Generality & ExistenceIII Substitution& Identity Greg - PowerPoint PPT Presentation

22 of 42 [ Cut ] Generality & Existence III Greg Restall [ L ] [ Cut ] From [ = Df ] to [ = L ] s = t � s = t [ = Df ] s = t, Fs � Ft [ Spec Fx A ( x ) ] Γ � A ( s ) , ∆ s = t, A ( s ) � A ( t ) s = t, Γ � A ( t ) , ∆ Γ, A ( t ) � ∆ s = t, Γ � ∆

22 of 42 [ Cut ] Generality & Existence III Greg Restall [ L ] [ Cut ] From [ = Df ] to [ = L ] s = t � s = t [ = Df ] s = t, Fs � Ft [ Spec Fx A ( x ) ] Γ � A ( s ) , ∆ s = t, A ( s ) � A ( t ) s = t, Γ � A ( t ) , ∆ Γ, A ( t ) � ∆ s = t, Γ � ∆

22 of 42 [ Cut ] Generality & Existence III Greg Restall [ L ] [ Cut ] From [ = Df ] to [ = L ] s = t � s = t [ = Df ] s = t, Fs � Ft [ Spec Fx A ( x ) ] Γ � A ( s ) , ∆ s = t, A ( s ) � A ( t ) s = t, Γ � A ( t ) , ∆ Γ, A ( t ) � ∆ s = t, Γ � ∆

22 of 42 [ Cut ] Generality & Existence III Greg Restall [ L ] [ Cut ] From [ = Df ] to [ = L ] s = t � s = t [ = Df ] s = t, Fs � Ft [ Spec Fx A ( x ) ] Γ � A ( s ) , ∆ s = t, A ( s ) � A ( t ) s = t, Γ � A ( t ) , ∆ Γ, A ( t ) � ∆ s = t, Γ � ∆

22 of 42 [ Cut ] Generality & Existence III Greg Restall [ Cut ] From [ = Df ] to [ = L ] s = t � s = t [ = Df ] s = t, Fs � Ft [ Spec Fx A ( x ) ] Γ � A ( s ) , ∆ s = t, A ( s ) � A ( t ) s = t, Γ � A ( t ) , ∆ Γ, A ( t ) � ∆ s = t, Γ � ∆ Γ � A ( s ) , ∆ Γ, A ( t ) � ∆ [ = L ] s = t, Γ � ∆

Proof search? This is valid , but ugly . Subformula property? Greg Restall Generality & Existence III 23 of 42 [ = L ] Γ � A ( s ) , ∆ Γ, A ( t ) � ∆ [ = L ] s = t, Γ � ∆

Proof search? This is valid , but ugly . Subformula property? Greg Restall Generality & Existence III 23 of 42 [ = L ] Γ � A ( s ) , ∆ Γ, A ( t ) � ∆ [ = L ] s = t, Γ � ∆

Proof search? This is valid , but ugly . Subformula property? Greg Restall Generality & Existence III 23 of 42 [ = L ] Γ � A ( s ) , ∆ Γ, A ( t ) � ∆ [ = L ] s = t, Γ � ∆

Proof search? This is valid , but ugly . Subformula property? Greg Restall Generality & Existence III 23 of 42 [ = L ] Γ � A ( s ) , ∆ Γ, A ( t ) � ∆ [ = L ] s = t, Γ � ∆

Backtracking a little [ Cut ] Generality & Existence III Greg Restall [ L ] [ Cut ] 24 of 42 s = t � s = t [ = Df ] s = t, Fs � Ft [ Spec Fx A ( x ) ] Γ � A ( s ) , ∆ s = t, A ( s ) � A ( t ) s = t, Γ � A ( t ) , ∆ Γ, A ( t ) � ∆ s = t, Γ � ∆

Backtracking a little [ Cut ] Generality & Existence III Greg Restall [ Cut ] 24 of 42 s = t � s = t [ = Df ] s = t, Fs � Ft [ Spec Fx A ( x ) ] Γ � A ( s ) , ∆ s = t, A ( s ) � A ( t ) s = t, Γ � A ( t ) , ∆ Γ, A ( t ) � ∆ s = t, Γ � ∆ Γ � A ( s ) , ∆ [ = L ′ ] s = t, Γ � A ( t ) , ∆

[ K ] [ Id ] [ K ] [ Cut ] Greg Restall Generality & Existence III 25 of 42 [ = L ′ ] is Enough to recover [ = Df ] Ft � Ft [ = L ′ ] Γ � s = t, ∆ s = t, Fs � Ft Γ, Fs � s = t, Ft, ∆ Γ, s = t, Fs � Ft, ∆ Γ, Fs � Ft, ∆

This is better … But it is still strange . It operates at two places in the concluding sequent. This puts compositionality in question. Greg Restall Generality & Existence III 26 of 42 [ = L ′ ] Γ � A ( s ) , ∆ [ = L ′ ] s = t, Γ � A ( t ) , ∆

This is better … But it is still strange . It operates at two places in the concluding sequent. This puts compositionality in question. Greg Restall Generality & Existence III 26 of 42 [ = L ′ ] Γ � A ( s ) , ∆ [ = L ′ ] s = t, Γ � A ( t ) , ∆

This is better … But it is still strange . It operates at two places in the concluding sequent. This puts compositionality in question. Greg Restall Generality & Existence III 26 of 42 [ = L ′ ] Γ � A ( s ) , ∆ [ = L ′ ] s = t, Γ � A ( t ) , ∆

This is better … But it is still strange . It operates at two places in the concluding sequent. This puts compositionality in question. Greg Restall Generality & Existence III 26 of 42 [ = L ′ ] Γ � A ( s ) , ∆ [ = L ′ ] s = t, Γ � A ( t ) , ∆

identity & predication

28 of 42 [ L ] on conjunctions is given by [ L ] on its conjuncts . Generality & Existence III 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 ) .)

28 of 42 Greg Restall Generality & Existence III 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 .

[ L ] on disjunctions is given by [ L ] on its disjuncts . [ W ] Greg Restall Generality & Existence III 29 of 42 Decomposing [ = L ′ ]: disjunctions Γ � A ( s ) ∨ B ( s ) , ∆ [ ∨ Df ] Γ � A ( s ) , B ( s ) , ∆ [ = L ′ ] s = t, Γ � A ( t ) , B ( s ) , ∆ [ = L ′ ] s = t, s = t, Γ � A ( t ) , B ( t ) , ∆ s = t, Γ � A ( t ) , B ( t ) , ∆ [ ∨ Df ] s = t, Γ � A ( t ) ∨ B ( t ) , ∆

[ W ] Greg Restall Generality & Existence III 29 of 42 Decomposing [ = L ′ ]: disjunctions Γ � A ( s ) ∨ B ( s ) , ∆ [ ∨ Df ] Γ � A ( s ) , B ( s ) , ∆ [ = L ′ ] s = t, Γ � A ( t ) , B ( s ) , ∆ [ = L ′ ] s = t, s = t, Γ � A ( t ) , B ( t ) , ∆ s = t, Γ � A ( t ) , B ( t ) , ∆ [ ∨ Df ] s = t, Γ � A ( t ) ∨ B ( t ) , ∆ [ = L ′ ] on disjunctions is given by [ = L ′ ] on its disjuncts .

Generality & Existence III Greg Restall 30 of 42 Decomposing [ = L ′ ]: universal quantifiers Γ � ( ∀ x ) A ( x, s ) , ∆ [ ∀ Df ] Γ � A ( n, s ) , ∆ [ = L ′ ] s = t, Γ � A ( n, t ) , ∆ [ ∀ Df ] s = t, Γ � ( ∀ x ) A ( x, t ) , ∆ [ = L ′ ] on a universally quantified statement is given by [ = L ′ ] on an instance .

[ Cut ] [ Id ] Greg Restall Generality & Existence III 31 of 42 Decomposing [ = L ′ ]: existential quantifiers A ( n, s ) � A ( n, s ) [ = L ′ ] s = t, A ( n, s ) � A ( n, t ) [ ∃ R ] s = t, A ( n, s ) � ( ∃ x ) A ( x, t ) [ ∃ Df ] s = t, ( ∃ x ) A ( x, s ) � ( ∃ x ) A ( x, t ) Γ � ( ∃ x ) A ( x, s ) , ∆ s = t, Γ � ( ∃ x ) A ( x, t ) , ∆ [ = L ′ ] on an existentially quantified statement is given by [ = L ′ ] on an instance .

Greg Restall But for negation … Generality & Existence III 32 of 42 Γ � ¬ A ( s ) , ∆ [ ¬ Df ] Γ, A ( s ) � ∆ [ = L ′ on the wrong side!] s = t, A ( t ) , Γ � ∆ [ ¬ Df ] s = t, Γ � ¬ A ( t ) , ∆

Different Identity Rules [ L ] [ R ] [ L ] [ L ] [ Df ] [ Spec ] Greg Restall Generality & Existence III 33 of 42 Γ � A ( s ) , ∆ Γ, A ( t ) � ∆ [ = L ] s = t, Γ � ∆

Different Identity Rules [ L ] Generality & Existence III Greg Restall ] [ Spec [ Df ] [ L ] [ R ] 33 of 42 Γ � A ( s ) , ∆ Γ, A ( t ) � ∆ [ = L ] s = t, Γ � ∆ Γ � A ( s ) , ∆ [ = L f r ] s = t, Γ � A ( t ) , ∆

Different Identity Rules [ L ] Generality & Existence III Greg Restall ] [ Spec [ Df ] [ L ] 33 of 42 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, ∆

Different Identity Rules [ L ] Generality & Existence III Greg Restall ] [ Spec [ Df ] 33 of 42 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, ∆ [ = L p r ] s = t, Γ � Ft, ∆

Different Identity Rules [ Df ] Generality & Existence III Greg Restall ] [ Spec 33 of 42 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 � ∆

Different Identity Rules [ Spec Generality & Existence III Greg Restall ] 33 of 42 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, Fa � Fb, ∆ = = = = = = = = = = [ = Df ] Γ � a = b, ∆

Different Identity Rules Greg Restall Generality & Existence III 33 of 42 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 III Greg Restall Each system gives you classical first-order predicate logic with identity. L /L /R L /L /R Cut L /R Cut 34 of 42 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 III 34 of 42 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 III Greg Restall Each system gives you classical first-order predicate logic with identity. L /L /R 34 of 42 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 III Greg Restall Each system gives you classical first-order predicate logic with identity. 34 of 42 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 III Greg Restall 34 of 42 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 III Greg Restall 34 of 42 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 ]

Non-Symmetric ‘Identity’ Human . Generality & Existence III Greg Restall . or but not , , , Spec : holds for atomic predicates, closed under . atomic predicates : closed upward under Mammal domain : Animal is . is There are models of this system in which [ isR ] 35 of 42 Γ � Fs, ∆ Γ, Fs � Ft, ∆ [ isL p r ] t is s, Γ � Ft, ∆ Γ � t is s, ∆

Non-Symmetric ‘Identity’ atomic predicates : closed upward under Generality & Existence III Greg Restall . or but not , , , Spec : holds for atomic predicates, closed under . Human . Mammal domain : Animal [ isR ] 35 of 42 Γ � Fs, ∆ Γ, Fs � Ft, ∆ [ isL p r ] t is s, Γ � Ft, ∆ Γ � t is s, ∆ There are models of this system in which s is t ̸ � t is s .

[ isR ] Non-Symmetric ‘Identity’ Greg Restall Generality & Existence III 35 of 42 Γ � Fs, ∆ Γ, Fs � Ft, ∆ [ isL p r ] t is s, Γ � Ft, ∆ Γ � t is s, ∆ There are models of this system in which s is t ̸ � t is s . domain : Animal < Mammal < Human . atomic predicates : closed upward under < . Spec : holds for atomic predicates, closed under ∧ , ∨ , ∀ , ∃ but not ¬ or ⊃ .

free logic & identity

Free Quantification [ Df ] Generality & Existence III Greg Restall 37 of 42 Γ, n � A ( n ) , ∆ Γ, n, A ( n ) � ∆ = = = = = = = = = = = = = [ ∀ Df ] = = = = = = = = = = = = [ ∃ Df ] Γ � ( ∀ x ) A ( x ) , ∆ Γ, ( ∃ x ) A ( x ) � ∆

Free Quantification Greg Restall Generality & Existence III 37 of 42 Γ, n � A ( n ) , ∆ Γ, n, A ( n ) � ∆ = = = = = = = = = = = = = [ ∀ Df ] = = = = = = = = = = = = [ ∃ Df ] Γ � ( ∀ x ) A ( x ) , ∆ Γ, ( ∃ x ) A ( x ) � ∆ Γ, t � ∆ = = = = = = = [ ↓ Df ] Γ, t ↓ � ∆

Free Quantification Greg Restall Generality & Existence III 37 of 42 Γ, n � A ( n ) , ∆ Γ, n, A ( n ) � ∆ = = = = = = = = = = = = = [ ∀ Df ] = = = = = = = = = = = = [ ∃ Df ] Γ � ( ∀ x ) A ( x ) , ∆ Γ, ( ∃ x ) A ( x ) � ∆ Γ, t � ∆ = = = = = = = [ ↓ Df ] Γ, t ↓ � ∆ ( ∀ x ) Fx ̸ � Ft

Free Quantification Greg Restall Generality & Existence III 37 of 42 Γ, n � A ( n ) , ∆ Γ, n, A ( n ) � ∆ = = = = = = = = = = = = = [ ∀ Df ] = = = = = = = = = = = = [ ∃ Df ] Γ � ( ∀ x ) A ( x ) , ∆ Γ, ( ∃ x ) A ( x ) � ∆ Γ, t � ∆ = = = = = = = [ ↓ Df ] Γ, t ↓ � ∆ ( ∀ x ) Fx ̸ � Ft A ( t ) ̸ � ( ∃ x ) A ( x )

Free Quantification Greg Restall Generality & Existence III 37 of 42 Γ, n � A ( n ) , ∆ Γ, n, A ( n ) � ∆ = = = = = = = = = = = = = [ ∀ Df ] = = = = = = = = = = = = [ ∃ Df ] Γ � ( ∀ x ) A ( x ) , ∆ Γ, ( ∃ x ) A ( x ) � ∆ Γ, t � ∆ = = = = = = = [ ↓ Df ] Γ, t ↓ � ∆ ( ∀ x ) Fx ̸ � Ft A ( t ) ̸ � ( ∃ x ) A ( x ) ( ∀ x ) Fx, t � Ft

Free Quantification Greg Restall Generality & Existence III 37 of 42 Γ, n � A ( n ) , ∆ Γ, n, A ( n ) � ∆ = = = = = = = = = = = = = [ ∀ Df ] = = = = = = = = = = = = [ ∃ Df ] Γ � ( ∀ x ) A ( x ) , ∆ Γ, ( ∃ x ) A ( x ) � ∆ Γ, t � ∆ = = = = = = = [ ↓ Df ] Γ, t ↓ � ∆ ( ∀ x ) Fx ̸ � Ft A ( t ) ̸ � ( ∃ x ) A ( x ) ( ∀ x ) Fx, t � Ft A ( t ) , t ↓ � ( ∃ x ) A ( t )

Is Predication Existentially Committing? Greg Restall Generality & Existence III 38 of 42 t i , Γ � ∆ [ F L ] Ft 1 · · · t n , Γ � ∆

Which Identity Rule? non-commital: Generality & Existence III Greg Restall Df ] [ committal: 39 of 42 Γ, Fs � Ft, ∆ = = = = = = = = = = [ = n Df ] Γ � s = n t, ∆

Which Identity Rule? committal: Generality & Existence III Greg Restall non-commital: 39 of 42 Γ, Fs � Ft, ∆ = = = = = = = = = = [ = n Df ] Γ � s = n t, ∆ Γ � s, ∆ Γ � t, ∆ Γ, Fs � Ft, ∆ = = = = = = = = = = = = = = = = = = = = = = = = = [ = c Df ] Γ � s = c t, ∆

Which Identity Rule? committal: Generality & Existence III Greg Restall non-commital: 39 of 42 Γ, Fs � Ft, ∆ = = = = = = = = = = [ = n Df ] Γ � s = n t, ∆ Γ � s, ∆ Γ � t, ∆ Γ, Fs � Ft, ∆ = = = = = = = = = = = = = = = = = = = = = = = = = [ = c Df ] Γ � s = c t, ∆ � t = n t

Which Identity Rule? committal: Generality & Existence III Greg Restall non-commital: 39 of 42 Γ, Fs � Ft, ∆ = = = = = = = = = = [ = n Df ] Γ � s = n t, ∆ Γ � s, ∆ Γ � t, ∆ Γ, Fs � Ft, ∆ = = = = = = = = = = = = = = = = = = = = = = = = = [ = c Df ] Γ � s = c t, ∆ � t = n t ̸ � t = c t

Which Identity Rule? committal: Generality & Existence III Greg Restall non-commital: 39 of 42 Γ, Fs � Ft, ∆ = = = = = = = = = = [ = n Df ] Γ � s = n t, ∆ Γ � s, ∆ Γ � t, ∆ Γ, Fs � Ft, ∆ = = = = = = = = = = = = = = = = = = = = = = = = = [ = c Df ] Γ � s = c t, ∆ � t = n t ̸ � t = c t s = c t � t ↓ s = c t � s ↓

moral: For non-committal identity, allow Non-committal identity clashes with committing predication to be negative as well as positive (e.g., nonexistence ) so L might fail for this predicate. Greg Restall Generality & Existence III 40 of 42 s � s, Ft [ F L ] Fs � s, Ft [ ↓ Df ] Fs � s ↓ , Ft [ ¬ Df ] ¬ s ↓ , Fs � Ft [ = n Df ] ¬ s ↓ � s = n t

Greg Restall Non-committal identity clashes with committing predication Generality & Existence III 40 of 42 s � s, Ft [ F L ] Fs � s, Ft [ ↓ Df ] Fs � s ↓ , Ft [ ¬ Df ] ¬ s ↓ , Fs � Ft [ = n Df ] ¬ s ↓ � s = n t moral: For non-committal identity, allow F to be negative as well as positive (e.g., nonexistence ) so F L might fail for this predicate.