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

Pertinence Construed Modally Arina Britz 1 , 2 Johannes Heidema 2 Ivan Varzinczak 1 1 Meraka Institute, CSIR 2 University 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 Usually W Extra information expressed either as β Syntactic rules , or as α 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 M β ) iff for every w ∈ W , if α entails β in M = � W , R , V � (denoted α | = M α , then w � M β . w � Definition Let C be a class of models C β ) iff α | M β for every M ∈ C α entails β in C (denoted α | = = Validity and satisfiability in C defined as usual C β When C is clear from the context, we write α | = β instead of α | = 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 β • α • Definition M β ) iff α | M β and β | M ✸ ˘ α pertinently entails β in M (denoted α | = = < α Definition C β ) iff for α pertinently entails β in the class C of models (denoted α | < M β every M ∈ C , α | < C β When C is clear from the context, we write α | < β instead of α | < Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally AiML 2010 7 / 16

Pertinence in the Meta-Level • ✸ ˘  α ˘ α ✸      W    β  . . . | < • • β •     α      • 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 : id W ⊆ R ⊆ W × W (modal logic KT) The minimum (w.r.t. ⊆ ) case: R = id W ◮ 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 C β . Then if | C α → ⊥ , then | C β → ⊥ Let α | < = = | < 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 ∧ α �| < β ) < 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 } Question: Is α the pertinent cause of β ? M : ¬ s , a , w w 3 ¬ a ∧ ¬ w | < ¬ a ; ¬ a ∧ ¬ w �| < ¬ w a ∧ ✷ ¬ s �| < a ; a ∧ ✷✸ s | < a w 2 ¬ s , a , ¬ w s , ¬ a , ¬ w w 4 s �| < ¬ a ; s ∨ ¬ a | < ¬ a w 1 ¬ s , ¬ a , ¬ w 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