NormalityQualitative Normality

Quantitative NormalityConclusion

A Quantitative Logic of Normality

Corina Stroßner

25/07/2011

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Outline

1 NormalityGrammatical and Logical ClassificationIdeas and Philosophical Concepts

2 Qualitative NormalityBoutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

3 Quantitative NormalityThe Idea of Quantitative NormalityDefinitionsResults

4 Conclusion

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Grammatical and Logical ClassificationIdeas and Philosophical Concepts

What is a Normality Statement?

A statement with “normally” as sentential modifier

A statement which is weaker than universal but stronger thanparticular

A statements which allows defeasible predictions

NOTE: “It is normal that p” is not a normality statement

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Grammatical and Logical ClassificationIdeas and Philosophical Concepts

What is a Normality Statement?

A statement with “normally” as sentential modifier

A statement which is weaker than universal but stronger thanparticular

A statements which allows defeasible predictions

NOTE: “It is normal that p” is not a normality statement

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Grammatical and Logical ClassificationIdeas and Philosophical Concepts

What is a Normality Statement?

A statement with “normally” as sentential modifier

A statement which is weaker than universal but stronger thanparticular

A statements which allows defeasible predictions

NOTE: “It is normal that p” is not a normality statement

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Grammatical and Logical ClassificationIdeas and Philosophical Concepts

What is a Normality Statement?

A statement with “normally” as sentential modifier

A statement which is weaker than universal but stronger thanparticular

A statements which allows defeasible predictions

NOTE: “It is normal that p” is not a normality statement

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Grammatical and Logical ClassificationIdeas and Philosophical Concepts

Logical Classication of Normality

Generality?

Normality StatementNormality Statement Universal StatementUniversal Statement

Prediction Defeasible? Particular StatementParticular Statement

Prediction Possible? Singular StatementSingular Statement

YES

YES

YES

NO

NO

NO

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Grammatical and Logical ClassificationIdeas and Philosophical Concepts

Quality

Ignorance

Explaining by ignoring the diversity: ceteris paribus assumption(Cartwright)Keeping things as simple as possible: Circumscription(McCarthy)

Typicality

Categorization by best specimen: Prototype Semantics (Rosch,Lakoff et. al.)Typicality as linguistic minimal knowledge: Knowing themeaning by knowing stereotype (Putnam)

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Grammatical and Logical ClassificationIdeas and Philosophical Concepts

Quality

Ignorance

Explaining by ignoring the diversity: ceteris paribus assumption(Cartwright)Keeping things as simple as possible: Circumscription(McCarthy)

Typicality

Categorization by best specimen: Prototype Semantics (Rosch,Lakoff et. al.)Typicality as linguistic minimal knowledge: Knowing themeaning by knowing stereotype (Putnam)

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Grammatical and Logical ClassificationIdeas and Philosophical Concepts

Quality

Ignorance

Explaining by ignoring the diversity: ceteris paribus assumption(Cartwright)Keeping things as simple as possible: Circumscription(McCarthy)

Typicality

Categorization by best specimen: Prototype Semantics (Rosch,Lakoff et. al.)Typicality as linguistic minimal knowledge: Knowing themeaning by knowing stereotype (Putnam)

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Grammatical and Logical ClassificationIdeas and Philosophical Concepts

Quantity

Statistics

Normal distributionExpected valueStandard deviation

Majority

Laws with exceptions: hos epi to poly - in most cases(Aristotle)Quasi-universality: “almost every” ...

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Grammatical and Logical ClassificationIdeas and Philosophical Concepts

Quantity

Statistics

Normal distributionExpected valueStandard deviation

Majority

Laws with exceptions: hos epi to poly - in most cases(Aristotle)Quasi-universality: “almost every” ...

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Grammatical and Logical ClassificationIdeas and Philosophical Concepts

Quantity

Statistics

Normal distributionExpected valueStandard deviation

Majority

Laws with exceptions: hos epi to poly - in most cases(Aristotle)Quasi-universality: “almost every” ...

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Boutilier’s Definitions

A modal logic with a transitive, reflexive accessibility relation:wRv iff v is at least as normal as w .−→�A is true in w iff A is true in every v such that wRv .

←−�A is true in w iff A is true in every v such that not wRv .

�A :=−→�A ∧

←−A�.

Normality Conditional: If A then normally B

A B := �(A ⊃−→♦(A ∧ −→�(A ⊃ B)))

For every A-world there is a not less normal A-world such thatall accessible worlds fulfil: if A then B.The most normal A-worlds (if there are any) are also B-worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Boutilier’s Definitions

A modal logic with a transitive, reflexive accessibility relation:wRv iff v is at least as normal as w .−→�A is true in w iff A is true in every v such that wRv .←−�A is true in w iff A is true in every v such that not wRv .

�A :=−→�A ∧

←−A�.

Normality Conditional: If A then normally B

A B := �(A ⊃−→♦(A ∧ −→�(A ⊃ B)))

For every A-world there is a not less normal A-world such thatall accessible worlds fulfil: if A then B.The most normal A-worlds (if there are any) are also B-worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Boutilier’s Definitions

A modal logic with a transitive, reflexive accessibility relation:wRv iff v is at least as normal as w .−→�A is true in w iff A is true in every v such that wRv .←−�A is true in w iff A is true in every v such that not wRv .

�A :=−→�A ∧

←−A�.

Normality Conditional: If A then normally B

A B := �(A ⊃−→♦(A ∧ −→�(A ⊃ B)))

For every A-world there is a not less normal A-world such thatall accessible worlds fulfil: if A then B.The most normal A-worlds (if there are any) are also B-worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Boutilier’s Definitions

A modal logic with a transitive, reflexive accessibility relation:wRv iff v is at least as normal as w .−→�A is true in w iff A is true in every v such that wRv .←−�A is true in w iff A is true in every v such that not wRv .

�A :=−→�A ∧

←−A�.

Normality Conditional: If A then normally B

A B := �(A ⊃−→♦(A ∧ −→�(A ⊃ B)))

For every A-world there is a not less normal A-world such thatall accessible worlds fulfil: if A then B.The most normal A-worlds (if there are any) are also B-worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Boutilier’s Definitions

A modal logic with a transitive, reflexive accessibility relation:wRv iff v is at least as normal as w .−→�A is true in w iff A is true in every v such that wRv .←−�A is true in w iff A is true in every v such that not wRv .

�A :=−→�A ∧

←−A�.

Normality Conditional: If A then normally B

A B := �(A ⊃−→♦(A ∧ −→�(A ⊃ B)))

For every A-world there is a not less normal A-world such thatall accessible worlds fulfil: if A then B.

The most normal A-worlds (if there are any) are also B-worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Boutilier’s Definitions

A modal logic with a transitive, reflexive accessibility relation:wRv iff v is at least as normal as w .−→�A is true in w iff A is true in every v such that wRv .←−�A is true in w iff A is true in every v such that not wRv .

�A :=−→�A ∧

←−A�.

Normality Conditional: If A then normally B

A B := �(A ⊃−→♦(A ∧ −→�(A ⊃ B)))

For every A-world there is a not less normal A-world such thatall accessible worlds fulfil: if A then B.The most normal A-worlds (if there are any) are also B-worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Boutilier’s Definitions

A modal logic with a transitive, reflexive accessibility relation:wRv iff v is at least as normal as w .−→�A is true in w iff A is true in every v such that wRv .←−�A is true in w iff A is true in every v such that not wRv .

�A :=−→�A ∧

←−A�.

Normality Conditional: If A then normally B

A B := �(A ⊃−→♦(A ∧ −→�(A ⊃ B)))

For every A-world there is a not less normal A-world such thatall accessible worlds fulfil: if A then B.The most normal A-worlds (if there are any) are also B-worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Boutilier’s Results

Conjunction of Normality: A B, A C |= A B ∧ C

Antecedent strenghtening with a consequent (ASC):A B, A C |= A ∧ B C

Prediction: No defeasible inference, normality conditionals|= (A ∧ (A B)) B (“weak modus ponus”)

Preliminarity: 6|= (A ∧ (A B) ∧ ¬B) B

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Boutilier’s Results

Conjunction of Normality: A B, A C |= A B ∧ C

Antecedent strenghtening with a consequent (ASC):A B, A C |= A ∧ B C

Prediction: No defeasible inference, normality conditionals|= (A ∧ (A B)) B (“weak modus ponus”)

Preliminarity: 6|= (A ∧ (A B) ∧ ¬B) B

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Boutilier’s Results

Conjunction of Normality: A B, A C |= A B ∧ C

Antecedent strenghtening with a consequent (ASC):A B, A C |= A ∧ B C

Prediction: No defeasible inference, normality conditionals|= (A ∧ (A B)) B (“weak modus ponus”)

Preliminarity: 6|= (A ∧ (A B) ∧ ¬B) B

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Boutilier’s Results

Conjunction of Normality: A B, A C |= A B ∧ C

Antecedent strenghtening with a consequent (ASC):A B, A C |= A ∧ B C

Prediction: No defeasible inference, normality conditionals|= (A ∧ (A B)) B (“weak modus ponus”)

Preliminarity: 6|= (A ∧ (A B) ∧ ¬B) B

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Veltman’s Definitions

A dynamic framework where information changes epistemicpossibilties and expectations.

A |= B iff (after ignorance) accepting A results in theacception of B.

The expectation pattern gives for every set of worlds atransitive and reflexive order of normality.

A B results in removing ¬B-worlds from the normal worldsin the expectation pattern for A-worlds.

Presumably A is accepted iff A is true in all optimal (mostnormal not excluded) worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Veltman’s Definitions

A dynamic framework where information changes epistemicpossibilties and expectations.

A |= B iff (after ignorance) accepting A results in theacception of B.

The expectation pattern gives for every set of worlds atransitive and reflexive order of normality.

A B results in removing ¬B-worlds from the normal worldsin the expectation pattern for A-worlds.

Presumably A is accepted iff A is true in all optimal (mostnormal not excluded) worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Veltman’s Definitions

A dynamic framework where information changes epistemicpossibilties and expectations.

A |= B iff (after ignorance) accepting A results in theacception of B.

The expectation pattern gives for every set of worlds atransitive and reflexive order of normality.

A B results in removing ¬B-worlds from the normal worldsin the expectation pattern for A-worlds.

Presumably A is accepted iff A is true in all optimal (mostnormal not excluded) worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Veltman’s Definitions

A dynamic framework where information changes epistemicpossibilties and expectations.

A |= B iff (after ignorance) accepting A results in theacception of B.

The expectation pattern gives for every set of worlds atransitive and reflexive order of normality.

A B results in removing ¬B-worlds from the normal worldsin the expectation pattern for A-worlds.

Presumably A is accepted iff A is true in all optimal (mostnormal not excluded) worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Veltman’s Definitions

A dynamic framework where information changes epistemicpossibilties and expectations.

A |= B iff (after ignorance) accepting A results in theacception of B.

The expectation pattern gives for every set of worlds atransitive and reflexive order of normality.

A B results in removing ¬B-worlds from the normal worldsin the expectation pattern for A-worlds.

Presumably A is accepted iff A is true in all optimal (mostnormal not excluded) worlds.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Veltman’s Results

Conjunction of normality: A B, A C |= A B ∧ C

NO antecedent strenghtening with a consequent (ASC):A B, A C 6|= A ∧ B C

Prediction: A B, A |= presumably B

Preliminarity: A B, A, ¬B 6|= presumably B

Relevance: A C , A ∧ B ¬C |= presumably ¬C

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Veltman’s Results

Conjunction of normality: A B, A C |= A B ∧ C

NO antecedent strenghtening with a consequent (ASC):A B, A C 6|= A ∧ B C

Prediction: A B, A |= presumably B

Preliminarity: A B, A, ¬B 6|= presumably B

Relevance: A C , A ∧ B ¬C |= presumably ¬C

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Veltman’s Results

Conjunction of normality: A B, A C |= A B ∧ C

NO antecedent strenghtening with a consequent (ASC):A B, A C 6|= A ∧ B C

Prediction: A B, A |= presumably B

Preliminarity: A B, A, ¬B 6|= presumably B

Relevance: A C , A ∧ B ¬C |= presumably ¬C

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Veltman’s Results

Conjunction of normality: A B, A C |= A B ∧ C

NO antecedent strenghtening with a consequent (ASC):A B, A C 6|= A ∧ B C

Prediction: A B, A |= presumably B

Preliminarity: A B, A, ¬B 6|= presumably B

Relevance: A C , A ∧ B ¬C |= presumably ¬C

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Veltman’s Results

Conjunction of normality: A B, A C |= A B ∧ C

NO antecedent strenghtening with a consequent (ASC):A B, A C 6|= A ∧ B C

Prediction: A B, A |= presumably B

Preliminarity: A B, A, ¬B 6|= presumably B

Relevance: A C , A ∧ B ¬C |= presumably ¬C

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

General Remarks

Idea of defeasible predictions

Two kinds of qualitative normality: general order or localorder

General order (Boutilier): Some valid implications about avaried antecedent (e.g. ASC)Idealizations like ceteris paribus assumptionsLocal order (Veltman): No valid implications about a variedantecedentClassifications like stereotypes and prototypes

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

General Remarks

Idea of defeasible predictions

Two kinds of qualitative normality: general order or localorder

General order (Boutilier): Some valid implications about avaried antecedent (e.g. ASC)Idealizations like ceteris paribus assumptions

Local order (Veltman): No valid implications about a variedantecedentClassifications like stereotypes and prototypes

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

General Remarks

Idea of defeasible predictions

Two kinds of qualitative normality: general order or localorder

General order (Boutilier): Some valid implications about avaried antecedent (e.g. ASC)Idealizations like ceteris paribus assumptionsLocal order (Veltman): No valid implications about a variedantecedentClassifications like stereotypes and prototypes

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

General Remarks

Idea of defeasible predictions

Two kinds of qualitative normality: general order or localorder

General order (Boutilier): Some valid implications about avaried antecedent (e.g. ASC)Idealizations like ceteris paribus assumptionsLocal order (Veltman): No valid implications about a variedantecedentClassifications like stereotypes and prototypes

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Critical Remarks

Conjunction of normality valid in qualitative logic:Usually no statistical justification

Is there really so little quantity in normality assumptions?

Could we really accept “Normally A” when we find out“Normally not A”?How can we presume that things resemble our stereotypes,prototypes or epistemic idealziation if they are not statisticaljustified?

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Critical Remarks

Conjunction of normality valid in qualitative logic:Usually no statistical justification

Is there really so little quantity in normality assumptions?

Could we really accept “Normally A” when we find out“Normally not A”?

How can we presume that things resemble our stereotypes,prototypes or epistemic idealziation if they are not statisticaljustified?

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Critical Remarks

Conjunction of normality valid in qualitative logic:Usually no statistical justification

Is there really so little quantity in normality assumptions?

Could we really accept “Normally A” when we find out“Normally not A”?How can we presume that things resemble our stereotypes,prototypes or epistemic idealziation if they are not statisticaljustified?

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Boutilier: “Conditional Logics of Normality”Veltman: “Defaults in Update Semantics”Discussion

Critical Remarks

Conjunction of normality valid in qualitative logic:Usually no statistical justification

Is there really so little quantity in normality assumptions?

Could we really accept “Normally A” when we find out“Normally not A”?How can we presume that things resemble our stereotypes,prototypes or epistemic idealziation if they are not statisticaljustified?

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Main Ideas

Quantitative interpretation“Normally B if A” to mean / imply “Mostly B if A”

Formal approach: Determiners from generalized quantifiertheoty (GQT)

Probabilistic inference“Mostly B if A” and “A” supports “B” (more than “B”)Formal approach: Symmetrical measurement (Carnap)

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Main Ideas

Quantitative interpretation“Normally B if A” to mean / imply “Mostly B if A”Formal approach: Determiners from generalized quantifiertheoty (GQT)

Probabilistic inference“Mostly B if A” and “A” supports “B” (more than “B”)Formal approach: Symmetrical measurement (Carnap)

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Main Ideas

Quantitative interpretation“Normally B if A” to mean / imply “Mostly B if A”Formal approach: Determiners from generalized quantifiertheoty (GQT)

Probabilistic inference

“Mostly B if A” and “A” supports “B” (more than “B”)Formal approach: Symmetrical measurement (Carnap)

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Main Ideas

Quantitative interpretation“Normally B if A” to mean / imply “Mostly B if A”Formal approach: Determiners from generalized quantifiertheoty (GQT)

Probabilistic inference“Mostly B if A” and “A” supports “B” (more than “B”)

Formal approach: Symmetrical measurement (Carnap)

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Main Ideas

Quantitative interpretation“Normally B if A” to mean / imply “Mostly B if A”Formal approach: Determiners from generalized quantifiertheoty (GQT)

Probabilistic inference“Mostly B if A” and “A” supports “B” (more than “B”)Formal approach: Symmetrical measurement (Carnap)

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Main Ideas

Quantitative interpretation“Normally B if A” to mean / imply “Mostly B if A”Formal approach: Determiners from generalized quantifiertheoty (GQT)

Probabilistic inference“Mostly B if A” and “A” supports “B” (more than “B”)Formal approach: Symmetrical measurement (Carnap)

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Normality

Foundation: A propositional modal logic with universalaccessibility

Normality: VM,w (A B) = 1 iff|[A]M ∩ [B]M| > |[A]M − [B]M|, where[A]M = {v : VM,v (A) = 1}

MOST A-worlds are B-world: Determiner MOST (interpretedas “more than a half”) from generalized quantifier theory usedfor possible worlds

Variation: Stronger definition of normality statementspossible, e.g. 95%: VM,w (A B) = 1 iff|[A]M ∩ [B]M| ≥ 19|[A]M − [B]M|

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Normality

Foundation: A propositional modal logic with universalaccessibility

Normality: VM,w (A B) = 1 iff|[A]M ∩ [B]M| > |[A]M − [B]M|, where[A]M = {v : VM,v (A) = 1}MOST A-worlds are B-world: Determiner MOST (interpretedas “more than a half”) from generalized quantifier theory usedfor possible worlds

Variation: Stronger definition of normality statementspossible, e.g. 95%: VM,w (A B) = 1 iff|[A]M ∩ [B]M| ≥ 19|[A]M − [B]M|

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Normality

Foundation: A propositional modal logic with universalaccessibility

Normality: VM,w (A B) = 1 iff|[A]M ∩ [B]M| > |[A]M − [B]M|, where[A]M = {v : VM,v (A) = 1}MOST A-worlds are B-world: Determiner MOST (interpretedas “more than a half”) from generalized quantifier theory usedfor possible worlds

Variation: Stronger definition of normality statementspossible, e.g. 95%: VM,w (A B) = 1 iff|[A]M ∩ [B]M| ≥ 19|[A]M − [B]M|

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Normality

Foundation: A propositional modal logic with universalaccessibility

Normality: VM,w (A B) = 1 iff|[A]M ∩ [B]M| > |[A]M − [B]M|, where[A]M = {v : VM,v (A) = 1}MOST A-worlds are B-world: Determiner MOST (interpretedas “more than a half”) from generalized quantifier theory usedfor possible worlds

Variation: Stronger definition of normality statementspossible, e.g. 95%: VM,w (A B) = 1 iff|[A]M ∩ [B]M| ≥ 19|[A]M − [B]M|

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Probabilistic Inference

For every model M there is a set of logical possibilities LM

containing all pairs 〈l , L〉 with L ∈ POW (W ) and l ∈ L

A measure function m yields for 〈l , L〉 ∈ LM a value0 < m(〈l , L〉) ≤ 1 such that:

If LM = {〈l1, L1〉, ..., 〈ln, Ln〉} thenm(〈l1, L1〉) + ... + m(〈l1, L1〉) = 1The sum of the probability of all possibilities is 1.m(〈l1, L1〉) = m(〈l2, L2〉) iff L1 = L2

Symmetry: Isomorphic structures which do not differ incardinalities are prima facie treated eqally.The modal case: Logical possibilities which differ only inactuality are prima facie treated eqally.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Probabilistic Inference

For every model M there is a set of logical possibilities LM

containing all pairs 〈l , L〉 with L ∈ POW (W ) and l ∈ L

A measure function m yields for 〈l , L〉 ∈ LM a value0 < m(〈l , L〉) ≤ 1 such that:

If LM = {〈l1, L1〉, ..., 〈ln, Ln〉} thenm(〈l1, L1〉) + ... + m(〈l1, L1〉) = 1

The sum of the probability of all possibilities is 1.m(〈l1, L1〉) = m(〈l2, L2〉) iff L1 = L2

Symmetry: Isomorphic structures which do not differ incardinalities are prima facie treated eqally.The modal case: Logical possibilities which differ only inactuality are prima facie treated eqally.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Probabilistic Inference

For every model M there is a set of logical possibilities LM

containing all pairs 〈l , L〉 with L ∈ POW (W ) and l ∈ L

A measure function m yields for 〈l , L〉 ∈ LM a value0 < m(〈l , L〉) ≤ 1 such that:

If LM = {〈l1, L1〉, ..., 〈ln, Ln〉} thenm(〈l1, L1〉) + ... + m(〈l1, L1〉) = 1The sum of the probability of all possibilities is 1.

m(〈l1, L1〉) = m(〈l2, L2〉) iff L1 = L2

Symmetry: Isomorphic structures which do not differ incardinalities are prima facie treated eqally.The modal case: Logical possibilities which differ only inactuality are prima facie treated eqally.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Probabilistic Inference

For every model M there is a set of logical possibilities LM

containing all pairs 〈l , L〉 with L ∈ POW (W ) and l ∈ L

A measure function m yields for 〈l , L〉 ∈ LM a value0 < m(〈l , L〉) ≤ 1 such that:

If LM = {〈l1, L1〉, ..., 〈ln, Ln〉} thenm(〈l1, L1〉) + ... + m(〈l1, L1〉) = 1The sum of the probability of all possibilities is 1.m(〈l1, L1〉) = m(〈l2, L2〉) iff L1 = L2

Symmetry: Isomorphic structures which do not differ incardinalities are prima facie treated eqally.The modal case: Logical possibilities which differ only inactuality are prima facie treated eqally.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Probabilistic Inference

For every model M there is a set of logical possibilities LM

containing all pairs 〈l , L〉 with L ∈ POW (W ) and l ∈ L

A measure function m yields for 〈l , L〉 ∈ LM a value0 < m(〈l , L〉) ≤ 1 such that:

If LM = {〈l1, L1〉, ..., 〈ln, Ln〉} thenm(〈l1, L1〉) + ... + m(〈l1, L1〉) = 1The sum of the probability of all possibilities is 1.m(〈l1, L1〉) = m(〈l2, L2〉) iff L1 = L2

Symmetry: Isomorphic structures which do not differ incardinalities are prima facie treated eqally.

The modal case: Logical possibilities which differ only inactuality are prima facie treated eqally.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Probabilistic Inference

For every model M there is a set of logical possibilities LM

containing all pairs 〈l , L〉 with L ∈ POW (W ) and l ∈ L

A measure function m yields for 〈l , L〉 ∈ LM a value0 < m(〈l , L〉) ≤ 1 such that:

If LM = {〈l1, L1〉, ..., 〈ln, Ln〉} thenm(〈l1, L1〉) + ... + m(〈l1, L1〉) = 1The sum of the probability of all possibilities is 1.m(〈l1, L1〉) = m(〈l2, L2〉) iff L1 = L2

Symmetry: Isomorphic structures which do not differ incardinalities are prima facie treated eqally.The modal case: Logical possibilities which differ only inactuality are prima facie treated eqally.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Probabilistic Inference

For every model M there is a set of logical possibilities LM

containing all pairs 〈l , L〉 with L ∈ POW (W ) and l ∈ L

A measure function m yields for 〈l , L〉 ∈ LM a value0 < m(〈l , L〉) ≤ 1 such that:

If LM = {〈l1, L1〉, ..., 〈ln, Ln〉} thenm(〈l1, L1〉) + ... + m(〈l1, L1〉) = 1The sum of the probability of all possibilities is 1.m(〈l1, L1〉) = m(〈l2, L2〉) iff L1 = L2

Symmetry: Isomorphic structures which do not differ incardinalities are prima facie treated eqally.The modal case: Logical possibilities which differ only inactuality are prima facie treated eqally.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Probabilistic Inference

The measure function m yields for a sentence A a value m(A)such that: m(A) is the sum of all m(〈l , L〉 ∈ LM) such thatVML/W ,l(A) = 1 where ML/W = 〈L, I 〉 if M = 〈W , I 〉 andL ⊆WThe probability of a sentence is the probability of thepossibilities (Carnaps state descriptions) in which it is true.

A1, ...,An |=prob B iff m(A1∧...∧An∧B)m(A1∧...∧An) > 1

2 for all measurefunctions m and models MAn inference is probabilistically valid iff the conditionalprobability of the conclusion given the conjunction of premises(e.g. its confirmation by the conjunction of the premises) isgreater than a half.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Definition of Probabilistic Inference

The measure function m yields for a sentence A a value m(A)such that: m(A) is the sum of all m(〈l , L〉 ∈ LM) such thatVML/W ,l(A) = 1 where ML/W = 〈L, I 〉 if M = 〈W , I 〉 andL ⊆WThe probability of a sentence is the probability of thepossibilities (Carnaps state descriptions) in which it is true.

A1, ...,An |=prob B iff m(A1∧...∧An∧B)m(A1∧...∧An) > 1

2 for all measurefunctions m and models MAn inference is probabilistically valid iff the conditionalprobability of the conclusion given the conjunction of premises(e.g. its confirmation by the conjunction of the premises) isgreater than a half.

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Results

NO conjunction of normality: A B, A C 6|= A B ∧ C

Prediction: A B, A |= presumably B

Preliminarity: A B, A, ¬B 6|= presumably B

Relevance: A C , A ∧ B ¬C |= presumably ¬C

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Results

NO conjunction of normality: A B, A C 6|= A B ∧ C

Prediction: A B, A |= presumably B

Preliminarity: A B, A, ¬B 6|= presumably B

Relevance: A C , A ∧ B ¬C |= presumably ¬C

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Results

NO conjunction of normality: A B, A C 6|= A B ∧ C

Prediction: A B, A |= presumably B

Preliminarity: A B, A, ¬B 6|= presumably B

Relevance: A C , A ∧ B ¬C |= presumably ¬C

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

The Idea of Quantitative NormalityDefinitionsResults

Results

NO conjunction of normality: A B, A C 6|= A B ∧ C

Prediction: A B, A |= presumably B

Preliminarity: A B, A, ¬B 6|= presumably B

Relevance: A C , A ∧ B ¬C |= presumably ¬C

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Critical Discussion of Quantitative Normality

Is majority really a necessary condition for normality?Usually it is! Exceptions: Norms

Is normality really just quantity?Of course not! Majority as the logically relevant aspect

Is a quanitative logic of normality better than a qualitativeone?To a certain extent! Justification of presumption prima faciemore intuitive in a quantitative frameworkBut: Qualitative concepts also related to normality!

Is there another way to justify presumptionsYes, by pragmatic rules about what is worth mentioning incommunicationNote, that such rules are also influenced by known quanitities!

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Critical Discussion of Quantitative Normality

Is majority really a necessary condition for normality?Usually it is! Exceptions: Norms

Is normality really just quantity?

Of course not! Majority as the logically relevant aspect

Is a quanitative logic of normality better than a qualitativeone?To a certain extent! Justification of presumption prima faciemore intuitive in a quantitative frameworkBut: Qualitative concepts also related to normality!

Is there another way to justify presumptionsYes, by pragmatic rules about what is worth mentioning incommunicationNote, that such rules are also influenced by known quanitities!

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Critical Discussion of Quantitative Normality

Is majority really a necessary condition for normality?Usually it is! Exceptions: Norms

Is normality really just quantity?Of course not! Majority as the logically relevant aspect

Is a quanitative logic of normality better than a qualitativeone?To a certain extent! Justification of presumption prima faciemore intuitive in a quantitative frameworkBut: Qualitative concepts also related to normality!

Is there another way to justify presumptionsYes, by pragmatic rules about what is worth mentioning incommunicationNote, that such rules are also influenced by known quanitities!

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Critical Discussion of Quantitative Normality

Is majority really a necessary condition for normality?Usually it is! Exceptions: Norms

Is normality really just quantity?Of course not! Majority as the logically relevant aspect

Is a quanitative logic of normality better than a qualitativeone?

To a certain extent! Justification of presumption prima faciemore intuitive in a quantitative frameworkBut: Qualitative concepts also related to normality!

Is there another way to justify presumptionsYes, by pragmatic rules about what is worth mentioning incommunicationNote, that such rules are also influenced by known quanitities!

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Critical Discussion of Quantitative Normality

Is majority really a necessary condition for normality?Usually it is! Exceptions: Norms

Is normality really just quantity?Of course not! Majority as the logically relevant aspect

Is a quanitative logic of normality better than a qualitativeone?To a certain extent! Justification of presumption prima faciemore intuitive in a quantitative framework

But: Qualitative concepts also related to normality!

Is there another way to justify presumptionsYes, by pragmatic rules about what is worth mentioning incommunicationNote, that such rules are also influenced by known quanitities!

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Critical Discussion of Quantitative Normality

Is majority really a necessary condition for normality?Usually it is! Exceptions: Norms

Is normality really just quantity?Of course not! Majority as the logically relevant aspect

Is a quanitative logic of normality better than a qualitativeone?To a certain extent! Justification of presumption prima faciemore intuitive in a quantitative frameworkBut: Qualitative concepts also related to normality!

Is there another way to justify presumptionsYes, by pragmatic rules about what is worth mentioning incommunicationNote, that such rules are also influenced by known quanitities!

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Critical Discussion of Quantitative Normality

Is majority really a necessary condition for normality?Usually it is! Exceptions: Norms

Is normality really just quantity?Of course not! Majority as the logically relevant aspect

Is a quanitative logic of normality better than a qualitativeone?To a certain extent! Justification of presumption prima faciemore intuitive in a quantitative frameworkBut: Qualitative concepts also related to normality!

Is there another way to justify presumptions

Yes, by pragmatic rules about what is worth mentioning incommunicationNote, that such rules are also influenced by known quanitities!

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Critical Discussion of Quantitative Normality

Is majority really a necessary condition for normality?Usually it is! Exceptions: Norms

Is normality really just quantity?Of course not! Majority as the logically relevant aspect

Is a quanitative logic of normality better than a qualitativeone?To a certain extent! Justification of presumption prima faciemore intuitive in a quantitative frameworkBut: Qualitative concepts also related to normality!

Is there another way to justify presumptionsYes, by pragmatic rules about what is worth mentioning incommunication

Note, that such rules are also influenced by known quanitities!

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Critical Discussion of Quantitative Normality

Is majority really a necessary condition for normality?Usually it is! Exceptions: Norms

Is normality really just quantity?Of course not! Majority as the logically relevant aspect

Is a quanitative logic of normality better than a qualitativeone?To a certain extent! Justification of presumption prima faciemore intuitive in a quantitative frameworkBut: Qualitative concepts also related to normality!

Is there another way to justify presumptionsYes, by pragmatic rules about what is worth mentioning incommunicationNote, that such rules are also influenced by known quanitities!

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Critical Discussion of Quantitative Normality

Is majority really a necessary condition for normality?Usually it is! Exceptions: Norms

Is normality really just quantity?Of course not! Majority as the logically relevant aspect

Is a quanitative logic of normality better than a qualitativeone?To a certain extent! Justification of presumption prima faciemore intuitive in a quantitative frameworkBut: Qualitative concepts also related to normality!

Is there another way to justify presumptionsYes, by pragmatic rules about what is worth mentioning incommunicationNote, that such rules are also influenced by known quanitities!

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Overview on Normality Logic and Concepts

SimplificationCeteris paribus

TypicalityStereo- & Prototypes

Pragmatic Rule

Influence

CATEGORIZATIONIDENTIFICATION

ABSTRACTIONIDEALIZATION

NORMALITY

PREDICTION

Justification

QuantityStatistical Information

Influence

Corina Stroßner A Quantitative Logic of Normality

NormalityQualitative Normality

Quantitative NormalityConclusion

Literature

Boutilier, Craig: Conditional logics of normality: a modalapproach. IN: Artificial Intelligence 68 1994, 87-154

Carnap, Rudolf: Logical Foundations of Probability. TheUniversity of Chicago Press 1950

Peters, Stanley / Westerstahl, Dag: Quantifiers in Languageand Logic. Clarendon Press Oxford 2006

Veltman, Frank: Defaults in update semantics. IN: Journal ofPhilosophical Logic 25 1996, 221-261

Corina Stroßner A Quantitative Logic of Normality