Generality & ExistenceIII Substitution& Identity Greg - - PowerPoint PPT Presentation
Generality & ExistenceIII Substitution& Identity Greg Restall melbourne logic workshop 11 december 2015 My Aim To analyse the quantifiers using the tools of proof theory in order to better understand existence and identity . Greg
Generality & ExistenceIII
Substitution& Identity Greg Restall
melbourne logic workshop · 11 december 2015
My Aim
To analyse the quantifiers using the tools of proof theory in order to better understand existence and identity.
Greg Restall Generality & Existence III 2 of 42
My Aim
To analyse the quantifiers using the tools of proof theory in order to better understand existence and identity.
Greg Restall Generality & Existence III 2 of 42
My Aim
To analyse the quantifiers using the tools of proof theory in order to better understand existence and identity.
Greg Restall Generality & Existence III 2 of 42
Today's Plan
Sequents & Defining Rules Identity & Indistinguishability Defining Rules & Left/Right Rules Identity & Predication Free Logic & Identity
Greg Restall Generality & Existence III 3 of 42
Sequents
Don’t assert each element of Γ and deny each element of ∆.
Greg Restall Generality & Existence III 5 of 42
Structural Rules Identity: A A Weakening: Contraction: Cut:
Structural rules govern declarative sentences as such.
Greg Restall Generality & Existence III 6 of 42
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 6 of 42
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 6 of 42
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 6 of 42
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 6 of 42
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 7 of 42
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 7 of 42
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 8 of 42
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 8 of 42
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 8 of 42
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 9 of 42
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 9 of 42
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 9 of 42
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 10 of 42
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 10 of 42
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 10 of 42
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 10 of 42
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 10 of 42
And Back
A A B B
[∧R]
A, B A ∧ B Γ, A ∧ B ∆
[Cut]
Γ, A, B ∆
Greg Restall Generality & Existence III 11 of 42
Quantifier Rules
Γ A(n), ∆ = = = = = = = = = = = = = [∀Df ] Γ (∀x)A(x), ∆ Γ, A(n) ∆ = = = = = = = = = = = = [∃Df ] Γ, (∃x)A(x) ∆
Greg Restall Generality & Existence III 12 of 42
Deductive Generality
Greg Restall Generality & Existence III 13 of 42
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 14 of 42
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 14 of 42
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 14 of 42
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 14 of 42
Identity and Harmony
Greg Restall Generality & Existence III 16 of 42
Identity Axioms
t = t
Greg Restall Generality & Existence III 17 of 42
Identity Axioms
t = t s = t ⊃ t = s s = t ⊃ (t = u ⊃ s = u)
Greg Restall Generality & Existence III 17 of 42
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 17 of 42
Identity Rules in Natural Deduction
[Fs] · · · Ft
[=I]
s = t
[ E]
Greg Restall Generality & Existence III 18 of 42
Identity Rules in Natural Deduction
[Fs] · · · Ft
[=I]
s = t s = t A(s)
[=E]
A(t)
Greg Restall Generality & Existence III 18 of 42
Defining Rule for Identity
Γ, Fs Ft, ∆ = = = = = = = = = [=Df ] Γ s = t, ∆
Greg Restall Generality & Existence III 19 of 42
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 20 of 42
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 20 of 42
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 22 of 42
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 22 of 42
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 22 of 42
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 22 of 42
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 22 of 42
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 22 of 42
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 22 of 42
[=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 23 of 42
[=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 23 of 42
[=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 23 of 42
[=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 23 of 42
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 24 of 42
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 24 of 42
[=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 25 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), ∆ 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
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 28 of 42
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 28 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), ∆
[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 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), ∆
[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 29 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.
Greg Restall Generality & Existence III 30 of 42
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 31 of 42
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 32 of 42
Different Identity Rules
Γ A(s), ∆ Γ, A(t) ∆
[=L]
s = t, Γ ∆
[ L ] [ R] [ L ] [ L ] [ Df ] [Spec ]
Greg Restall Generality & Existence III 33 of 42
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 33 of 42
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 33 of 42
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 33 of 42
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 33 of 42
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 33 of 42
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 33 of 42
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 34 of 42
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 34 of 42
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 34 of 42
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 34 of 42
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 34 of 42
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 34 of 42
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 35 of 42
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 35 of 42
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 35 of 42
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 37 of 42
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 37 of 42
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 37 of 42
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 37 of 42
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 37 of 42
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 37 of 42
Is Predication Existentially Committing?
ti, Γ ∆
[FL]
Ft1 · · · tn, Γ ∆
Greg Restall Generality & Existence III 38 of 42
Which Identity Rule?
non-commital: Γ, Fs Ft, ∆ = = = = = = = = = = [=nDf ] Γ s =n t, ∆ committal:
[ Df ]
Greg Restall Generality & Existence III 39 of 42
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 39 of 42
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 39 of 42
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 39 of 42
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 39 of 42
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 40 of 42
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 40 of 42
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 41 of 42
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 41 of 42
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