DefiningRules, Proofsand Counterexamples Greg Restall vii workshop - - PowerPoint PPT Presentation
DefiningRules, Proofsand Counterexamples Greg Restall vii workshop on philosophical logic buenos aires august 3, 2018 My Aim To present an account of defining rules , with the aim of explaining these rules they play a central role in
DefiningRules, Proofsand Counterexamples
Greg Restall
vii workshop on philosophical logic · buenos aires · august 3, 2018
My Aim
To present an account of defining rules, with the aim of explaining these rules they play a central role in analytic proofs.
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My Aim
Along the way, I’ll explain how Kreisel’s squeezing argument helps us understand the connection between an informal notion of validity and the notions formalised in our accounts of proofs and models, and the relationship between proof-theoretic and model-theoretic analyses of logical consequence.
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Outline
Positions and Bounds Definitions What Proofs Are & What They Do Counterexamples & Kreisel’s Squeeze
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Positions …
Assertions and Denials [X : Y]
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… in a communicative practice Assertions and denials are moves in a practice. I can deny what you assert. I can retract an assertion or a denial. I can ‘try on’ assertion or denial hypothetically. They are connected to other speech acts, too, like imperatives, interrogatives, recognitives, observatives, etc.
Greg Restall Defining Rules, Proofs and Counterexamples 7 of 43
… in a communicative practice Assertions and denials are moves in a practice. I can deny what you assert. I can retract an assertion or a denial. I can ‘try on’ assertion or denial hypothetically. They are connected to other speech acts, too, like imperatives, interrogatives, recognitives, observatives, etc.
Greg Restall Defining Rules, Proofs and Counterexamples 7 of 43
… in a communicative practice Assertions and denials are moves in a practice. I can deny what you assert. I can retract an assertion or a denial. I can ‘try on’ assertion or denial hypothetically. They are connected to other speech acts, too, like imperatives, interrogatives, recognitives, observatives, etc.
Greg Restall Defining Rules, Proofs and Counterexamples 7 of 43
… in a communicative practice Assertions and denials are moves in a practice. I can deny what you assert. I can retract an assertion or a denial. I can ‘try on’ assertion or denial hypothetically. They are connected to other speech acts, too, like imperatives, interrogatives, recognitives, observatives, etc.
Greg Restall Defining Rules, Proofs and Counterexamples 7 of 43
… in a communicative practice Assertions and denials are moves in a practice. I can deny what you assert. I can retract an assertion or a denial. I can ‘try on’ assertion or denial hypothetically. They are connected to other speech acts, too, like imperatives, interrogatives, recognitives, observatives, etc.
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Norms for Assertion and Denial Assertions and denials take a stand (pro or con) on something. denial clashes with assertion. assertion clashes with denial.
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Yes/No Questions
Ask: p? yes no
Assert Deny
These two answers clash.
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Yes/No Questions
Ask: p? yes no
Assert Deny
These two answers clash.
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Yes/No Questions
Ask: p? yes no
Assert p Deny
These two answers clash.
Greg Restall Defining Rules, Proofs and Counterexamples 9 of 43
Yes/No Questions
Ask: p? yes no
Assert p Deny
These two answers clash.
Greg Restall Defining Rules, Proofs and Counterexamples 9 of 43
Yes/No Questions
Ask: p? yes no
Assert p Deny p
These two answers clash.
Greg Restall Defining Rules, Proofs and Counterexamples 9 of 43
Yes/No Questions
Ask: p? yes no
Assert p Deny p
These two answers clash.
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Strong and Weak Denial Not all ‘no’s have the same force.
Greg: Is Jen in the study? Greg: Jen is in the study. Lesley: No. Lesley: No. Lesley: She’s outside. Lesley: She’s either in the study or outside. Strong denial Weak denial Adds to the common ground Retracts from the common ground
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Strong and Weak Denial Not all ‘no’s have the same force.
Greg: Is Jen in the study? Greg: Jen is in the study. Lesley: No. Lesley: No. Lesley: She’s outside. Lesley: She’s either in the study or outside. Strong denial Weak denial Adds to the common ground Retracts from the common ground
Greg Restall Defining Rules, Proofs and Counterexamples 10 of 43
Strong and Weak Denial Not all ‘no’s have the same force.
Greg: Is Jen in the study? Greg: Jen is in the study. Lesley: No. Lesley: No. Lesley: She’s outside. Lesley: She’s either in the study or outside. Strong denial Weak denial Adds to the common ground Retracts from the common ground
Greg Restall Defining Rules, Proofs and Counterexamples 10 of 43
Strong and Weak Denial Not all ‘no’s have the same force.
Greg: Is Jen in the study? Greg: Jen is in the study. Lesley: No. Lesley: No. Lesley: She’s outside. Lesley: She’s either in the study or outside. Strong denial Weak denial Adds to the common ground Retracts from the common ground
Greg Restall Defining Rules, Proofs and Counterexamples 10 of 43
Strong and Weak Denial Not all ‘no’s have the same force.
Greg: Is Jen in the study? Greg: Jen is in the study. Lesley: No. Lesley: No. Lesley: She’s outside. Lesley: She’s either in the study or outside. Strong denial Weak denial Adds to the common ground Retracts from the common ground
Greg Restall Defining Rules, Proofs and Counterexamples 10 of 43
Strong and Weak Denial Not all ‘no’s have the same force.
Greg: Is Jen in the study? Greg: Jen is in the study. Lesley: No. Lesley: No. Lesley: She’s outside. Lesley: She’s either in the study or outside. Strong denial Weak denial Adds to the common ground Retracts from the common ground
Greg Restall Defining Rules, Proofs and Counterexamples 10 of 43
Strong and Weak Denial Not all ‘no’s have the same force.
Greg: Is Jen in the study? Greg: Jen is in the study. Lesley: No. Lesley: No. Lesley: She’s outside. Lesley: She’s either in the study or outside. Strong denial Weak denial Adds to the common ground Retracts from the common ground
Greg Restall Defining Rules, Proofs and Counterexamples 10 of 43
Strong and Weak Denial Not all ‘no’s have the same force.
Greg: Is Jen in the study? Greg: Jen is in the study. Lesley: No. Lesley: No. Lesley: She’s outside. Lesley: She’s either in the study or outside. Strong denial Weak denial Adds to the common ground Retracts from the common ground
Greg Restall Defining Rules, Proofs and Counterexamples 10 of 43
Strong and Weak Denial Not all ‘no’s have the same force.
Greg: Is Jen in the study? Greg: Jen is in the study. Lesley: No. Lesley: No. Lesley: She’s outside. Lesley: She’s either in the study or outside. Strong denial Weak denial Adds to the common ground Retracts from the common ground
Greg Restall Defining Rules, Proofs and Counterexamples 10 of 43
Strong and Weak Denial Not all ‘no’s have the same force.
Greg: Is Jen in the study? Greg: Jen is in the study. Lesley: No. Lesley: No. Lesley: She’s outside. Lesley: She’s either in the study or outside. Strong denial Weak denial Adds p to the common ground Retracts from the common ground
Greg Restall Defining Rules, Proofs and Counterexamples 10 of 43
Strong and Weak Denial Not all ‘no’s have the same force.
Greg: Is Jen in the study? Greg: Jen is in the study. Lesley: No. Lesley: No. Lesley: She’s outside. Lesley: She’s either in the study or outside. Strong denial Weak denial Adds p to the common ground Retracts p from the common ground
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Weak Assertion?
Perhaps p.
Retracts from the common ground on the negative side?
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Weak Assertion?
Perhaps p.
Retracts p from the common ground on the negative side?
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Bounds for Positions
▶ identity: [A : A] is out of bounds.
weakening: If is out of bounds, then and are also out of bounds. cut: If and are out of bounds, then so is . A position that is out of bounds is overcommitted.
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Bounds for Positions
▶ identity: [A : A] is out of bounds. ▶ weakening: If [X : Y] is out of bounds, then [X, A : Y] and
[X : A, Y] are also out of bounds. cut: If and are out of bounds, then so is . A position that is out of bounds is overcommitted.
Greg Restall Defining Rules, Proofs and Counterexamples 12 of 43
Bounds for Positions
▶ identity: [A : A] is out of bounds. ▶ weakening: If [X : Y] is out of bounds, then [X, A : Y] and
[X : A, Y] are also out of bounds.
▶ cut: If [X, A : Y] and [X : A, Y] are out of bounds, then so is [X : Y].
A position that is out of bounds is overcommitted.
Greg Restall Defining Rules, Proofs and Counterexamples 12 of 43
Bounds for Positions
▶ identity: [A : A] is out of bounds. ▶ weakening: If [X : Y] is out of bounds, then [X, A : Y] and
[X : A, Y] are also out of bounds.
▶ cut: If [X, A : Y] and [X : A, Y] are out of bounds, then so is [X : Y]. ▶ A position that is out of bounds is overcommitted.
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On Cut Suppose is not out of bounds. Suppose is out of bounds. Ask the question: ? The answer no is forced, as a yes answer is excluded (given our other commitments).
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On Cut Suppose [X : Y] is not out of bounds. Suppose is out of bounds. Ask the question: ? The answer no is forced, as a yes answer is excluded (given our other commitments).
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On Cut Suppose [X : Y] is not out of bounds. Suppose [X, A : Y] is out of bounds. Ask the question: ? The answer no is forced, as a yes answer is excluded (given our other commitments).
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On Cut Suppose [X : Y] is not out of bounds. Suppose [X, A : Y] is out of bounds. Ask the question: A? The answer no is forced, as a yes answer is excluded (given our other commitments).
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On Cut Suppose [X : Y] is not out of bounds. Suppose [X, A : Y] is out of bounds. Ask the question: A? The answer no is forced, as a yes answer is excluded (given our other commitments).
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Structural Rules A A Id X A, Y X′, A Y ′
Cut
X, X′ Y, Y ′ X Y
K
X A, Y X Y
K
X, A Y X A, A, Y
W
X A, Y X, A, A Y
W
X, A Y
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The Power of Bounds: Comparatives
Fs s >F t s ⩾F t
strong transitivity: weak transitivity: strong irreflexivity: weak reflexivity: contraries: subcontraries: strength: preservation:
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The Power of Bounds: Comparatives
Fs s >F t s ⩾F t
strong transitivity: s >F t, t >F u s >F u weak transitivity: strong irreflexivity: weak reflexivity: contraries: subcontraries: strength: preservation:
Greg Restall Defining Rules, Proofs and Counterexamples 15 of 43
The Power of Bounds: Comparatives
Fs s >F t s ⩾F t
strong transitivity: s >F t, t >F u s >F u weak transitivity: s ⩾F t, t ⩾F u s ⩾F u strong irreflexivity: weak reflexivity: contraries: subcontraries: strength: preservation:
Greg Restall Defining Rules, Proofs and Counterexamples 15 of 43
The Power of Bounds: Comparatives
Fs s >F t s ⩾F t
strong transitivity: s >F t, t >F u s >F u weak transitivity: s ⩾F t, t ⩾F u s ⩾F u strong irreflexivity: s >F s weak reflexivity: contraries: subcontraries: strength: preservation:
Greg Restall Defining Rules, Proofs and Counterexamples 15 of 43
The Power of Bounds: Comparatives
Fs s >F t s ⩾F t
strong transitivity: s >F t, t >F u s >F u weak transitivity: s ⩾F t, t ⩾F u s ⩾F u strong irreflexivity: s >F s weak reflexivity: s ⩾F s contraries: subcontraries: strength: preservation:
Greg Restall Defining Rules, Proofs and Counterexamples 15 of 43
The Power of Bounds: Comparatives
Fs s >F t s ⩾F t
strong transitivity: s >F t, t >F u s >F u weak transitivity: s ⩾F t, t ⩾F u s ⩾F u strong irreflexivity: s >F s weak reflexivity: s ⩾F s contraries: s >F t, t ⩾F s subcontraries: strength: preservation:
Greg Restall Defining Rules, Proofs and Counterexamples 15 of 43
The Power of Bounds: Comparatives
Fs s >F t s ⩾F t
strong transitivity: s >F t, t >F u s >F u weak transitivity: s ⩾F t, t ⩾F u s ⩾F u strong irreflexivity: s >F s weak reflexivity: s ⩾F s contraries: s >F t, t ⩾F s subcontraries: s >F t, t ⩾F s strength: preservation:
Greg Restall Defining Rules, Proofs and Counterexamples 15 of 43
The Power of Bounds: Comparatives
Fs s >F t s ⩾F t
strong transitivity: s >F t, t >F u s >F u weak transitivity: s ⩾F t, t ⩾F u s ⩾F u strong irreflexivity: s >F s weak reflexivity: s ⩾F s contraries: s >F t, t ⩾F s subcontraries: s >F t, t ⩾F s strength: s >F t s ⩾F t preservation:
Greg Restall Defining Rules, Proofs and Counterexamples 15 of 43
The Power of Bounds: Comparatives
Fs s >F t s ⩾F t
strong transitivity: s >F t, t >F u s >F u weak transitivity: s ⩾F t, t ⩾F u s ⩾F u strong irreflexivity: s >F s weak reflexivity: s ⩾F s contraries: s >F t, t ⩾F s subcontraries: s >F t, t ⩾F s strength: s >F t s ⩾F t preservation: Fs, t ⩾F s Ft
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How do you define a concept?
By showing people how to use it.
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Explicit Definition
Define a concept by showing how you can compose that concept out of more primitive concepts. isasquare
df
isarectangle all sidesof are equal inlength. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions.
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Explicit Definition
Define a concept by showing how you can compose that concept out of more primitive concepts. x isasquare =df x isarectangle ∧ all sidesof x are equal inlength. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions.
Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43
Explicit Definition
Define a concept by showing how you can compose that concept out of more primitive concepts. x isasquare =df x isarectangle ∧ all sidesof x are equal inlength. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions.
Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43
Explicit Definition
Define a concept by showing how you can compose that concept out of more primitive concepts. x isasquare =df x isarectangle ∧ all sidesof x are equal inlength. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions.
Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43
Explicit Definition
Define a concept by showing how you can compose that concept out of more primitive concepts. x isasquare =df x isarectangle ∧ all sidesof x are equal inlength. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions.
Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43
Explicit Definition
Define a concept by showing how you can compose that concept out of more primitive concepts. x isasquare =df x isarectangle ∧ all sidesof x are equal inlength. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions.
Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43
Explicit Definition
Define a concept by showing how you can compose that concept out of more primitive concepts. x isasquare =df x isarectangle ∧ all sidesof x are equal inlength. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions.
Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43
Definition through a ruleforuse
[X, A ⊗ B : Y] is out of bounds if and only if [X, A, B : Y] is out of bounds
Df
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Definition through a ruleforuse
[X, A ⊗ B : Y] is out of bounds if and only if [X, A, B : Y] is out of bounds X, A, B Y = = = = = = = = = = ⊗Df X, A ⊗ B Y
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What about when to deny a conjunction?
When do we have X A ⊗ B, Y?
Id Df Cut Cut
So, we have
R
Greg Restall Defining Rules, Proofs and Counterexamples 20 of 43
What about when to deny a conjunction?
When do we have X A ⊗ B, Y?
X A, Y X′ B, Y ′
Id
A ⊗ B A ⊗ B
⊗Df
A, B A ⊗ B
Cut
X′, A A ⊗ B, Y ′
Cut
X, X′ A ⊗ B, Y, Y ′ So, we have
R
Greg Restall Defining Rules, Proofs and Counterexamples 20 of 43
What about when to deny a conjunction?
When do we have X A ⊗ B, Y?
X A, Y X′ B, Y ′
Id
A ⊗ B A ⊗ B
⊗Df
A, B A ⊗ B
Cut
X′, A A ⊗ B, Y ′
Cut
X, X′ A ⊗ B, Y, Y ′ So, we have X A, Y X′ B, Y ′
⊗R
X, X′ A ⊗ B, Y, Y ′
Greg Restall Defining Rules, Proofs and Counterexamples 20 of 43
What we've done We have given norms governing ⊗ judgements in terms of norms governing simpler judgements.
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Definitions for other logical concepts
X A, Y = = = = = = = = ¬Df X, ¬A Y X, A B, Y = = = = = = = = = = = →Df X A → B, Y X A, B, Y = = = = = = = = = = ⊕Df X A ⊕ B, Y
Df Df Df Df Df
(Where and are not present in and .)
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Definitions for other logical concepts
X A, Y = = = = = = = = ¬Df X, ¬A Y X, A B, Y = = = = = = = = = = = →Df X A → B, Y X A, B, Y = = = = = = = = = = ⊕Df X A ⊕ B, Y X A, Y X B, Y = = = = = = = = = = = = = = = ∧Df X A ∧ B, Y X, A Y X, B Y = = = = = = = = = = = = = = = ∨Df X, A ∨ B Y
Df Df Df
(Where and are not present in and .)
Greg Restall Defining Rules, Proofs and Counterexamples 22 of 43
Definitions for other logical concepts
X A, Y = = = = = = = = ¬Df X, ¬A Y X, A B, Y = = = = = = = = = = = →Df X A → B, Y X A, B, Y = = = = = = = = = = ⊕Df X A ⊕ B, Y X A, Y X B, Y = = = = = = = = = = = = = = = ∧Df X A ∧ B, Y X, A Y X, B Y = = = = = = = = = = = = = = = ∨Df X, A ∨ B Y X A|x
n, Y
= = = = = = = = = = ∀Df X (∀x)A, Y X, A|x
n Y
= = = = = = = = = = ∀Df X, (∃x)A Y X, Fs Ft, Y = = = = = = = = = = =Df X s = t, Y
(Where n and F are not present in X and Y.)
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How does this work? How do concepts defined in this way work?
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Transforming Systems of Rules ∗Df + Cut + Id ⇔ ∗L/R + Cut + Id
Df L/R
L/R + Cut + Id L/R + Cut
Id Elimination
L/R + Cut L/R
Cut Elimination
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Transforming Systems of Rules ∗Df + Cut + Id ⇔ ∗L/R + Cut + Id
∗Df ↔ ∗L/R
L/R + Cut + Id L/R + Cut
Id Elimination
L/R + Cut L/R
Cut Elimination
Greg Restall Defining Rules, Proofs and Counterexamples 24 of 43
Transforming Systems of Rules ∗Df + Cut + Id ⇔ ∗L/R + Cut + Id
∗Df ↔ ∗L/R
∗L/R + Cut + Id ⇔ ∗L/R + Cut
Id Elimination
L/R + Cut L/R
Cut Elimination
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Transforming Systems of Rules ∗Df + Cut + Id ⇔ ∗L/R + Cut + Id
∗Df ↔ ∗L/R
∗L/R + Cut + Id ⇔ ∗L/R + Cut
Id Elimination
L/R + Cut L/R
Cut Elimination
Greg Restall Defining Rules, Proofs and Counterexamples 24 of 43
Transforming Systems of Rules ∗Df + Cut + Id ⇔ ∗L/R + Cut + Id
∗Df ↔ ∗L/R
∗L/R + Cut + Id ⇔ ∗L/R + Cut
Id Elimination
∗L/R + Cut ⇔ ∗L/R
Cut Elimination
Greg Restall Defining Rules, Proofs and Counterexamples 24 of 43
Transforming Systems of Rules ∗Df + Cut + Id ⇔ ∗L/R + Cut + Id
∗Df ↔ ∗L/R
∗L/R + Cut + Id ⇔ ∗L/R + Cut
Id Elimination
∗L/R + Cut ⇔ ∗L/R
Cut Elimination
Greg Restall Defining Rules, Proofs and Counterexamples 24 of 43
Concepts defined in this way…
▶ Are uniquely defined. (If you and I use the same rule, we define the same
concept.) Are conservatively extending. (Adding a logical concept to your vocabulary in this way doesn’t constrain the bounds in the original language.) Play useful dialogical roles. (You can do things with these concepts that you cannot do without. Denying a conjunction does something different to denying the conjuncts.) Are subject matter neutral. (They work wherever you assert and deny—and have singular terms and predicates.) In Brandom’s terms, they make explicit some of what was implicit in the practice of assertion and denial.
Greg Restall Defining Rules, Proofs and Counterexamples 25 of 43
Concepts defined in this way…
▶ Are uniquely defined. (If you and I use the same rule, we define the same
concept.)
▶ Are conservatively extending. (Adding a logical concept to your vocabulary
in this way doesn’t constrain the bounds in the original language.) Play useful dialogical roles. (You can do things with these concepts that you cannot do without. Denying a conjunction does something different to denying the conjuncts.) Are subject matter neutral. (They work wherever you assert and deny—and have singular terms and predicates.) In Brandom’s terms, they make explicit some of what was implicit in the practice of assertion and denial.
Greg Restall Defining Rules, Proofs and Counterexamples 25 of 43
Concepts defined in this way…
▶ Are uniquely defined. (If you and I use the same rule, we define the same
concept.)
▶ Are conservatively extending. (Adding a logical concept to your vocabulary
in this way doesn’t constrain the bounds in the original language.)
▶ Play useful dialogical roles. (You can do things with these concepts that you
cannot do without. Denying a conjunction does something different to denying the conjuncts.) Are subject matter neutral. (They work wherever you assert and deny—and have singular terms and predicates.) In Brandom’s terms, they make explicit some of what was implicit in the practice of assertion and denial.
Greg Restall Defining Rules, Proofs and Counterexamples 25 of 43
Concepts defined in this way…
▶ Are uniquely defined. (If you and I use the same rule, we define the same
concept.)
▶ Are conservatively extending. (Adding a logical concept to your vocabulary
in this way doesn’t constrain the bounds in the original language.)
▶ Play useful dialogical roles. (You can do things with these concepts that you
cannot do without. Denying a conjunction does something different to denying the conjuncts.)
▶ Are subject matter neutral. (They work wherever you assert and deny—and
have singular terms and predicates.) In Brandom’s terms, they make explicit some of what was implicit in the practice of assertion and denial.
Greg Restall Defining Rules, Proofs and Counterexamples 25 of 43
Concepts defined in this way…
▶ Are uniquely defined. (If you and I use the same rule, we define the same
concept.)
▶ Are conservatively extending. (Adding a logical concept to your vocabulary
in this way doesn’t constrain the bounds in the original language.)
▶ Play useful dialogical roles. (You can do things with these concepts that you
cannot do without. Denying a conjunction does something different to denying the conjuncts.)
▶ Are subject matter neutral. (They work wherever you assert and deny—and
have singular terms and predicates.)
▶ In Brandom’s terms, they make explicit some of what was implicit in the
practice of assertion and denial.
Greg Restall Defining Rules, Proofs and Counterexamples 25 of 43
A Tiny Proof
If it’s Thursday, I’m in Melbourne. It’s Thursday. Therefore, I’m in Melbourne.
Id Df
It’s Thursday I’m in Melbourne It’s Thursday I’m in Melbourne
(This is out of bounds.)
Greg Restall Defining Rules, Proofs and Counterexamples 27 of 43
A Tiny Proof
If it’s Thursday, I’m in Melbourne. It’s Thursday. Therefore, I’m in Melbourne.
Id
A → B A → B
→Df
A → B, A B It’s Thursday I’m in Melbourne It’s Thursday I’m in Melbourne
(This is out of bounds.)
Greg Restall Defining Rules, Proofs and Counterexamples 27 of 43
A Tiny Proof
If it’s Thursday, I’m in Melbourne. It’s Thursday. Therefore, I’m in Melbourne.
Id
A → B A → B
→Df
A → B, A B [It’s Thursday → I’m in Melbourne, It’s Thursday : I’m in Melbourne]
(This is out of bounds.)
Greg Restall Defining Rules, Proofs and Counterexamples 27 of 43
The Undeniable
Take a context in which I’ve asserted it’s Thursday → I’m in Melbourne and I’ve asserted it’s Thursday, then I’m in Melbourne is undeniable. Adding the assertion makes explicit what was implicit before that assertion. The stance (pro or con)
Greg Restall Defining Rules, Proofs and Counterexamples 28 of 43
The Undeniable
Take a context in which I’ve asserted it’s Thursday → I’m in Melbourne and I’ve asserted it’s Thursday, then I’m in Melbourne is undeniable. Adding the assertion makes explicit what was implicit before that assertion. The stance (pro or con)
Greg Restall Defining Rules, Proofs and Counterexamples 28 of 43
The Undeniable
Take a context in which I’ve asserted it’s Thursday → I’m in Melbourne and I’ve asserted it’s Thursday, then I’m in Melbourne is undeniable. Adding the assertion makes explicit what was implicit before that assertion. The stance (pro or con)
Greg Restall Defining Rules, Proofs and Counterexamples 28 of 43
Proofs
A proof for X Y shows that the position [X : Y] is out of bounds, by way of the defining rules for the concepts involved in the proof. In this sense, proofs are analytic. They apply, given the definitions, independently
Greg Restall Defining Rules, Proofs and Counterexamples 29 of 43
Proofs
A proof for X Y shows that the position [X : Y] is out of bounds, by way of the defining rules for the concepts involved in the proof. In this sense, proofs are analytic. They apply, given the definitions, independently
Greg Restall Defining Rules, Proofs and Counterexamples 29 of 43
What Proofs Prove A proof of A, B C, D can be seen as a proof of C from [A, B : D], and a refutation of from , and more.
Greg Restall Defining Rules, Proofs and Counterexamples 30 of 43
What Proofs Prove A proof of A, B C, D can be seen as a proof of C from [A, B : D], and a refutation of A from [B : C, D], and more.
Greg Restall Defining Rules, Proofs and Counterexamples 30 of 43
Enlarging Positions X A, Y X, A Y
Cut
X Y If is not derivable then one of and is also not derivable. If is available, then so is either
Greg Restall Defining Rules, Proofs and Counterexamples 32 of 43
Enlarging Positions X A, Y X, A Y
Cut
X Y If X Y is not derivable then one of X, A Y and X A Y is also not derivable. If is available, then so is either
Greg Restall Defining Rules, Proofs and Counterexamples 32 of 43
Enlarging Positions X A, Y X, A Y
Cut
X Y If X Y is not derivable then one of X, A Y and X A Y is also not derivable. If [X : Y] is available, then so is either [X, A : Y] or [X : A, Y]
Greg Restall Defining Rules, Proofs and Counterexamples 32 of 43
Keep Going …
If [X : Y] is available, we can extend it into a partition [X′ : Y ′]
is not derivable for any finite and .
Greg Restall Defining Rules, Proofs and Counterexamples 33 of 43
Keep Going …
If [X : Y] is available, we can extend it into a partition [X′ : Y ′]
U V is not derivable for any finite U ⊆ X′ and V ⊆ Y ′.
Greg Restall Defining Rules, Proofs and Counterexamples 33 of 43
Adding Witnesses If (∃x)A is added on the left, we also add a witness A|x
n, where n is fresh
and similarly when is added on the right.
Df,W Df,W
Greg Restall Defining Rules, Proofs and Counterexamples 34 of 43
Adding Witnesses If (∃x)A is added on the left, we also add a witness A|x
n, where n is fresh
and similarly when is added on the right.
X, A|x
n, (∃x)A Y
∃Df,W
X, (∃x)A Y
Df,W
Greg Restall Defining Rules, Proofs and Counterexamples 34 of 43
Adding Witnesses If (∃x)A is added on the left, we also add a witness A|x
n, where n is fresh
and similarly when (∀x)A is added on the right.
X, A|x
n, (∃x)A Y
∃Df,W
X, (∃x)A Y X (∀x)A, A|x
n, Y
∀Df,W
X (∀x)A, Y
Greg Restall Defining Rules, Proofs and Counterexamples 34 of 43
Witnessed Limit Positions give rise to Models
iff iff , iff and . iff
. iff
. iff for each name . iff for some name . This is a model, where the true formulas are in and the false formulas are in , and whose domain is the set of names.
(Things are little more delicate when the language contains the identity predicate.)
Greg Restall Defining Rules, Proofs and Counterexamples 35 of 43
Witnessed Limit Positions give rise to Models
A ∈ X′ iff ¬A ̸∈ X′ iff ¬A ∈ Y ′, A ∧ B ∈ X′ iff A ∈ X′ and B ∈ X′. A ∨ B ∈ X′ iff A ∈ X′ or B ∈ X′. A → B ∈ X′ iff A ∈ Y ′ or B ∈ X′. (∀x)A ∈ X′ iff A|x
n ∈ X′ for each name n.
(∃x)A ∈ X′ iff A|x
n ∈ X′ for some name n.
This is a model, where the true formulas are in and the false formulas are in , and whose domain is the set of names.
(Things are little more delicate when the language contains the identity predicate.)
Greg Restall Defining Rules, Proofs and Counterexamples 35 of 43
Witnessed Limit Positions give rise to Models
A ∈ X′ iff ¬A ̸∈ X′ iff ¬A ∈ Y ′, A ∧ B ∈ X′ iff A ∈ X′ and B ∈ X′. A ∨ B ∈ X′ iff A ∈ X′ or B ∈ X′. A → B ∈ X′ iff A ∈ Y ′ or B ∈ X′. (∀x)A ∈ X′ iff A|x
n ∈ X′ for each name n.
(∃x)A ∈ X′ iff A|x
n ∈ X′ for some name n.
This is a model, where the true formulas are in X′ and the false formulas are in Y ′, and whose domain is the set of names.
(Things are little more delicate when the language contains the identity predicate.)
Greg Restall Defining Rules, Proofs and Counterexamples 35 of 43
Witnessed Limit Positions give rise to Models
A ∈ X′ iff ¬A ̸∈ X′ iff ¬A ∈ Y ′, A ∧ B ∈ X′ iff A ∈ X′ and B ∈ X′. A ∨ B ∈ X′ iff A ∈ X′ or B ∈ X′. A → B ∈ X′ iff A ∈ Y ′ or B ∈ X′. (∀x)A ∈ X′ iff A|x
n ∈ X′ for each name n.
(∃x)A ∈ X′ iff A|x
n ∈ X′ for some name n.
This is a model, where the true formulas are in X′ and the false formulas are in Y ′, and whose domain is the set of names.
(Things are little more delicate when the language contains the identity predicate.)
Greg Restall Defining Rules, Proofs and Counterexamples 35 of 43
Soundness and Completeness
X Y is derivable iff there is no model in which each member of X is true and each member of Y is false.
Greg Restall Defining Rules, Proofs and Counterexamples 36 of 43
Kreisel's Squeeze has a derivation X Y is informally valid has no countermodel has a derivation.
Greg Restall Defining Rules, Proofs and Counterexamples 37 of 43
Kreisel's Squeeze X Y has a derivation ⇓ X Y is informally valid has no countermodel has a derivation.
Greg Restall Defining Rules, Proofs and Counterexamples 37 of 43
Kreisel's Squeeze X Y has a derivation ⇓ X Y is informally valid ⇓ X Y has no countermodel has a derivation.
Greg Restall Defining Rules, Proofs and Counterexamples 37 of 43
Kreisel's Squeeze X Y has a derivation ⇓ X Y is informally valid ⇓ X Y has no countermodel ⇓ X Y has a derivation.
Greg Restall Defining Rules, Proofs and Counterexamples 37 of 43
(1) From Derivability to Informal Validity
▶ To say that X Y is informally valid means that is a clash involved in
asserting each member of X and denying each member of Y. Axiomatic sequents ( ) are informally valid in this sense. Structural rules preserve informal validity. Defining rules define the connectives/quantifiers.
Greg Restall Defining Rules, Proofs and Counterexamples 38 of 43
(1) From Derivability to Informal Validity
▶ To say that X Y is informally valid means that is a clash involved in
asserting each member of X and denying each member of Y.
▶ Axiomatic sequents (A A) are informally valid in this sense.
Structural rules preserve informal validity. Defining rules define the connectives/quantifiers.
Greg Restall Defining Rules, Proofs and Counterexamples 38 of 43
(1) From Derivability to Informal Validity
▶ To say that X Y is informally valid means that is a clash involved in
asserting each member of X and denying each member of Y.
▶ Axiomatic sequents (A A) are informally valid in this sense. ▶ Structural rules preserve informal validity.
Defining rules define the connectives/quantifiers.
Greg Restall Defining Rules, Proofs and Counterexamples 38 of 43
(1) From Derivability to Informal Validity
▶ To say that X Y is informally valid means that is a clash involved in
asserting each member of X and denying each member of Y.
▶ Axiomatic sequents (A A) are informally valid in this sense. ▶ Structural rules preserve informal validity. ▶ Defining rules define the connectives/quantifiers.
Greg Restall Defining Rules, Proofs and Counterexamples 38 of 43
(2) From Informal Validity to Absence of Countermodel
Refine our notion of informal validity: Literals ( , , etc.) are informally logically independent. We ignore logical connections between literals—we fix on informal validity in virtue of first order logical form. Given a witnessed partition position (i.e., given a model), there is no informal clash (in virtue of logical form) involved in asserting any of the literals in and denying any in . So, there is no clash involved in asserting any formulas in and denying any formulas in , by appeal to the defining rules. (This is an induction
clashes involving formulas into clashes involving subformulas.) So, a countermodel for a sequent shows how there is no clash involved in asserting each member of and denying each member of .
Greg Restall Defining Rules, Proofs and Counterexamples 39 of 43
(2) From Countermodel to Informal Invalidity
Refine our notion of informal validity: Literals ( , , etc.) are informally logically independent. We ignore logical connections between literals—we fix on informal validity in virtue of first order logical form. Given a witnessed partition position (i.e., given a model), there is no informal clash (in virtue of logical form) involved in asserting any of the literals in and denying any in . So, there is no clash involved in asserting any formulas in and denying any formulas in , by appeal to the defining rules. (This is an induction
clashes involving formulas into clashes involving subformulas.) So, a countermodel for a sequent shows how there is no clash involved in asserting each member of and denying each member of .
Greg Restall Defining Rules, Proofs and Counterexamples 39 of 43
(2) From Countermodel to Informal Invalidity
▶ Refine our notion of informal validity: Literals (Fa, Gbc, etc.) are
informally logically independent. We ignore logical connections between literals—we fix on informal validity in virtue of first order logical form. Given a witnessed partition position (i.e., given a model), there is no informal clash (in virtue of logical form) involved in asserting any of the literals in and denying any in . So, there is no clash involved in asserting any formulas in and denying any formulas in , by appeal to the defining rules. (This is an induction
clashes involving formulas into clashes involving subformulas.) So, a countermodel for a sequent shows how there is no clash involved in asserting each member of and denying each member of .
Greg Restall Defining Rules, Proofs and Counterexamples 39 of 43
(2) From Countermodel to Informal Invalidity
▶ Refine our notion of informal validity: Literals (Fa, Gbc, etc.) are
informally logically independent. We ignore logical connections between literals—we fix on informal validity in virtue of first order logical form.
▶ Given a witnessed partition position [X : Y] (i.e., given a model), there is
no informal clash (in virtue of logical form) involved in asserting any of the literals in X and denying any in Y. So, there is no clash involved in asserting any formulas in and denying any formulas in , by appeal to the defining rules. (This is an induction
clashes involving formulas into clashes involving subformulas.) So, a countermodel for a sequent shows how there is no clash involved in asserting each member of and denying each member of .
Greg Restall Defining Rules, Proofs and Counterexamples 39 of 43
(2) From Countermodel to Informal Invalidity
▶ Refine our notion of informal validity: Literals (Fa, Gbc, etc.) are
informally logically independent. We ignore logical connections between literals—we fix on informal validity in virtue of first order logical form.
▶ Given a witnessed partition position [X : Y] (i.e., given a model), there is
no informal clash (in virtue of logical form) involved in asserting any of the literals in X and denying any in Y.
▶ So, there is no clash involved in asserting any formulas in X and denying
any formulas in Y, by appeal to the defining rules. (This is an induction
clashes involving formulas into clashes involving subformulas.) So, a countermodel for a sequent shows how there is no clash involved in asserting each member of and denying each member of .
Greg Restall Defining Rules, Proofs and Counterexamples 39 of 43
(2) From Countermodel to Informal Invalidity
▶ Refine our notion of informal validity: Literals (Fa, Gbc, etc.) are
informally logically independent. We ignore logical connections between literals—we fix on informal validity in virtue of first order logical form.
▶ Given a witnessed partition position [X : Y] (i.e., given a model), there is
no informal clash (in virtue of logical form) involved in asserting any of the literals in X and denying any in Y.
▶ So, there is no clash involved in asserting any formulas in X and denying
any formulas in Y, by appeal to the defining rules. (This is an induction
clashes involving formulas into clashes involving subformulas.) So, a countermodel for a sequent shows how there is no clash involved in asserting each member of and denying each member of .
Greg Restall Defining Rules, Proofs and Counterexamples 39 of 43
(2) From Countermodel to Informal Invalidity
▶ Refine our notion of informal validity: Literals (Fa, Gbc, etc.) are
informally logically independent. We ignore logical connections between literals—we fix on informal validity in virtue of first order logical form.
▶ Given a witnessed partition position [X : Y] (i.e., given a model), there is
no informal clash (in virtue of logical form) involved in asserting any of the literals in X and denying any in Y.
▶ So, there is no clash involved in asserting any formulas in X and denying
any formulas in Y, by appeal to the defining rules. (This is an induction
clashes involving formulas into clashes involving subformulas.)
▶ So, a countermodel for a sequent shows how there is no clash involved in
asserting each member of X and denying each member of Y.
Greg Restall Defining Rules, Proofs and Counterexamples 39 of 43
(3) From Absence of Countermodel to Derivability That’s the Completeness Theorem.
Greg Restall Defining Rules, Proofs and Counterexamples 40 of 43
Kreisel's Squeeze X Y has a derivation ⇓ X Y is informally valid ⇓ X Y has no countermodel ⇓ X Y has a derivation.
Greg Restall Defining Rules, Proofs and Counterexamples 41 of 43
The Result
Informal validity (in virtue of first order logical form), for the language given by the defining rules, is first order classical logic, as given by the sequent calculus and Tarski’s models.
Greg Restall Defining Rules, Proofs and Counterexamples 42 of 43
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