Generality & ExistenceIII Predication& Identity Greg - - PowerPoint PPT Presentation
Generality & ExistenceIII Predication& Identity Greg Restall arch, st andrews 2 december 2015 My Aim To analyse the quantifiers (including their interactions with modals ) using the tools of proof theory in order to better
Generality & ExistenceIII
Predication& Identity Greg Restall
arché, st andrews · 2 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 III 2 of 43
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 III 2 of 43
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 III 2 of 43
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 III 2 of 43
My Aim for This Talk
Understanding the behaviour
Greg Restall Generality & Existence III 3 of 43
Today's Plan
Sequents & Defining Rules Identity & Indistinguishability Defining Rules & Left/Right Rules Identity & Predication Free Logic & Identity
Greg Restall Generality & Existence III 4 of 43
Sequents
Don’t assert each element of Γ and deny each element of ∆.
Greg Restall Generality & Existence III 6 of 43
Structural Rules Identity: A A Weakening: Contraction: Cut:
Structural rules govern declarative sentences as such.
Greg Restall Generality & Existence III 7 of 43
Structural Rules Identity: A A Weakening: Γ ∆ Γ, A ∆ Γ ∆ Γ A, ∆ Contraction: Cut:
Structural rules govern declarative sentences as such.
Greg Restall Generality & Existence III 7 of 43
Structural Rules Identity: A A Weakening: Γ ∆ Γ, A ∆ Γ ∆ Γ A, ∆ Contraction: Γ, A, A ∆ Γ, A ∆ Γ A, A, ∆ Γ A, ∆ Cut:
Structural rules govern declarative sentences as such.
Greg Restall Generality & Existence III 7 of 43
Structural Rules Identity: A A Weakening: Γ ∆ Γ, A ∆ Γ ∆ Γ A, ∆ Contraction: Γ, A, A ∆ Γ, A ∆ Γ A, A, ∆ Γ A, ∆ Cut: Γ A, ∆ Γ, A ∆ Γ ∆
Structural rules govern declarative sentences as such.
Greg Restall Generality & Existence III 7 of 43
Structural Rules Identity: A A Weakening: Γ ∆ Γ, A ∆ Γ ∆ Γ A, ∆ Contraction: Γ, A, A ∆ Γ, A ∆ Γ A, A, ∆ Γ A, ∆ Cut: Γ A, ∆ Γ, A ∆ Γ ∆
Structural rules govern declarative sentences as such.
Greg Restall Generality & Existence III 7 of 43
Extending a Language with Specific Vocabulary With Left/Right rules? Γ, A, B ∆
[∧L]
Γ, A ∧ B ∆ Γ A, ∆ Γ B, ∆
[∧R]
Γ A ∧ B, ∆
[tonkL]
tonk
[tonkR]
tonk
Greg Restall Generality & Existence III 8 of 43
Extending a Language with Specific Vocabulary With Left/Right rules? Γ, A, B ∆
[∧L]
Γ, A ∧ B ∆ Γ A, ∆ Γ B, ∆
[∧R]
Γ A ∧ B, ∆ Γ, B ∆
[tonkL]
Γ, A tonk B ∆ Γ A, ∆
[tonkR]
Γ A tonk B, ∆
Greg Restall Generality & Existence III 8 of 43
What is involved in going from L to L′?
Use L to define L′.
Desideratum #1: is conservative: is . Desideratum #2: Concepts are defined uniquely.
Greg Restall Generality & Existence III 9 of 43
What is involved in going from L to L′?
Use L to define L′.
Desideratum #1: L′ is conservative: (L′)|L is L. Desideratum #2: Concepts are defined uniquely.
Greg Restall Generality & Existence III 9 of 43
What is involved in going from L to L′?
Use L to define L′.
Desideratum #1: L′ is conservative: (L′)|L is L. Desideratum #2: Concepts are defined uniquely.
Greg Restall Generality & Existence III 9 of 43
A Defining Rule Γ, A, B ∆ = = = = = = = = = = [∧Df ] Γ, A ∧ B ∆ Fully specifies norms governing conjunctions
Identity and Cut determine the behaviour
Greg Restall Generality & Existence III 10 of 43
A Defining Rule Γ, A, B ∆ = = = = = = = = = = [∧Df ] Γ, A ∧ B ∆ Fully specifies norms governing conjunctions
Identity and Cut determine the behaviour
Greg Restall Generality & Existence III 10 of 43
A Defining Rule Γ, A, B ∆ = = = = = = = = = = [∧Df ] Γ, A ∧ B ∆ Fully specifies norms governing conjunctions
Identity and Cut determine the behaviour
Greg Restall Generality & Existence III 10 of 43
From [∧Df] to [∧L/R]
Γ A, ∆ Γ B, ∆
[Id]
A ∧ B A ∧ B
[∧Df ]
A, B A ∧ B
[Cut]
Γ, A A ∧ B, ∆
[Cut]
Γ A ∧ B, ∆
[ R]
Greg Restall Generality & Existence III 11 of 43
From [∧Df] to [∧L/R]
Γ A, ∆ Γ B, ∆
[Id]
A ∧ B A ∧ B
[∧Df ]
A, B A ∧ B
[Cut]
Γ, A A ∧ B, ∆
[Cut]
Γ A ∧ B, ∆
[ R]
Greg Restall Generality & Existence III 11 of 43
From [∧Df] to [∧L/R]
Γ A, ∆ Γ B, ∆
[Id]
A ∧ B A ∧ B
[∧Df ]
A, B A ∧ B
[Cut]
Γ, A A ∧ B, ∆
[Cut]
Γ A ∧ B, ∆
[ R]
Greg Restall Generality & Existence III 11 of 43
From [∧Df] to [∧L/R]
Γ A, ∆ Γ B, ∆
[Id]
A ∧ B A ∧ B
[∧Df ]
A, B A ∧ B
[Cut]
Γ, A A ∧ B, ∆
[Cut]
Γ A ∧ B, ∆
[ R]
Greg Restall Generality & Existence III 11 of 43
From [∧Df] to [∧L/R]
Γ A, ∆ Γ B, ∆
[Id]
A ∧ B A ∧ B
[∧Df ]
A, B A ∧ B
[Cut]
Γ, A A ∧ B, ∆
[Cut]
Γ A ∧ B, ∆ Γ A, ∆ Γ B, ∆
[∧R]
Γ A ∧ B, ∆
Greg Restall Generality & Existence III 11 of 43
And Back
A A B B
[∧R]
A, B A ∧ B Γ, A ∧ B ∆
[Cut]
Γ, A, B ∆
Greg Restall Generality & Existence III 12 of 43
Quantifier Rules
Γ A(n), ∆ = = = = = = = = = = = = = [∀Df ] Γ (∀x)A(x), ∆ Γ, A(n) ∆ = = = = = = = = = = = = [∃Df ] Γ, (∃x)A(x) ∆
Greg Restall Generality & Existence III 13 of 43
Deductive Generality
Greg Restall Generality & Existence III 14 of 43
Equivalence
L[∧Df, Cut] = L[∧L/R, Cut] L/R This generalises: , , , work in the same way. I want to see how this works for identity.
Greg Restall Generality & Existence III 15 of 43
Equivalence
L[∧Df, Cut] = L[∧L/R, Cut] = L[∧L/R] This generalises: , , , work in the same way. I want to see how this works for identity.
Greg Restall Generality & Existence III 15 of 43
Equivalence
L[∧Df, Cut] = L[∧L/R, Cut] = L[∧L/R] This generalises: ∧, ∨, ⊃, ¬ work in the same way. I want to see how this works for identity.
Greg Restall Generality & Existence III 15 of 43
Equivalence
L[∧Df, Cut] = L[∧L/R, Cut] = L[∧L/R] This generalises: ∧, ∨, ⊃, ¬ work in the same way. I want to see how this works for identity.
Greg Restall Generality & Existence III 15 of 43
Identity and Harmony
Greg Restall Generality & Existence III 17 of 43
Identity Axioms
t = t
Greg Restall Generality & Existence III 18 of 43
Identity Axioms
t = t s = t ⊃ t = s s = t ⊃ (t = u ⊃ s = u)
Greg Restall Generality & Existence III 18 of 43
Identity Axioms
t = t s = t ⊃ t = s s = t ⊃ (t = u ⊃ s = u) s = t ⊃ (A(s) ≡ A(t))
Greg Restall Generality & Existence III 18 of 43
Identity Rules in Natural Deduction
[Fs] · · · Ft
[=I]
s = t
[ E]
Greg Restall Generality & Existence III 19 of 43
Identity Rules in Natural Deduction
[Fs] · · · Ft
[=I]
s = t s = t A(s)
[=E]
A(t)
Greg Restall Generality & Existence III 19 of 43
Defining Rule for Identity
Γ, Fs Ft, ∆ = = = = = = = = = [=Df ] Γ s = t, ∆
Greg Restall Generality & Existence III 20 of 43
Generality in Predicate Position
Γ ∆
[SpecFx
A(x)]
Γ|Fx
A(x) ∆|Fx A(x)
No norm holds of that doesn’t also hold of the sentence context .
Greg Restall Generality & Existence III 21 of 43
Generality in Predicate Position
Γ ∆
[SpecFx
A(x)]
Γ|Fx
A(x) ∆|Fx A(x)
No norm holds of Fx that doesn’t also hold of the sentence context A(x).
Greg Restall Generality & Existence III 21 of 43
From [=Df] to [=L]
Γ A(s), ∆ s = t s = t
[=Df ]
s = t, Fs Ft
[SpecFx
A(x)]
s = t, A(s) A(t)
[Cut]
s = t, Γ A(t), ∆ Γ, A(t) ∆
[Cut]
s = t, Γ ∆
[ L]
Greg Restall Generality & Existence III 23 of 43
From [=Df] to [=L]
Γ A(s), ∆ s = t s = t
[=Df ]
s = t, Fs Ft
[SpecFx
A(x)]
s = t, A(s) A(t)
[Cut]
s = t, Γ A(t), ∆ Γ, A(t) ∆
[Cut]
s = t, Γ ∆
[ L]
Greg Restall Generality & Existence III 23 of 43
From [=Df] to [=L]
Γ A(s), ∆ s = t s = t
[=Df ]
s = t, Fs Ft
[SpecFx
A(x)]
s = t, A(s) A(t)
[Cut]
s = t, Γ A(t), ∆ Γ, A(t) ∆
[Cut]
s = t, Γ ∆
[ L]
Greg Restall Generality & Existence III 23 of 43
From [=Df] to [=L]
Γ A(s), ∆ s = t s = t
[=Df ]
s = t, Fs Ft
[SpecFx
A(x)]
s = t, A(s) A(t)
[Cut]
s = t, Γ A(t), ∆ Γ, A(t) ∆
[Cut]
s = t, Γ ∆
[ L]
Greg Restall Generality & Existence III 23 of 43
From [=Df] to [=L]
Γ A(s), ∆ s = t s = t
[=Df ]
s = t, Fs Ft
[SpecFx
A(x)]
s = t, A(s) A(t)
[Cut]
s = t, Γ A(t), ∆ Γ, A(t) ∆
[Cut]
s = t, Γ ∆
[ L]
Greg Restall Generality & Existence III 23 of 43
From [=Df] to [=L]
Γ A(s), ∆ s = t s = t
[=Df ]
s = t, Fs Ft
[SpecFx
A(x)]
s = t, A(s) A(t)
[Cut]
s = t, Γ A(t), ∆ Γ, A(t) ∆
[Cut]
s = t, Γ ∆
[ L]
Greg Restall Generality & Existence III 23 of 43
From [=Df] to [=L]
Γ A(s), ∆ s = t s = t
[=Df ]
s = t, Fs Ft
[SpecFx
A(x)]
s = t, A(s) A(t)
[Cut]
s = t, Γ A(t), ∆ Γ, A(t) ∆
[Cut]
s = t, Γ ∆ Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆
Greg Restall Generality & Existence III 23 of 43
[=L]
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆ This is valid, but ugly. Proof search? Subformula property?
Greg Restall Generality & Existence III 24 of 43
[=L]
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆ This is valid, but ugly. Proof search? Subformula property?
Greg Restall Generality & Existence III 24 of 43
[=L]
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆ This is valid, but ugly. Proof search? Subformula property?
Greg Restall Generality & Existence III 24 of 43
[=L]
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆ This is valid, but ugly. Proof search? Subformula property?
Greg Restall Generality & Existence III 24 of 43
Backtracking a little
Γ A(s), ∆ s = t s = t
[=Df ]
s = t, Fs Ft
[SpecFx
A(x)]
s = t, A(s) A(t)
[Cut]
s = t, Γ A(t), ∆ Γ, A(t) ∆
[Cut]
s = t, Γ ∆
[ L ]
Greg Restall Generality & Existence III 25 of 43
Backtracking a little
Γ A(s), ∆ s = t s = t
[=Df ]
s = t, Fs Ft
[SpecFx
A(x)]
s = t, A(s) A(t)
[Cut]
s = t, Γ A(t), ∆ Γ, A(t) ∆
[Cut]
s = t, Γ ∆ Γ A(s), ∆
[=L′]
s = t, Γ A(t), ∆
Greg Restall Generality & Existence III 25 of 43
[=L′] is Enough to recover [=Df]
Γ s = t, ∆
[K]
Γ, Fs s = t, Ft, ∆
[Id]
Ft Ft
[=L′]
s = t, Fs Ft
[K]
Γ, s = t, Fs Ft, ∆
[Cut]
Γ, Fs Ft, ∆
Greg Restall Generality & Existence III 26 of 43
[=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 27 of 43
[=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 27 of 43
[=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 27 of 43
[=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 27 of 43
Decomposing [=L′]: conjunctions
Γ A(s) ∧ B(s), ∆
[∧E]
Γ A(s), ∆
[=L′]
s = t, Γ A(t), ∆ Γ A(s) ∧ B(s), ∆
[∧E]
Γ B(s), ∆
[=L′]
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 Generality & Existence III 29 of 43
Decomposing [=L′]: conjunctions
Γ A(s) ∧ B(s), ∆
[∧E]
Γ A(s), ∆
[=L′]
s = t, Γ A(t), ∆ Γ A(s) ∧ B(s), ∆
[∧E]
Γ B(s), ∆
[=L′]
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 Generality & Existence III 29 of 43
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), ∆
[W]
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.
Greg Restall Generality & Existence III 30 of 43
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), ∆
[W]
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.
Greg Restall Generality & Existence III 30 of 43
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.
Greg Restall Generality & Existence III 31 of 43
Decomposing [=L′]: existential quantifiers
[Id]
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), ∆
[Cut]
s = t, Γ (∃x)A(x, t), ∆ [=L′] on an existentially quantified statement is given by [=L′] on an instance.
Greg Restall Generality & Existence III 32 of 43
But for negation…
Γ ¬A(s), ∆
[¬Df ]
Γ, A(s) ∆
[=L′ on the wrong side!]
s = t, A(t), Γ ∆
[¬Df ]
s = t, Γ ¬A(t), ∆
Greg Restall Generality & Existence III 33 of 43
Different Identity Rules
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆
[ L ] [ R] [ L ] [ L ] [ Df ] [Spec ]
Greg Restall Generality & Existence III 34 of 43
Different Identity Rules
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆ Γ A(s), ∆
[=Lf
r]
s = t, Γ A(t), ∆
[ R] [ L ] [ L ] [ Df ] [Spec ]
Greg Restall Generality & Existence III 34 of 43
Different Identity Rules
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆ Γ A(s), ∆
[=Lf
r]
s = t, Γ A(t), ∆ Γ, Fa Fb, ∆
[=R]
Γ a = b, ∆
[ L ] [ L ] [ Df ] [Spec ]
Greg Restall Generality & Existence III 34 of 43
Different Identity Rules
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆ Γ A(s), ∆
[=Lf
r]
s = t, Γ A(t), ∆ Γ, Fa Fb, ∆
[=R]
Γ a = b, ∆ Γ Fs, ∆
[=Lp
r ]
s = t, Γ Ft, ∆
[ L ] [ Df ] [Spec ]
Greg Restall Generality & Existence III 34 of 43
Different Identity Rules
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆ Γ A(s), ∆
[=Lf
r]
s = t, Γ A(t), ∆ Γ, Fa Fb, ∆
[=R]
Γ a = b, ∆ Γ Fs, ∆
[=Lp
r ]
s = t, Γ Ft, ∆ Fs, Γ ∆
[=Lp
l ]
s = t, Ft, Γ ∆
[ Df ] [Spec ]
Greg Restall Generality & Existence III 34 of 43
Different Identity Rules
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆ Γ A(s), ∆
[=Lf
r]
s = t, Γ A(t), ∆ Γ, Fa Fb, ∆
[=R]
Γ a = b, ∆ Γ Fs, ∆
[=Lp
r ]
s = t, Γ Ft, ∆ Fs, Γ ∆
[=Lp
l ]
s = t, Ft, Γ ∆ Γ, Fa Fb, ∆ = = = = = = = = = = [=Df ] Γ a = b, ∆
[Spec ]
Greg Restall Generality & Existence III 34 of 43
Different Identity Rules
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆ Γ A(s), ∆
[=Lf
r]
s = t, Γ A(t), ∆ Γ, Fa Fb, ∆
[=R]
Γ a = b, ∆ Γ Fs, ∆
[=Lp
r ]
s = t, Γ Ft, ∆ Fs, Γ ∆
[=Lp
l ]
s = t, Ft, Γ ∆ Γ, Fa Fb, ∆ = = = = = = = = = = [=Df ] Γ a = b, ∆ Γ ∆
[SpecFx
A(x)]
Γ|Fx
A(x) ∆|Fx A(x)
Greg Restall Generality & Existence III 34 of 43
Equivalences
Γ, Fa Fb, ∆ = = = = = = = = = = [=Df ] Γ a = b, ∆ Γ ∆
[SpecFx
A(x)]
Γ|Fx
A(x) ∆|Fx A(x)
L[=Df, Spec, Cut] L/R Cut L /R Cut L /L /R Cut L /L /R Each system gives you classical first-order predicate logic with identity.
Greg Restall Generality & Existence III 35 of 43
Equivalences
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆ L[=Df, Spec, Cut] = L[=L/R, Cut] L /R Cut L /L /R Cut L /L /R Each system gives you classical first-order predicate logic with identity.
Greg Restall Generality & Existence III 35 of 43
Equivalences
Γ A(s), ∆
[=Lf
r]
s = t, Γ A(t), ∆ L[=Df, Spec, Cut] = L[=L/R, Cut] = L[=Lf
r/R, Cut]
L /L /R Cut L /L /R Each system gives you classical first-order predicate logic with identity.
Greg Restall Generality & Existence III 35 of 43
Equivalences
Γ Fs, ∆
[=Lp
r ]
s = t, Γ Ft, ∆ Fs, Γ ∆
[=Lp
l ]
s = t, Ft, Γ ∆ Γ, Fa Fb, ∆
[=R]
Γ a = b, ∆ L[=Df, Spec, Cut] = L[=L/R, Cut] = L[=Lf
r/R, Cut]
= L[=Lp
r /Lp l /R, Cut]
L /L /R Each system gives you classical first-order predicate logic with identity.
Greg Restall Generality & Existence III 35 of 43
Equivalences
Γ Fs, ∆
[=Lp
r ]
s = t, Γ Ft, ∆ Fs, Γ ∆
[=Lp
l ]
s = t, Ft, Γ ∆ Γ, Fa Fb, ∆
[=R]
Γ a = b, ∆ L[=Df, Spec, Cut] = L[=L/R, Cut] = L[=Lf
r/R, Cut]
= L[=Lp
r /Lp l /R, Cut]
= L[=Lp
r /Lp l /R]
Each system gives you classical first-order predicate logic with identity.
Greg Restall Generality & Existence III 35 of 43
Equivalences
Γ Fs, ∆
[=Lp
r ]
s = t, Γ Ft, ∆ Fs, Γ ∆
[=Lp
l ]
s = t, Ft, Γ ∆ Γ, Fa Fb, ∆
[=R]
Γ a = b, ∆ L[=Df, Spec, Cut] = L[=L/R, Cut] = L[=Lf
r/R, Cut]
= L[=Lp
r /Lp l /R, Cut]
= L[=Lp
r /Lp l /R]
Each system gives you classical first-order predicate logic with identity.
Greg Restall Generality & Existence III 35 of 43
Non-Symmetric ‘Identity’
Γ Fs, ∆
[isLp
r ]
t is s, Γ Ft, ∆ Γ, Fs Ft, ∆
[isR]
Γ t is s, ∆ There are models of this system in which is is . domain: Animal Mammal Human. atomic predicates: closed upward under . Spec: holds for atomic predicates, closed under , , , but not
.
Greg Restall Generality & Existence III 36 of 43
Non-Symmetric ‘Identity’
Γ Fs, ∆
[isLp
r ]
t is s, Γ Ft, ∆ Γ, Fs Ft, ∆
[isR]
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
.
Greg Restall Generality & Existence III 36 of 43
Non-Symmetric ‘Identity’
Γ Fs, ∆
[isLp
r ]
t is s, Γ Ft, ∆ Γ, Fs Ft, ∆
[isR]
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 ⊃.
Greg Restall Generality & Existence III 36 of 43
Free Quantification
Γ, n A(n), ∆ = = = = = = = = = = = = = [∀Df ] Γ (∀x)A(x), ∆ Γ, n, A(n) ∆ = = = = = = = = = = = = [∃Df ] Γ, (∃x)A(x) ∆
[ Df ]
Greg Restall Generality & Existence III 38 of 43
Free Quantification
Γ, n A(n), ∆ = = = = = = = = = = = = = [∀Df ] Γ (∀x)A(x), ∆ Γ, n, A(n) ∆ = = = = = = = = = = = = [∃Df ] Γ, (∃x)A(x) ∆ Γ, t ∆ = = = = = = = [↓Df ] Γ, t↓ ∆
Greg Restall Generality & Existence III 38 of 43
Free Quantification
Γ, n A(n), ∆ = = = = = = = = = = = = = [∀Df ] Γ (∀x)A(x), ∆ Γ, n, A(n) ∆ = = = = = = = = = = = = [∃Df ] Γ, (∃x)A(x) ∆ Γ, t ∆ = = = = = = = [↓Df ] Γ, t↓ ∆
(∀x)Fx ̸ Ft
Greg Restall Generality & Existence III 38 of 43
Free Quantification
Γ, n A(n), ∆ = = = = = = = = = = = = = [∀Df ] Γ (∀x)A(x), ∆ Γ, n, A(n) ∆ = = = = = = = = = = = = [∃Df ] Γ, (∃x)A(x) ∆ Γ, t ∆ = = = = = = = [↓Df ] Γ, t↓ ∆
(∀x)Fx ̸ Ft A(t) ̸ (∃x)A(x)
Greg Restall Generality & Existence III 38 of 43
Free Quantification
Γ, n A(n), ∆ = = = = = = = = = = = = = [∀Df ] Γ (∀x)A(x), ∆ Γ, n, A(n) ∆ = = = = = = = = = = = = [∃Df ] Γ, (∃x)A(x) ∆ Γ, t ∆ = = = = = = = [↓Df ] Γ, t↓ ∆
(∀x)Fx ̸ Ft A(t) ̸ (∃x)A(x) (∀x)Fx, t Ft
Greg Restall Generality & Existence III 38 of 43
Free Quantification
Γ, n A(n), ∆ = = = = = = = = = = = = = [∀Df ] Γ (∀x)A(x), ∆ Γ, n, A(n) ∆ = = = = = = = = = = = = [∃Df ] 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)
Greg Restall Generality & Existence III 38 of 43
Is Predication Existentially Committing?
ti, Γ ∆
[FL]
Ft1 · · · tn, Γ ∆
Greg Restall Generality & Existence III 39 of 43
Which Identity Rule?
non-commital: Γ, Fs Ft, ∆ = = = = = = = = = = [=nDf ] Γ s =n t, ∆ committal:
[ Df ]
Greg Restall Generality & Existence III 40 of 43
Which Identity Rule?
non-commital: Γ, Fs Ft, ∆ = = = = = = = = = = [=nDf ] Γ s =n t, ∆ committal: Γ s, ∆ Γ t, ∆ Γ, Fs Ft, ∆ = = = = = = = = = = = = = = = = = = = = = = = = = [=cDf ] Γ s =c t, ∆
Greg Restall Generality & Existence III 40 of 43
Which Identity Rule?
non-commital: Γ, Fs Ft, ∆ = = = = = = = = = = [=nDf ] Γ s =n t, ∆ committal: Γ s, ∆ Γ t, ∆ Γ, Fs Ft, ∆ = = = = = = = = = = = = = = = = = = = = = = = = = [=cDf ] Γ s =c t, ∆ t =n t
Greg Restall Generality & Existence III 40 of 43
Which Identity Rule?
non-commital: Γ, Fs Ft, ∆ = = = = = = = = = = [=nDf ] Γ s =n t, ∆ committal: Γ s, ∆ Γ t, ∆ Γ, Fs Ft, ∆ = = = = = = = = = = = = = = = = = = = = = = = = = [=cDf ] Γ s =c t, ∆ t =n t ̸ t =c t
Greg Restall Generality & Existence III 40 of 43
Which Identity Rule?
non-commital: Γ, Fs Ft, ∆ = = = = = = = = = = [=nDf ] Γ s =n t, ∆ committal: Γ s, ∆ Γ t, ∆ Γ, Fs Ft, ∆ = = = = = = = = = = = = = = = = = = = = = = = = = [=cDf ] Γ s =c t, ∆ t =n t ̸ t =c t s =c t t↓ s =c t s↓
Greg Restall Generality & Existence III 40 of 43
Non-committal identity clashes with committing predication
s s, Ft
[FL]
Fs s, Ft
[↓Df ]
Fs s↓, Ft
[¬Df ]
¬s↓, Fs Ft
[=nDf ]
¬s↓ s =n t
moral: For non-committal identity, allow to be negative as well as positive (e.g., nonexistence) so L might fail for this predicate.
Greg Restall Generality & Existence III 41 of 43
Non-committal identity clashes with committing predication
s s, Ft
[FL]
Fs s, Ft
[↓Df ]
Fs s↓, Ft
[¬Df ]
¬s↓, Fs Ft
[=nDf ]
¬s↓ s =n t
moral: For non-committal identity, allow F to be negative as well as positive (e.g., nonexistence) so FL might fail for this predicate.
Greg Restall Generality & Existence III 41 of 43
The Generality of Predication Matters
Γ, Fs Ft, ∆ = = = = = = = = = [=Df ] Γ s = t, ∆ What can be substituted for the F here makes a real difference. I’ll consider this more on this in the next talk, when I consider the interaction with modality.
Greg Restall Generality & Existence III 42 of 43
The Generality of Predication Matters
Γ, Fs Ft, ∆ = = = = = = = = = [=Df ] Γ s = t, ∆ What can be substituted for the F here makes a real difference. I’ll consider this more on this in the next talk, when I consider the interaction with modality.
Greg Restall Generality & Existence III 42 of 43
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