Post on 06-Mar-2023
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