ p) [ : □([K]p ⊃ p) ]@ [ : ]@ | [ : [K]p ⊃ p ]
@ @
(For more details on this construction, see tomorrow.)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 40 of 47
Proof Search for invalid sequents generates models
̸⊢@ □([K]p ⊃ p) [ : □([K]p ⊃ p) ]@ [ : ]@ | [ : [K]p ⊃ p ] [ : ]@ | [ [K]p : p ]
@
(For more details on this construction, see tomorrow.)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 40 of 47
Proof Search for invalid sequents generates models
̸⊢@ □([K]p ⊃ p) [ : □([K]p ⊃ p) ]@ [ : ]@ | [ : [K]p ⊃ p ] [ : ]@ | [ [K]p : p ] [ p : ]@ | [ [K]p : p ] (For more details on this construction, see tomorrow.)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 40 of 47
Proof Search for invalid sequents generates models
̸⊢@ □([K]p ⊃ p) [ : □([K]p ⊃ p) ]@ [ : ]@ | [ : [K]p ⊃ p ] [ : ]@ | [ [K]p : p ] [ p : ]@ | [ [K]p : p ] (For more details on this construction, see tomorrow.)
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 40 of 47
This is (and is not) Davies and Humberstone's Logic
In the propositional language, the sequent calculus is sound and complete for the @ fragment of Davies and Humberstone’s logic. But the models are different—they are ragged, not square. (A simple model construction shows that anything in the propositional language refutable in a ragged model is refutable in a square model too.) In the natural extension to propositional (or second order) quantification, D&H logic validates this principle: (where is the modal fatalist claim: ), while the sequent system does not validate this. Is this a virtue or a vice?
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 41 of 47
This is (and is not) Davies and Humberstone's Logic
▶ In the propositional language, the sequent calculus is sound and complete
for the ¬, ∧, □, @, [K] fragment of Davies and Humberstone’s logic. But the models are different—they are ragged, not square. (A simple model construction shows that anything in the propositional language refutable in a ragged model is refutable in a square model too.) In the natural extension to propositional (or second order) quantification, D&H logic validates this principle: (where is the modal fatalist claim: ), while the sequent system does not validate this. Is this a virtue or a vice?
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 41 of 47
This is (and is not) Davies and Humberstone's Logic
▶ In the propositional language, the sequent calculus is sound and complete
for the ¬, ∧, □, @, [K] fragment of Davies and Humberstone’s logic.
▶ But the models are different—they are ragged, not square.
(A simple model construction shows that anything in the propositional language refutable in a ragged model is refutable in a square model too.) In the natural extension to propositional (or second order) quantification, D&H logic validates this principle: (where is the modal fatalist claim: ), while the sequent system does not validate this. Is this a virtue or a vice?
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 41 of 47
This is (and is not) Davies and Humberstone's Logic
▶ In the propositional language, the sequent calculus is sound and complete
for the ¬, ∧, □, @, [K] fragment of Davies and Humberstone’s logic.
▶ But the models are different—they are ragged, not square. ▶ (A simple model construction shows that anything in the propositional
language refutable in a ragged model is refutable in a square model too.) In the natural extension to propositional (or second order) quantification, D&H logic validates this principle: (where is the modal fatalist claim: ), while the sequent system does not validate this. Is this a virtue or a vice?
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 41 of 47
This is (and is not) Davies and Humberstone's Logic
▶ In the propositional language, the sequent calculus is sound and complete
for the ¬, ∧, □, @, [K] fragment of Davies and Humberstone’s logic.
▶ But the models are different—they are ragged, not square. ▶ (A simple model construction shows that anything in the propositional
language refutable in a ragged model is refutable in a square model too.)
▶ In the natural extension to propositional (or second order)
quantification, D&H logic validates this principle: ⊢ (f → [K]f) ∧ (¬f → [K]¬f) (where f is the modal fatalist claim: (∀p)(p ↔ □p)), while the sequent system does not validate this. Is this a virtue or a vice?
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 41 of 47
This is (and is not) Davies and Humberstone's Logic
▶ In the propositional language, the sequent calculus is sound and complete
for the ¬, ∧, □, @, [K] fragment of Davies and Humberstone’s logic.
▶ But the models are different—they are ragged, not square. ▶ (A simple model construction shows that anything in the propositional
language refutable in a ragged model is refutable in a square model too.)
▶ In the natural extension to propositional (or second order)
quantification, D&H logic validates this principle: ⊢ (f → [K]f) ∧ (¬f → [K]¬f) (where f is the modal fatalist claim: (∀p)(p ↔ □p)), while the sequent system does not validate this.
▶ Is this a virtue or a vice?
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 41 of 47
What we've done
We’ve seen how the hypersequent calculus is not only a general technique for giving a sequent style proof theory for a range of propositional modal logics, but it can also be tailored to give simple proof systems for specific modal logics, with separable rules, and structural features neatly matched to the frame conditions for those logics.
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 42 of 47
Tomorrow
Semantics and beyond
Speech Acts and Norms Proofs and Models Where to go from here
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 43 of 47
Tomorrow
Semantics and beyond
Speech Acts and Norms Proofs and Models Where to go from here
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 43 of 47
Hypersequents for Modal Logic
francesca poggiolesi “A Cut-Free Simple Sequent Calculus for Modal Logic S5.” Bulletin of Symbolic Logic 1:1, 3–15, 2008. francesca poggiolesi Gentzen Calculi for Modal Propositional Logic Springer, 2011.
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 44 of 47
Hypersequents for Modal Logic
greg restall “Proofnets for s5: sequents and circuits for modal logic.” Logic Colloquium 2005, ed. C. Dimitracopoulos, L. Newelski, D. Normann and J. R. Steel, Cambridge University Press, 2008. http://consequently.org/writing/s5nets kaja bednarska and andrzej indrzejczak “Hypersequent Calculi for s5: the methods of cut elimination.” Logic and Logical Philosophy, 24 277–311, 2015. greg restall “A Cut-Free Sequent System for Two Dimensional Modal Logic, and why it matters.” Annals of Pure and Applied Logic, 163:11, 1611–1623, 2012. http://consequently.org/writing/cfss2dml
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 45 of 47
Two-Dimensional Modal Logic
martin davies “Reference, Contingency, and the Two-Dimensional Framework.” Philosophical Studies, 118(1):83–131, 2004. martin davies and lloyd humberstone “Two Notions of Necessity.” Philosophical Studies, 38(1):1–30, 1980. lloyd humberstone “Two-Dimensional Adventures.” Philosohical Studies, 118(1):257–277, 2004.
Greg Restall and Shawn Standefer Proof Theory:, Logical and Philosophical Aspects 46 of 47
https://consequently.org/class/2016/PTPLA-NASSLLI/ @consequently / @standefer on Twitter