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)

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)

In Classical Logic

¬p ∧ q |= q

¬p |= ¬p ∨ q

¬p |= >⊥ |= ¬p

|= p → (q → p)

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)

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?

Classical Logic: the Logic of ‘Complete Ignorance’

α |= β

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

α |= β

One intuitive connotation of entailment is that more,some notion of relevance or pertinence, should holdbetween α and β

α |= β

Usually

Extra information expressed either as

Syntactic rules, or as

Semantic constraintsI Binary relation on sets of sentences

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

Moreover

Outline

1 Logical PreliminariesModal Logic

2 Pertinent EntailmentInfra-Modal EntailmentPropertiesExamples

3 Conclusion

Outline

3 Conclusion

Outline

3 Conclusion

Outline

3 Conclusion

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

Modal Logic

α ::= p | > | ¬α | α ∧ α | 2α

Modal Logic

α ::= p | > | ¬α | α ∧ α | 2α

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

¬p, qw2 p, q w3

¬p,¬qw1 p,¬q w4

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

¬p, qw2 p, q w3

¬p,¬qw1 p,¬q w4

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

Modal Logic

Standard semantics

Definition

w M¬α iff w 6 Mα

Modal Logic

Standard semantics

Definition

w M¬α iff w 6 Mα

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

Modal Logic

Classes of models

I Modal logic KT

Modal Logic

Classes of models

I Modal logic KT

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

Modal Logic

Local consequence

Definition

w Mα, then w Mβ.

Definition

Modal Logic

Local consequence

Definition

w Mα, then w Mβ.

Definition

Outline

3 Conclusion

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

However

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.

β •

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.

β •

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

Definition

• 3̆α

• β

• α

• •. . .

Clearly, |< is infra-modal: if α |< β, then α |= β

‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’

Proposition

α |< β iff α ∨ β ≡ β ∧ 3̆α

• 3̆α

• β

• α

• •. . .

‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’

Proposition

α |< β iff α ∨ β ≡ β ∧ 3̆α

• 3̆α

• β

• α

• •. . .

‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’

Proposition

α |< β iff α ∨ β ≡ β ∧ 3̆α

• 3̆α

• β

• α

• •. . .

‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’

Proposition

α |< β iff α ∨ β ≡ β ∧ 3̆α

• 3̆α

• β

• α

• •. . .

‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’

Proposition

α |< β iff α ∨ β ≡ β ∧ 3̆α

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]

Theorem

Outline

3 Conclusion

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 → ⊥

Non-explosiveness

if ⊥ |< α, then α ≡ ⊥

More generally

Theorem

Non-explosiveness

if ⊥ |< α, then α ≡ ⊥

More generally

Theorem

Properties of |<|< is paratrivial

verum is not omnigenerated

α 6|< > in general

|< preserves valid modal formulas

Theorem

> |< α iff > |= α

Properties of |<|< is paratrivial

verum is not omnigenerated

α 6|< > in general

|< preserves valid modal formulas

Theorem

> |< α iff > |= α

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

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: OK

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

Definition

α �→ β ≡def (α→ β) ∧ (β → 3̆α)

Theorem

α |< β iff |< α �→ β

Proposition

6|< α �→ (β �→ α)

Corollary

α 6|< β �→ α

Definition

α �→ β ≡def (α→ β) ∧ (β → 3̆α)

Theorem

α |< β iff |< α �→ β

Proposition

6|< α �→ (β �→ α)

Corollary

α 6|< β �→ α

Definition

α �→ β ≡def (α→ β) ∧ (β → 3̆α)

Theorem

α |< β iff |< α �→ β

Proposition

6|< α �→ (β �→ α)

Corollary

α 6|< β �→ α

Properties of |<Modus Ponens

|< α, |< α→ 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

|< α, |< α→ β

α |< β, γ |= α

γ |< β

α |< β, β |< γ

α |< γ

α |< β, α |< β �→ γ

α |< γ

|< α, |< α→ β

α |< β, γ |= α

γ |< β

α |< β, β |< γ

α |< γ

α |< β, α |< β �→ γ

α |< γ

|< α, |< α→ β

α |< β, γ |= α

γ |< β

α |< β, β |< γ

α |< γ

α |< β, α |< β �→ γ

α |< γ

Outline

3 Conclusion

‘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}

¬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

Example

B = {¬p → 2¬p}

¬p, qw2 p, q w3

¬p,¬qw1 p,¬q w4

Example

B = {¬p → 2¬p}

¬p, qw2 p, q w3

¬p,¬qw1 p,¬q w4

Example

B = {¬p → 2¬p}

¬p, qw2 p, q w3

¬p,¬qw1 p,¬q w4

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}

¬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

Example

B = {w → a, s → ¬a,3s}

¬s, a,¬ww2

¬s, a,w w3

¬s,¬a,¬ww1

s,¬a,¬w w4

¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w

a ∧2¬s 6|< a ; a ∧23s |< a

s 6|< ¬a ; s ∨ ¬a |< ¬a

Example

B = {w → a, s → ¬a,3s}

¬s, a,¬ww2

¬s, a,w w3

¬s,¬a,¬ww1

s,¬a,¬w w4

¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w

a ∧2¬s 6|< a ; a ∧23s |< a

s 6|< ¬a ; s ∨ ¬a |< ¬a

Example

B = {w → a, s → ¬a,3s}

¬s, a,¬ww2

¬s, a,w w3

¬s,¬a,¬ww1

s,¬a,¬w w4

¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w

a ∧2¬s 6|< a ; a ∧23s |< a

s 6|< ¬a ; s ∨ ¬a |< ¬a

Example

B = {w → a, s → ¬a,3s}

¬s, a,¬ww2

¬s, a,w w3

¬s,¬a,¬ww1

s,¬a,¬w w4

¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w

a ∧2¬s 6|< a ; a ∧23s |< a

s 6|< ¬a ; s ∨ ¬a |< ¬a

Example

B = {w → a, s → ¬a,3s}

¬s, a,¬ww2

¬s, a,w w3

¬s,¬a,¬ww1

s,¬a,¬w w4

¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w

a ∧2¬s 6|< a ; a ∧23s |< a

s 6|< ¬a ; s ∨ ¬a |< ¬a

Example

B = {w → a, s → ¬a,3s}

¬s, a,¬ww2

¬s, a,w w3

¬s,¬a,¬ww1

s,¬a,¬w w4

¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w 6|< ¬w

a ∧2¬s 6|< a ; a ∧23s |< a

s 6|< ¬a ; s ∨ ¬a |< ¬a

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

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

Reference

K. Britz, J. Heidema, I. Varzinczak. Pertinent Reasoning. Workshopon Nonmonotonic Reasoning (NMR), 2010.

Thank you!

Reference

K. Britz, J. Heidema, I. Varzinczak. Pertinent Reasoning. Workshopon Nonmonotonic Reasoning (NMR), 2010.

Thank you!

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