Generality & ExistenceIV Modality& Identity Greg Restall - PowerPoint PPT Presentation

Generality & ExistenceIV Modality& Identity Greg Restall arché, st andrews · 3 december 2015

My Aim To analyse the quantifiers (including their interactions with modals ) using the tools of proof theory in order to better understand quantification , existence and identity . Greg Restall Generality & Existence IV 2 of 35

My Aim To analyse the quantifiers (including their interactions with modals ) using the tools of proof theory in order to better understand quantification , existence and identity . Greg Restall Generality & Existence IV 2 of 35

My Aim To analyse the quantifiers (including their interactions with modals ) using the tools of proof theory in order to better understand quantification , existence and identity . Greg Restall Generality & Existence IV 2 of 35

My Aim To analyse the quantifiers (including their interactions with modals ) using the tools of proof theory in order to better understand quantification , existence and identity . Greg Restall Generality & Existence IV 2 of 35

My Aim for This Talk Understanding the interaction between modality and identity. Greg Restall Generality & Existence IV 3 of 35

Today’s Plan Hypersequents & Defining Rules Identity Rules Subjunctive Alternatives Indicative Alternatives The Status of Worlds Semantics Greg Restall Generality & Existence IV 4 of 35

hypersequents & defining rules

6 of 35 Greg Restall Generality & Existence IV Subjunctive Alternatives and □ H [ Γ � ∆ | � A ] = = = = = = = = = = = = = = [ □ Df ] H [ Γ � □ A, ∆ ]

7 of 35 Greg Restall Generality & Existence IV Indicative Alternatives and [ e ] H [ Γ � ∆ ∥ � @ A ] = = = = = = = = = = = = = = = [ [ e ] Df ] H [ Γ � [ e ] A, ∆ ]

Actual zones and @ Greg Restall Generality & Existence IV 8 of 35 H [ Γ, A � @ ∆ | Γ ′ � ∆ ′ ] = = = = = = = = = = = = = = = = = = = = [@ Df ] H [ Γ � @ ∆ | Γ ′ , @ A � ∆ ′ ]

Two Dimensional Hypersequents . Generality & Existence IV Greg Restall Think of these as scorecards , keeping track of assertions and denials. . . . . . . . . 9 of 35 X 1 1 � @ Y 1 X 1 2 � Y 1 X 1 m 1 � Y 1 | 2 | · · · | ∥ 1 m 1 X 2 1 � @ Y 2 X 2 2 � Y 2 X 2 m 2 � Y 2 · · · ∥ | 2 | | 1 m 2 X n 1 � @ Y n X n 2 � Y n X n m n � Y n | 2 | · · · | 1 m n

Free Quantification [ Df ] Generality & Existence IV Greg Restall [ L ] 10 of 35 H [ Γ, n � A ( n ) , ∆ ] H [ Γ, n, A ( n ) � ∆ ] = = = = = = = = = = = = = = = [ ∀ Df ] = = = = = = = = = = = = = = = [ ∃ Df ] H [ Γ � ( ∀ x ) A ( x ) , ∆ ] H [ Γ, ( ∃ x ) A ( x ) � ∆ ]

Free Quantification Greg Restall Generality & Existence IV 10 of 35 H [ Γ, n � A ( n ) , ∆ ] H [ Γ, n, A ( n ) � ∆ ] = = = = = = = = = = = = = = = [ ∀ Df ] = = = = = = = = = = = = = = = [ ∃ Df ] H [ Γ � ( ∀ x ) A ( x ) , ∆ ] H [ Γ, ( ∃ x ) A ( x ) � ∆ ] H [ t, Γ � ∆ ] H [ t i , Γ � ∆ ] = = = = = = = = = = [ ↓ Df ] [ F L ] H [ t ↓ , Γ � ∆ ] H [ Ft 1 · · · t n , Γ � ∆ ]

identity rules

Identity Rules Greg Restall Generality & Existence IV 12 of 35 Γ � A ( s ) , ∆ Γ, A ( t ) � ∆ [ = L ] s = t, Γ � ∆ Γ � A ( s ) , ∆ Γ, Fa � Fb, ∆ [ = L f [ = R ] r ] s = t, Γ � A ( t ) , ∆ Γ � a = b, ∆ Γ � Fs, ∆ Fs, Γ � ∆ [ = L p [ = L p r ] l ] s = t, Γ � Ft, ∆ s = t, Ft, Γ � ∆ Γ � ∆ Γ, Fa � Fb, ∆ [ Spec Fx = = = = = = = = = = [ = Df ] A ( x ) ] Γ | Fx A ( x ) � ∆ | Fx Γ � a = b, ∆ A ( x )

Equivalences L/R Cut Generality & Existence IV Greg Restall Each system gives you classical first-order predicate logic with identity. L /L /R L /L /R Cut L /R Cut 13 of 35 Γ � ∆ Γ, Fa � Fb, ∆ [ Spec Fx = = = = = = = = = = [ = Df ] A ( x ) ] Γ | Fx A ( x ) � ∆ | Fx Γ � a = b, ∆ A ( x ) L [= Df , Spec , Cut ]

L /R Cut Equivalences L /L /R Cut L /L /R Each system gives you classical first-order predicate logic with identity. Greg Restall Generality & Existence IV 13 of 35 Γ � A ( s ) , ∆ Γ, A ( t ) � ∆ [ = L ] s = t, Γ � ∆ L [= Df , Spec , Cut ] = L [= L/R , Cut ]

Equivalences L /L /R Cut Generality & Existence IV Greg Restall Each system gives you classical first-order predicate logic with identity. L /L /R 13 of 35 Γ � A ( s ) , ∆ [ = L f r ] s = t, Γ � A ( t ) , ∆ L [= Df , Spec , Cut ] = L [= L/R , Cut ] L [= L f = r /R , Cut ]

Equivalences L /L /R Generality & Existence IV Greg Restall Each system gives you classical first-order predicate logic with identity. 13 of 35 Γ � Fs, ∆ Fs, Γ � ∆ Γ, Fa � Fb, ∆ [ = L p [ = L p [ = R ] r ] l ] s = t, Γ � Ft, ∆ s = t, Ft, Γ � ∆ Γ � a = b, ∆ L [= Df , Spec , Cut ] = L [= L/R , Cut ] L [= L f = r /R , Cut ] L [= L p r /L p = l /R , Cut ]

Equivalences Each system gives you classical first-order predicate logic with identity. Generality & Existence IV Greg Restall 13 of 35 Γ � Fs, ∆ Fs, Γ � ∆ Γ, Fa � Fb, ∆ [ = L p [ = L p [ = R ] r ] l ] s = t, Γ � Ft, ∆ s = t, Ft, Γ � ∆ Γ � a = b, ∆ L [= Df , Spec , Cut ] = L [= L/R , Cut ] L [= L f = r /R , Cut ] L [= L p r /L p = l /R , Cut ] L [= L p r /L p = l /R ]

Equivalences Each system gives you classical first-order predicate logic with identity. Generality & Existence IV Greg Restall 13 of 35 Γ � Fs, ∆ Fs, Γ � ∆ Γ, Fa � Fb, ∆ [ = L p [ = L p [ = R ] r ] l ] s = t, Γ � Ft, ∆ s = t, Ft, Γ � ∆ Γ � a = b, ∆ L [= Df , Spec , Cut ] = L [= L/R , Cut ] L [= L f = r /R , Cut ] L [= L p r /L p = l /R , Cut ] L [= L p r /L p = l /R ]

14 of 35 [ L ] on conjunctions is given by [ L ] on its conjuncts . Generality & Existence IV Greg Restall Decomposing [ = L ′ ]: conjunctions Γ � A ( s ) ∧ B ( s ) , ∆ Γ � A ( s ) ∧ B ( s ) , ∆ [ ∧ E ] [ ∧ E ] Γ � A ( s ) , ∆ Γ � B ( s ) , ∆ [ = L ′ ] [ = L ′ ] s = t, Γ � A ( t ) , ∆ s = t, Γ � B ( t ) , ∆ [ ∧ R ] s = t, Γ � A ( t ) ∧ B ( t ) , ∆ (Where the [ ∧ E ] is given by a Cut on A ( t ) ∧ B ( t ) � A ( t ) , or A ( t ) ∧ B ( t ) � B ( t ) .)

14 of 35 Greg Restall Generality & Existence IV Decomposing [ = L ′ ]: conjunctions Γ � A ( s ) ∧ B ( s ) , ∆ Γ � A ( s ) ∧ B ( s ) , ∆ [ ∧ E ] [ ∧ E ] Γ � A ( s ) , ∆ Γ � B ( s ) , ∆ [ = L ′ ] [ = L ′ ] s = t, Γ � A ( t ) , ∆ s = t, Γ � B ( t ) , ∆ [ ∧ R ] s = t, Γ � A ( t ) ∧ B ( t ) , ∆ (Where the [ ∧ E ] is given by a Cut on A ( t ) ∧ B ( t ) � A ( t ) , or A ( t ) ∧ B ( t ) � B ( t ) .) [ = L ′ ] on conjunctions is given by [ = L ′ ] on its conjuncts .

Greg Restall But for negation … Generality & Existence IV 15 of 35 Γ � ¬ A ( s ) , ∆ [ ¬ Df ] Γ, A ( s ) � ∆ [ = L ′ on the wrong side!] s = t, A ( t ) , Γ � ∆ [ ¬ Df ] s = t, Γ � ¬ A ( t ) , ∆

subjunctive alternatives

The Defining Rule for Identity in Hypersequents How general is the Generality & Existence IV Greg Restall Call this [ Modal Spec ]. . and to contain Let’s allow ] [ Spec in [ Df ]? 17 of 35 H [ Γ, Fa � Fb, ∆ ] = = = = = = = = = = = = = [ = Df ] H [ Γ � a = b, ∆ ]

The Defining Rule for Identity in Hypersequents [ Spec Generality & Existence IV Greg Restall Call this [ Modal Spec ]. . and to contain Let’s allow ] 17 of 35 H [ Γ, Fa � Fb, ∆ ] = = = = = = = = = = = = = [ = Df ] H [ Γ � a = b, ∆ ] How general is the F in [ = Df ]?

The Defining Rule for Identity in Hypersequents Call this [ Modal Spec ]. Generality & Existence IV Greg Restall 17 of 35 H [ Γ, Fa � Fb, ∆ ] = = = = = = = = = = = = = [ = Df ] H [ Γ � a = b, ∆ ] How general is the F in [ = Df ]? Γ � ∆ [ Spec Fx A ( x ) ] Γ | Fx A ( x ) � ∆ | Fx A ( x ) Let’s allow A ( x ) to contain □ and ♢ .

The Defining Rule for Identity in Hypersequents Call this [ Modal Spec ]. Generality & Existence IV Greg Restall 17 of 35 H [ Γ, Fa � Fb, ∆ ] = = = = = = = = = = = = = [ = Df ] H [ Γ � a = b, ∆ ] How general is the F in [ = Df ]? Γ � ∆ [ Spec Fx A ( x ) ] Γ | Fx A ( x ) � ∆ | Fx A ( x ) Let’s allow A ( x ) to contain □ and ♢ .

Generality & Existence IV Greg Restall 18 of 35 [ = L □ ] H [ Γ � A ( s ) , ∆ ] [ = L □ ] H [ s = t, Γ � A ( t ) , ∆ ] (Where A ( x ) can contain □ .)

Greg Restall Generality & Existence IV 19 of 35 Decomposing [ = L □ ] with [ Modal Spec ]: necessities H [ s = t, Γ � □ A ( s ) , ∆ ] [ □ Df ] H [ s = t, Γ � ∆ | � A ( s )] [ = L ′ ] H [ s = t, Γ � ∆ | � A ( t )] [ □ Df ] H [ s = t, Γ � □ A ( t ) , ∆ ]

Identity across Subjunctive Alternatives This makes sense in planning contexts. Greg Restall Generality & Existence IV 20 of 35 H [ Γ � ∆ | Γ ′ � Fs, ∆ ′ ] [ = L p | r ] H [ s = t, Γ � ∆ | Γ ′ � Ft, ∆ ′ ] H [ Γ � ∆ | Γ ′ , Fs � ∆ ′ ] [ = L p H [ s = t, Γ � ∆ | Γ ′ , Ft � ∆ ′ ] | l ]