Pertinence Construed Modally
-
Upload
ivan-varzinczak -
Category
Technology
-
view
423 -
download
0
description
Transcript of Pertinence Construed Modally
Pertinence Construed Modally
Arina Britz1,2 Johannes Heidema2 Ivan Varzinczak1
1Meraka Institute, CSIRPretoria, South Africa
2University of South AfricaPretoria, South Africa
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 1 / 29
A Simple Example (attributed to Russell)
Let
p: “Mars orbits the Sun”
q: “a red teapot is orbiting Mars”
In Classical Logic
¬p ∧ q |= q
¬p |= ¬p ∨ q
¬p |= >⊥ |= ¬p
|= p → (q → p)
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 2 / 29
A Simple Example (attributed to Russell)
Let
p: “Mars orbits the Sun”
q: “a red teapot is orbiting Mars”
In Classical Logic
¬p ∧ q |= q
¬p |= ¬p ∨ q
¬p |= >⊥ |= ¬p
|= p → (q → p)
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 2 / 29
A Simple Example (attributed to Russell)
Let
p: “Mars orbits the Sun”
q: “a red teapot is orbiting Mars”
In Classical Logic
¬p ∧ q |= q (disjunctive syllogism: ¬p ∧ (p ∨ q) |= q)
¬p |= ¬p ∨ q
¬p |= >⊥ |= ¬p (ex contradictione quodlibet)
|= p → (q → p) (positive paradox)
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 2 / 29
A Simple Example (attributed to Russell)
Let
p: “Mars orbits the Sun”
q: “a red teapot is orbiting Mars”
In Classical Logic
¬p ∧ q |= q (disjunctive syllogism: ¬p ∧ (p ∨ q) |= q)
¬p |= ¬p ∨ q
¬p |= >⊥ |= ¬p (ex contradictione quodlibet)
|= p → (q → p) (positive paradox)
Do we want this?
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 2 / 29
Classical Logic: the Logic of ‘Complete Ignorance’
α |= β
W
α
β
Fact
Every α-world is a β-world
β ∧ ¬α-worlds completely free and arbitraryI Nothing to do with α or any of the α-worlds
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 3 / 29
Classical Logic: the Logic of ‘Complete Ignorance’
α |= β
W
α
β
But
One intuitive connotation of entailment is that more,some notion of relevance or pertinence, should holdbetween α and β
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 3 / 29
Classical Logic: the Logic of ‘Complete Ignorance’
α |= β
W
α
β
Usually
Extra information expressed either as
Syntactic rules, or as
Semantic constraintsI Binary relation on sets of sentences
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 3 / 29
Less Attractive Features of Traditional Relevance Logics
Following Avron [1992]:
Conflation of |= with → [Anderson and Belnap, 1975, 1992]
Start with proof theory, then find a proper semantics
Moreover
Sometimes metaphysical ideas get admixed into the relevanceendeavour
Relevance logics traditionally pay scant attention to contexts
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 4 / 29
Less Attractive Features of Traditional Relevance Logics
Following Avron [1992]:
Conflation of |= with → [Anderson and Belnap, 1975, 1992]
Start with proof theory, then find a proper semantics
Moreover
Sometimes metaphysical ideas get admixed into the relevanceendeavour
Relevance logics traditionally pay scant attention to contexts
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 4 / 29
Less Attractive Features of Traditional Relevance Logics
Following Avron [1992]:
Conflation of |= with → [Anderson and Belnap, 1975, 1992]
Start with proof theory, then find a proper semantics
Moreover
Sometimes metaphysical ideas get admixed into the relevanceendeavour
Relevance logics traditionally pay scant attention to contexts
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 4 / 29
Less Attractive Features of Traditional Relevance Logics
Following Avron [1992]:
Conflation of |= with → [Anderson and Belnap, 1975, 1992]
Start with proof theory, then find a proper semantics
Moreover
Sometimes metaphysical ideas get admixed into the relevanceendeavour
Relevance logics traditionally pay scant attention to contexts
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 4 / 29
Less Attractive Features of Traditional Relevance Logics
Following Avron [1992]:
Conflation of |= with → [Anderson and Belnap, 1975, 1992]
Start with proof theory, then find a proper semantics
Moreover
Sometimes metaphysical ideas get admixed into the relevanceendeavour
Relevance logics traditionally pay scant attention to contexts
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 4 / 29
Outline
1 Logical PreliminariesModal Logic
2 Pertinent EntailmentInfra-Modal EntailmentPropertiesExamples
3 Conclusion
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 5 / 29
Outline
1 Logical PreliminariesModal Logic
2 Pertinent EntailmentInfra-Modal EntailmentPropertiesExamples
3 Conclusion
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 5 / 29
Outline
1 Logical PreliminariesModal Logic
2 Pertinent EntailmentInfra-Modal EntailmentPropertiesExamples
3 Conclusion
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 5 / 29
Outline
1 Logical PreliminariesModal Logic
2 Pertinent EntailmentInfra-Modal EntailmentPropertiesExamples
3 Conclusion
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 6 / 29
Modal Logic
Propositional modal language
Atoms: p, q, . . . and >Normal modal operator 2
Formulas: α, β, . . .
α ::= p | > | ¬α | α ∧ α | 2α
Other connectives defined as usual
3α ≡def ¬2¬αGiven 2 and 3, we can speak of their converses: 2̆ and 3̆
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 7 / 29
Modal Logic
Propositional modal language
Atoms: p, q, . . . and >Normal modal operator 2
Formulas: α, β, . . .
α ::= p | > | ¬α | α ∧ α | 2α
Other connectives defined as usual
3α ≡def ¬2¬αGiven 2 and 3, we can speak of their converses: 2̆ and 3̆
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 7 / 29
Modal Logic
Propositional modal language
Atoms: p, q, . . . and >Normal modal operator 2
Formulas: α, β, . . .
α ::= p | > | ¬α | α ∧ α | 2α
Other connectives defined as usual
3α ≡def ¬2¬αGiven 2 and 3, we can speak of their converses: 2̆ and 3̆
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 7 / 29
Modal Logic
Standard semantics
Definition
A model is a tuple M = 〈W,R,V〉, where
W is a set of worlds
R ⊆W×W is an accessibility relation on W
V : P×W −→ {0, 1} is a valuation
M :
¬p, qw2 p, q w3
¬p,¬qw1 p,¬q w4
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 8 / 29
Modal Logic
Standard semantics
Definition
A model is a tuple M = 〈W,R,V〉, where
W is a set of worlds
R ⊆W×W is an accessibility relation on W
V : P×W −→ {0, 1} is a valuation
M :
¬p, qw2 p, q w3
¬p,¬qw1 p,¬q w4
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 8 / 29
Modal Logic
Standard semantics
Definition
Given a model M = 〈W,R,V〉,w M p iff V(p,w) = 1
w M> for every w ∈W
w M¬α iff w 6 Mα
w Mα ∧ β iff w Mα and w Mβ
w M2α iff w ′ Mα for every w ′ such that (w ,w ′) ∈ R
truth conditions for the other connectives are as usual
w ′ M 2̆α iff w Mα for every w such that (w ,w ′) ∈ R
w ′ M 3̆α iff w Mα for some w such that (w ,w ′) ∈ R
If w Mα for every w ∈W, we say that α is valid in M , denoted |=Mα
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 9 / 29
Modal Logic
Standard semantics
Definition
Given a model M = 〈W,R,V〉,w M p iff V(p,w) = 1
w M> for every w ∈W
w M¬α iff w 6 Mα
w Mα ∧ β iff w Mα and w Mβ
w M2α iff w ′ Mα for every w ′ such that (w ,w ′) ∈ R
truth conditions for the other connectives are as usual
w ′ M 2̆α iff w Mα for every w such that (w ,w ′) ∈ R
w ′ M 3̆α iff w Mα for some w such that (w ,w ′) ∈ R
If w Mα for every w ∈W, we say that α is valid in M , denoted |=Mα
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 9 / 29
Modal Logic
Standard semantics
Definition
Given a model M = 〈W,R,V〉,w M p iff V(p,w) = 1
w M> for every w ∈W
w M¬α iff w 6 Mα
w Mα ∧ β iff w Mα and w Mβ
w M2α iff w ′ Mα for every w ′ such that (w ,w ′) ∈ R
truth conditions for the other connectives are as usual
w ′ M 2̆α iff w Mα for every w such that (w ,w ′) ∈ R
w ′ M 3̆α iff w Mα for some w such that (w ,w ′) ∈ R
If w Mα for every w ∈W, we say that α is valid in M , denoted |=Mα
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 9 / 29
Modal Logic
Classes of models
Sets of models we work with
Determined by additional constraints
I Axiom schemas (reflexivity, transitivity, etc.)
I Global axioms (see later)
Here we are interested in the class of reflexive models
I Given M = 〈W,R,V〉, idW ⊆ R
I Axiom schema 2α→ α
I Modal logic KT
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 10 / 29
Modal Logic
Classes of models
Sets of models we work with
Determined by additional constraints
I Axiom schemas (reflexivity, transitivity, etc.)
I Global axioms (see later)
Here we are interested in the class of reflexive models
I Given M = 〈W,R,V〉, idW ⊆ R
I Axiom schema 2α→ α
I Modal logic KT
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 10 / 29
Modal Logic
Classes of models
Sets of models we work with
Determined by additional constraints
I Axiom schemas (reflexivity, transitivity, etc.)
I Global axioms (see later)
Here we are interested in the class of reflexive models
I Given M = 〈W,R,V〉, idW ⊆ R
I Axiom schema 2α→ α
I Modal logic KT
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 10 / 29
Modal Logic
Local consequence
Definition
α entails β in M = 〈W,R,V〉 (denoted α |=Mβ) iff for every w ∈W, if
w Mα, then w Mβ.
Definition
Let C be a class of models
α entails β in C (denoted α |=Cβ) iff α |=Mβ for every M ∈ C
Validity and satisfiability in C defined as usual
When C is clear from the context, we write α |= β instead of α |=Cβ
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 11 / 29
Modal Logic
Local consequence
Definition
α entails β in M = 〈W,R,V〉 (denoted α |=Mβ) iff for every w ∈W, if
w Mα, then w Mβ.
Definition
Let C be a class of models
α entails β in C (denoted α |=Cβ) iff α |=Mβ for every M ∈ C
Validity and satisfiability in C defined as usual
When C is clear from the context, we write α |= β instead of α |=Cβ
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 11 / 29
Modal Logic
Local consequence
Definition
α entails β in M = 〈W,R,V〉 (denoted α |=Mβ) iff for every w ∈W, if
w Mα, then w Mβ.
Definition
Let C be a class of models
α entails β in C (denoted α |=Cβ) iff α |=Mβ for every M ∈ C
Validity and satisfiability in C defined as usual
When C is clear from the context, we write α |= β instead of α |=Cβ
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 11 / 29
Outline
1 Logical PreliminariesModal Logic
2 Pertinent EntailmentInfra-Modal EntailmentPropertiesExamples
3 Conclusion
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 12 / 29
The Flow of Entailment
Asymmetric, directed
Access from premiss to consequence
Entailment as ‘access’: natural analogue in the accessibility relation
However
Relevance cannot be captured by standard modalities [Meyer, 1975]
Relevance is a relation between sentences (sets of worlds), and notbetween worlds alone
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 13 / 29
The Flow of Entailment
Asymmetric, directed
Access from premiss to consequence
Entailment as ‘access’: natural analogue in the accessibility relation
However
Relevance cannot be captured by standard modalities [Meyer, 1975]
Relevance is a relation between sentences (sets of worlds), and notbetween worlds alone
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 13 / 29
The Flow of Entailment
Asymmetric, directed
Access from premiss to consequence
Entailment as ‘access’: natural analogue in the accessibility relation
However
Relevance cannot be captured by standard modalities [Meyer, 1975]
Relevance is a relation between sentences (sets of worlds), and notbetween worlds alone
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 13 / 29
The Flow of Entailment
Asymmetric, directed
Access from premiss to consequence
Entailment as ‘access’: natural analogue in the accessibility relation
However
Relevance cannot be captured by standard modalities [Meyer, 1975]
Relevance is a relation between sentences (sets of worlds), and notbetween worlds alone
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 13 / 29
Pertinence in the Meta-Level
Notion of pertinence in the meta-level
Pertinence of α and β to each other
In our new entailment of β by α, the condition that we impose upon the(previously wild) β ∧ ¬α-worlds is that now each of them must beaccessible from some α-world.
W
α
β •
•
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 14 / 29
Pertinence in the Meta-Level
Notion of pertinence in the meta-level
Pertinence of α and β to each other
In our new entailment of β by α, the condition that we impose upon the(previously wild) β ∧ ¬α-worlds is that now each of them must beaccessible from some α-world.
W
α
β •
•
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 14 / 29
Pertinence in the Meta-Level
Definition
α pertinently entails β in M (denoted α |<Mβ) iff α |=Mβ and β |=M 3̆α
Definition
α pertinently entails β in the class C of models (denoted α |<Cβ) iff for
every M ∈ C , α |<Mβ
When C is clear from the context, we write α |< β instead of α |<Cβ
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 15 / 29
Pertinence in the Meta-Level
Definition
α pertinently entails β in M (denoted α |<Mβ) iff α |=Mβ and β |=M 3̆α
Definition
α pertinently entails β in the class C of models (denoted α |<Cβ) iff for
every M ∈ C , α |<Mβ
When C is clear from the context, we write α |< β instead of α |<Cβ
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 15 / 29
Pertinence in the Meta-Level
Definition
α pertinently entails β in M (denoted α |<Mβ) iff α |=Mβ and β |=M 3̆α
Definition
α pertinently entails β in the class C of models (denoted α |<Cβ) iff for
every M ∈ C , α |<Mβ
When C is clear from the context, we write α |< β instead of α |<Cβ
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 15 / 29
Pertinence in the Meta-Level
W
α
β
3̆α
|<
• 3̆α
• β
• α
• •. . .
Clearly, |< is infra-modal: if α |< β, then α |= β
‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’
Proposition
α |< β iff α ∨ β ≡ β ∧ 3̆α
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 16 / 29
Pertinence in the Meta-Level
W
α
β
3̆α
|<
• 3̆α
• β
• α
• •. . .
Clearly, |< is infra-modal: if α |< β, then α |= β
‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’
Proposition
α |< β iff α ∨ β ≡ β ∧ 3̆α
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 16 / 29
Pertinence in the Meta-Level
W
α
β
3̆α
|<
• 3̆α
• β
• α
• •. . .
Clearly, |< is infra-modal: if α |< β, then α |= β
‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’
Proposition
α |< β iff α ∨ β ≡ β ∧ 3̆α
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 16 / 29
Pertinence in the Meta-Level
W
α
β
3̆α
|<
• 3̆α
• β
• α
• •. . .
Clearly, |< is infra-modal: if α |< β, then α |= β
‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’
Proposition
α |< β iff α ∨ β ≡ β ∧ 3̆α
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 16 / 29
Pertinence in the Meta-Level
W
α
β
3̆α
|<
• 3̆α
• β
• α
• •. . .
Clearly, |< is infra-modal: if α |< β, then α |= β
‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’
Proposition
α |< β iff α ∨ β ≡ β ∧ 3̆α
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 16 / 29
A Spectrum of Entailment Relations
Only restriction on R: idW ⊆ R ⊆W×W (modal logic KT)
The minimum (w.r.t. ⊆) case: R = idW
I maximum pertinence: |< = ≡
The maximum case: R = W×W (assume α 6≡ ⊥, cf. later)I minimum pertinence: |< = |=
Theorem
If the underlying modal logic is at least KT, then ≡ ⊆ |< ⊂ |=
Note how, psychologically speaking, with increased pertinencebetween premiss and consequence ‘if’ tends to drift in the direction of‘if and only if’ [Johnson-Laird & Savary, 1999]
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 17 / 29
A Spectrum of Entailment Relations
Only restriction on R: idW ⊆ R ⊆W×W (modal logic KT)
The minimum (w.r.t. ⊆) case: R = idW
I maximum pertinence: |< = ≡
The maximum case: R = W×W (assume α 6≡ ⊥, cf. later)I minimum pertinence: |< = |=
Theorem
If the underlying modal logic is at least KT, then ≡ ⊆ |< ⊂ |=
Note how, psychologically speaking, with increased pertinencebetween premiss and consequence ‘if’ tends to drift in the direction of‘if and only if’ [Johnson-Laird & Savary, 1999]
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 17 / 29
A Spectrum of Entailment Relations
Only restriction on R: idW ⊆ R ⊆W×W (modal logic KT)
The minimum (w.r.t. ⊆) case: R = idW
I maximum pertinence: |< = ≡
The maximum case: R = W×W (assume α 6≡ ⊥, cf. later)I minimum pertinence: |< = |=
Theorem
If the underlying modal logic is at least KT, then ≡ ⊆ |< ⊂ |=
Note how, psychologically speaking, with increased pertinencebetween premiss and consequence ‘if’ tends to drift in the direction of‘if and only if’ [Johnson-Laird & Savary, 1999]
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 17 / 29
A Spectrum of Entailment Relations
Only restriction on R: idW ⊆ R ⊆W×W (modal logic KT)
The minimum (w.r.t. ⊆) case: R = idW
I maximum pertinence: |< = ≡
The maximum case: R = W×W (assume α 6≡ ⊥, cf. later)I minimum pertinence: |< = |=
Theorem
If the underlying modal logic is at least KT, then ≡ ⊆ |< ⊂ |=
Note how, psychologically speaking, with increased pertinencebetween premiss and consequence ‘if’ tends to drift in the direction of‘if and only if’ [Johnson-Laird & Savary, 1999]
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 17 / 29
Outline
1 Logical PreliminariesModal Logic
2 Pertinent EntailmentInfra-Modal EntailmentPropertiesExamples
3 Conclusion
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 18 / 29
Properties of |<Decidability
Straightforward from definition
Non-explosiveness
falsum is not omnigenerating, in fact, only self-generating
if ⊥ |< α, then α ≡ ⊥
More generally
Theorem
Let α |<Cβ. Then if |=Cα→ ⊥, then |=Cβ → ⊥
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 19 / 29
Properties of |<Decidability
Straightforward from definition
Non-explosiveness
falsum is not omnigenerating, in fact, only self-generating
if ⊥ |< α, then α ≡ ⊥
More generally
Theorem
Let α |<Cβ. Then if |=Cα→ ⊥, then |=Cβ → ⊥
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 19 / 29
Properties of |<Decidability
Straightforward from definition
Non-explosiveness
falsum is not omnigenerating, in fact, only self-generating
if ⊥ |< α, then α ≡ ⊥
More generally
Theorem
Let α |<Cβ. Then if |=Cα→ ⊥, then |=Cβ → ⊥
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 19 / 29
Properties of |<|< is paratrivial
verum is not omnigenerated
α 6|< > in general
|< preserves valid modal formulas
Theorem
> |< α iff > |= α
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 20 / 29
Properties of |<|< is paratrivial
verum is not omnigenerated
α 6|< > in general
|< preserves valid modal formulas
Theorem
> |< α iff > |= α
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 20 / 29
Properties of |<|< rules out disjunctive syllogism
(¬α ∨ β) ∧ α |= β (equivalent to β ∧ α |= β)
β ∧ α |< β means that β ∧ α |= β and β |= 3̆(β ∧ α)
Every β-world, even if not an α-world, can be reached from someβ ∧ α-world
Does not hold in general
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 21 / 29
Properties of |<|< rules out disjunctive syllogism
(¬α ∨ β) ∧ α |= β (equivalent to β ∧ α |= β)
β ∧ α |< β means that β ∧ α |= β and β |= 3̆(β ∧ α)
Every β-world, even if not an α-world, can be reached from someβ ∧ α-world
Does not hold in general
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 21 / 29
Properties of |<|< rules out disjunctive syllogism
(¬α ∨ β) ∧ α |= β (equivalent to β ∧ α |= β)
β ∧ α |< β means that β ∧ α |= β and β |= 3̆(β ∧ α)
Every β-world, even if not an α-world, can be reached from someβ ∧ α-world
Does not hold in general
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 21 / 29
Properties of |<|< does not satisfy contraposition
Classically and modally we have contraposition:I α |= β is equivalent to ¬β |= ¬α
Not so for |<, and proof by contradiction does not hold in general
¬β |< ¬α says that α |= β and ¬α |= 3̆¬β: Every α-world is aβ-world and every ¬α-world can be reached from some ¬β-world
|< does not satisfy the deduction theorem α |< β iff > |< α→ β
(⇒) direction: OK
(⇐) direction: Fails! Let > |< α→ β, i.e., > |= α→ β andα→ β |= 3̆>. The latter is just the triviality α→ β |= >. We donot (in general) get the needed β |= 3̆α.
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 22 / 29
Properties of |<|< does not satisfy contraposition
Classically and modally we have contraposition:I α |= β is equivalent to ¬β |= ¬α
Not so for |<, and proof by contradiction does not hold in general
¬β |< ¬α says that α |= β and ¬α |= 3̆¬β: Every α-world is aβ-world and every ¬α-world can be reached from some ¬β-world
|< does not satisfy the deduction theorem α |< β iff > |< α→ β
(⇒) direction: OK
(⇐) direction: Fails! Let > |< α→ β, i.e., > |= α→ β andα→ β |= 3̆>. The latter is just the triviality α→ β |= >. We donot (in general) get the needed β |= 3̆α.
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 22 / 29
Properties of |<|< does not satisfy contraposition
Classically and modally we have contraposition:I α |= β is equivalent to ¬β |= ¬α
Not so for |<, and proof by contradiction does not hold in general
¬β |< ¬α says that α |= β and ¬α |= 3̆¬β: Every α-world is aβ-world and every ¬α-world can be reached from some ¬β-world
|< does not satisfy the deduction theorem α |< β iff > |< α→ β
(⇒) direction: OK
(⇐) direction: Fails! Let > |< α→ β, i.e., > |= α→ β andα→ β |= 3̆>. The latter is just the triviality α→ β |= >. We donot (in general) get the needed β |= 3̆α.
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 22 / 29
Properties of |<|< does not satisfy contraposition
Classically and modally we have contraposition:I α |= β is equivalent to ¬β |= ¬α
Not so for |<, and proof by contradiction does not hold in general
¬β |< ¬α says that α |= β and ¬α |= 3̆¬β: Every α-world is aβ-world and every ¬α-world can be reached from some ¬β-world
|< does not satisfy the deduction theorem α |< β iff > |< α→ β
(⇒) direction: OK
(⇐) direction: Fails! Let > |< α→ β, i.e., > |= α→ β andα→ β |= 3̆>. The latter is just the triviality α→ β |= >. We donot (in general) get the needed β |= 3̆α.
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 22 / 29
Properties of |<|< does not satisfy contraposition
Classically and modally we have contraposition:I α |= β is equivalent to ¬β |= ¬α
Not so for |<, and proof by contradiction does not hold in general
¬β |< ¬α says that α |= β and ¬α |= 3̆¬β: Every α-world is aβ-world and every ¬α-world can be reached from some ¬β-world
|< does not satisfy the deduction theorem α |< β iff > |< α→ β
(⇒) direction: OK
(⇐) direction: Fails! Let > |< α→ β, i.e., > |= α→ β andα→ β |= 3̆>. The latter is just the triviality α→ β |= >. We donot (in general) get the needed β |= 3̆α.
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 22 / 29
Pertinent Conditional
Definition
α �→ β ≡def (α→ β) ∧ (β → 3̆α)
Theorem
α |< β iff |< α �→ β
Positive paradox: α→ (β → α)
Proposition
6|< α �→ (β �→ α)
Corollary
α 6|< β �→ α
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 23 / 29
Pertinent Conditional
Definition
α �→ β ≡def (α→ β) ∧ (β → 3̆α)
Theorem
α |< β iff |< α �→ β
Positive paradox: α→ (β → α)
Proposition
6|< α �→ (β �→ α)
Corollary
α 6|< β �→ α
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 23 / 29
Pertinent Conditional
Definition
α �→ β ≡def (α→ β) ∧ (β → 3̆α)
Theorem
α |< β iff |< α �→ β
Positive paradox: α→ (β → α)
Proposition
6|< α �→ (β �→ α)
Corollary
α 6|< β �→ α
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 23 / 29
Pertinent Conditional
Definition
α �→ β ≡def (α→ β) ∧ (β → 3̆α)
Theorem
α |< β iff |< α �→ β
Positive paradox: α→ (β → α)
Proposition
6|< α �→ (β �→ α)
Corollary
α 6|< β �→ α
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 23 / 29
Properties of |<Modus Ponens
|< α, |< α→ β
|< β
Non-Monotonicity: For |<, the monotonicity rule fails:
α |< β, γ |= α
γ |< β
Substitution of Equivalents
Transitivity: If the underlying logic is at least S4
α |< β, β |< γ
α |< γ
α |< β, α |< β �→ γ
α |< γ
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 24 / 29
Properties of |<Modus Ponens
|< α, |< α→ β
|< β
Non-Monotonicity: For |<, the monotonicity rule fails:
α |< β, γ |= α
γ |< β
Substitution of Equivalents
Transitivity: If the underlying logic is at least S4
α |< β, β |< γ
α |< γ
α |< β, α |< β �→ γ
α |< γ
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 24 / 29
Properties of |<Modus Ponens
|< α, |< α→ β
|< β
Non-Monotonicity: For |<, the monotonicity rule fails:
α |< β, γ |= α
γ |< β
Substitution of Equivalents
Transitivity: If the underlying logic is at least S4
α |< β, β |< γ
α |< γ
α |< β, α |< β �→ γ
α |< γ
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 24 / 29
Properties of |<Modus Ponens
|< α, |< α→ β
|< β
Non-Monotonicity: For |<, the monotonicity rule fails:
α |< β, γ |= α
γ |< β
Substitution of Equivalents
Transitivity: If the underlying logic is at least S4
α |< β, β |< γ
α |< γ
α |< β, α |< β �→ γ
α |< γ
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 24 / 29
Outline
1 Logical PreliminariesModal Logic
2 Pertinent EntailmentInfra-Modal EntailmentPropertiesExamples
3 Conclusion
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 25 / 29
‘Paraconsistent’ Character of |<
Example
p: “Mars orbits the Sun”; q: “a red teapot is orbiting Mars”
Background assumption:p-worlds are ‘preferred’
B = {¬p → 2¬p}
M :
¬p, qw2 p, q w3
¬p,¬qw1 p,¬q w4
Premiss compatible with B: p ∧ q |< p; p |< p ∨ q; p |< >
Premiss incompatible with B: ¬p ∧3p 6|< ¬p; ⊥ 6|< p; 2p 6|< 2p ∨¬q
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 26 / 29
‘Paraconsistent’ Character of |<
Example
p: “Mars orbits the Sun”; q: “a red teapot is orbiting Mars”
Background assumption:p-worlds are ‘preferred’
B = {¬p → 2¬p}
M :
¬p, qw2 p, q w3
¬p,¬qw1 p,¬q w4
Premiss compatible with B: p ∧ q |< p; p |< p ∨ q; p |< >
Premiss incompatible with B: ¬p ∧3p 6|< ¬p; ⊥ 6|< p; 2p 6|< 2p ∨¬q
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 26 / 29
‘Paraconsistent’ Character of |<
Example
p: “Mars orbits the Sun”; q: “a red teapot is orbiting Mars”
Background assumption:p-worlds are ‘preferred’
B = {¬p → 2¬p}
M :
¬p, qw2 p, q w3
¬p,¬qw1 p,¬q w4
Premiss compatible with B: p ∧ q |< p; p |< p ∨ q; p |< >
Premiss incompatible with B: ¬p ∧3p 6|< ¬p; ⊥ 6|< p; 2p 6|< 2p ∨¬q
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 26 / 29
‘Paraconsistent’ Character of |<
Example
p: “Mars orbits the Sun”; q: “a red teapot is orbiting Mars”
Background assumption:p-worlds are ‘preferred’
B = {¬p → 2¬p}
M :
¬p, qw2 p, q w3
¬p,¬qw1 p,¬q w4
Premiss compatible with B: p ∧ q |< p; p |< p ∨ q; p |< >
Premiss incompatible with B: ¬p ∧3p 6|< ¬p; ⊥ 6|< p; 2p 6|< 2p ∨¬q
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 26 / 29
Pertinence and Causation
Example
s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking”
Background assumption:
B = {w → a, s → ¬a,3s}
M :
¬s, a,¬ww2
¬s, a,w w3
¬s,¬a,¬ww1
s,¬a,¬w w4
Question: Is α the pertinent cause of β?
¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w
a ∧2¬s 6|< a ; a ∧23s |< a
s 6|< ¬a ; s ∨ ¬a |< ¬a
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
Pertinence and Causation
Example
s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking”
Background assumption:
B = {w → a, s → ¬a,3s}
M :
¬s, a,¬ww2
¬s, a,w w3
¬s,¬a,¬ww1
s,¬a,¬w w4
Question: Is α the pertinent cause of β?
¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w
a ∧2¬s 6|< a ; a ∧23s |< a
s 6|< ¬a ; s ∨ ¬a |< ¬a
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
Pertinence and Causation
Example
s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking”
Background assumption:
B = {w → a, s → ¬a,3s}
M :
¬s, a,¬ww2
¬s, a,w w3
¬s,¬a,¬ww1
s,¬a,¬w w4
Question: Is α the pertinent cause of β?
¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w
a ∧2¬s 6|< a ; a ∧23s |< a
s 6|< ¬a ; s ∨ ¬a |< ¬a
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
Pertinence and Causation
Example
s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking”
Background assumption:
B = {w → a, s → ¬a,3s}
M :
¬s, a,¬ww2
¬s, a,w w3
¬s,¬a,¬ww1
s,¬a,¬w w4
Question: Is α the pertinent cause of β?
¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w
a ∧2¬s 6|< a ; a ∧23s |< a
s 6|< ¬a ; s ∨ ¬a |< ¬a
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
Pertinence and Causation
Example
s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking”
Background assumption:
B = {w → a, s → ¬a,3s}
M :
¬s, a,¬ww2
¬s, a,w w3
¬s,¬a,¬ww1
s,¬a,¬w w4
Question: Is α the pertinent cause of β?
¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w
a ∧2¬s 6|< a ; a ∧23s |< a
s 6|< ¬a ; s ∨ ¬a |< ¬a
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
Pertinence and Causation
Example
s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking”
Background assumption:
B = {w → a, s → ¬a,3s}
M :
¬s, a,¬ww2
¬s, a,w w3
¬s,¬a,¬ww1
s,¬a,¬w w4
Question: Is α the pertinent cause of β?
¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w
a ∧2¬s 6|< a ; a ∧23s |< a
s 6|< ¬a ; s ∨ ¬a |< ¬a
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
Pertinence and Causation
Example
s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking”
Background assumption:
B = {w → a, s → ¬a,3s}
M :
¬s, a,¬ww2
¬s, a,w w3
¬s,¬a,¬ww1
s,¬a,¬w w4
Question: Is α the pertinent cause of β?
¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w
a ∧2¬s 6|< a ; a ∧23s |< a
s 6|< ¬a ; s ∨ ¬a |< ¬a
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
Conclusion
Contributions
Semantic approach to the notion of pertinence
Pertinence captured in a simple modal logic
Whole spectrum of pertinent entailments, ranging between ≡ and |=
We restrict some paradoxes avoided by relevance logics
|< possesses other non-classical properties
Ongoing and Future Work
Other infra-modal entailment relations
Supra-modal entailment: prototypical and venturous reasoning
Relationship with contexts such as obligations, beliefs, etc
Pertinent subsumptions in Description Logics
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 28 / 29
Conclusion
Contributions
Semantic approach to the notion of pertinence
Pertinence captured in a simple modal logic
Whole spectrum of pertinent entailments, ranging between ≡ and |=
We restrict some paradoxes avoided by relevance logics
|< possesses other non-classical properties
Ongoing and Future Work
Other infra-modal entailment relations
Supra-modal entailment: prototypical and venturous reasoning
Relationship with contexts such as obligations, beliefs, etc
Pertinent subsumptions in Description Logics
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 28 / 29
Reference
K. Britz, J. Heidema, I. Varzinczak. Pertinent Reasoning. Workshopon Nonmonotonic Reasoning (NMR), 2010.
Thank you!
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 29 / 29
Reference
K. Britz, J. Heidema, I. Varzinczak. Pertinent Reasoning. Workshopon Nonmonotonic Reasoning (NMR), 2010.
Thank you!
Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 29 / 29