Pertinence Construed Modally Arina Britz 1 , 2 Johannes Heidema 2 - - PowerPoint PPT Presentation

Pertinence Construed Modally

Arina Britz1,2 Johannes Heidema2 Ivan Varzinczak1

1Meraka Institute, CSIR 2University of South Africa

Pretoria, South Africa http://krr.meraka.org.za

AiML 2010

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 1 / 16

A Simple Example (apology 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)

But

Some notion of relevance or pertinence, should hold between premiss and consequence

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 2 / 16

Classical Logic: the Logic of ‘Complete Ignorance’

α | = β

W α β

Usually

Extra information expressed either as Syntactic rules, or as Semantic constraints on sets of sentences Less Attractive Features of Traditional Relevance Logics [Avron, 1992] Conflation of | = with → [Anderson and Belnap, 1975, 1992] Start with proof theory, then find a proper semantics Sometimes metaphysical ideas get admixed into the relevance endeavour Relevance logics traditionally pay scant attention to contexts

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 3 / 16

Outline

1 Preliminaries 2 Pertinent Entailment 3 Conclusion

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 4 / 16

Modal Logic

Here: standard modal logics Propositional language Possible worlds semantics Classes of models Sets of models we work with Determined by additional constraints

◮ Axiom schemas (reflexivity, transitivity, etc.) ◮ Global axioms (see later)

Here we are interested in the class of reflexive models Axiom schema ✷α → α: Modal logic KT

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 5 / 16

Modal Logic

Local consequence

Definition

α entails β in M = W, R, V (denoted α | =

M β) iff for every w ∈ W, if

w

M α, then w M β.

Definition

Let C be a class of models α entails β in C (denoted α | =

C β) iff α |

=

M β 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 | =

C β

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 6 / 16

Pertinence in the Meta-Level

The consequent β should not run wild

W α β

• Definition

α pertinently entails β in M (denoted α | <

M β) iff α |

=

M β and β |

=

M ✸

˘ α

Definition

α pertinently entails β in the class C of models (denoted α | <

C β) iff for

every M ∈ C , α | <

M β

When C is clear from the context, we write α | < β instead of α | <

C β

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 7 / 16

Pertinence in the Meta-Level

W α β ✸ ˘ α

| <                   

• ✸

˘ α

• β
• α
• . . .

Clearly, | < is infra-modal: if α | < β, then α | = β ‘| <’ vs. ‘| =’ like ‘<’ vs. ‘=’

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 8 / 16

A Spectrum of Entailment Relations

Only restriction on R: idW ⊆ R ⊆ W × W (modal logic KT) The minimum (w.r.t. ⊆) case: R = idW

◮ maximum pertinence: |

< = ≡

The maximum case: R = W × W (assume α ≡ ⊥, cf. later)

◮ minimum pertinence: |

< = | =

Theorem

If the underlying modal logic is at least KT, then ≡ ⊆ | < ⊂ | = Note how, psychologically speaking, with increased pertinence between 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 AiML 2010 9 / 16

Properties of | <

Decidability

◮ Straightforward from definition

Non-explosiveness

◮ if ⊥ |

< α, then α ≡ ⊥

Theorem

Let α | <

C β. Then if |

=

C α → ⊥, then |

=

C β → ⊥

| < is paratrivial

◮ α |

< ⊤ in general

| < preserves validities

Theorem

⊤ | < α iff ⊤ | = α

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 10 / 16

Properties of | <

Disjunctive Syllogism: (¬α ∨ β) ∧ α | < β (β ∧ α | < β) | < does not satisfy contraposition | < does not satisfy the deduction theorem α | < β iff ⊤ | < α → β 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

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 11 / 16

Pertinent Conditional

Definition

α ⋄ → β ≡def (α → β) ∧ (β → ✸ ˘ α)

Theorem

α | < β iff | < α ⋄ → β

Proposition

| < α → (β → α) (positive paradox) | < α ⋄ → (β ⋄ → α) α | < β ⋄ → α

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 12 / 16

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, ✸s}

M :

¬s, a, ¬w w2 ¬s, a, w w3 ¬s, ¬a, ¬w w1 s, ¬a, ¬w w4

Question: Is α the pertinent cause of β? ¬a ∧ ¬w | < ¬a ; ¬a ∧ ¬w | < ¬w a ∧ ✷¬s | < a ; a ∧ ✷✸s | < a s | < ¬a ; s ∨ ¬a | < ¬a

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 13 / 16

General Setting

Definition of a weakening operator •

◮ Weakening α |

= •α

◮ Uniformity If α |

= β, then •α | = •β

Pertinence as additional constraint given by •

◮ α |

< β iff α | = β and β | = •α

General pertinent entailment relation

◮ Reflexivity α |

< α

◮ Infraclassicality If α |

< β, then α | = β

◮ Generalized Disjunction If α |

< β and γ | < δ, then α ∨ γ | < β ∨ δ

◮ Interpolation If α |

= β, β | = γ, and α | < γ, then α | < β and β | < γ

Here we have seen one case: • = ✸ ˘

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 14 / 16

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 AiML 2010 15 / 16

More: http://krr.meraka.org.za

Thank you!

Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 16 / 16