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Reasoning on Issues: One Logic and a Half

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Reasoning on Issues: One Logic and a Half Reasoning on issues: one logic and a half Ivano Ciardelli partly based on joint work with Jeroen Groenendijk and Floris Roelofsen KNAW Colloquium on Dependence Logic — 3 March 2014 Overview 1. Dichotomous inquisitive logic: reasoning with issues 2. Inquisitive epistemic logic: reasoning about entertaining issues 3. Inquisitive dynamic epistemic logic: reasoning about raising issues Overview 1. Dichotomous inquisitive logic: reasoning with issues 2. Inquisitive epistemic logic: reasoning about entertaining issues 3. Inquisitive dynamic epistemic logic: reasoning about raising issues Preliminaries Information states I Let W be a set of possible worlds. I Definition: an information state is a set of possible worlds. I We identify a body of information with the worlds compatible with it. I t is at least as informed as s in case t ⊆ s. I The state ; compatible with no worlds is called the absurd state. w1 w2 w1 w2 w1 w2 w1 w2 w3 w4 w3 w4 w3 w4 w3 w4 (a) (b) (c) (d) Preliminaries Issues I Definition: an issue is a non-empty, downward closed set of states. I An issue is identified with the information needed to resolve it. S I An issue I is an issue over a state s in case s = I. I The alternatives for an issue I are the maximal elements of I. w1 w2 w1 w2 w1 w2 w1 w2 w3 w4 w3 w4 w3 w4 w3 w4 (e) (f) (g) (h) Four issues over fw1; w2; w3; w4g: only alternatives are displayed. Part I Dichotomous inquisitive logic: reasoning with issues Dichotomous inquisitive semantics Definition (Syntax of InqDπ) LInqDπ consists of a set L! of declaratives and a set L? of interrogatives: 1. if p 2 P, then p 2 L! 2. ? 2 L! 3. if α1; : : : ; αn 2 L!, then ?fα1; : : : ; αng 2 L? 4. if '; 2 L◦, then ' ^ 2 L◦ 5. if ' 2 L! [L? and 2 L◦, then ' ! 2 L◦ Abbreviations I if α 2 L!, :α := α !? I if α, β 2 L!, α _ β := :(:α ^ :β) I if α 2 L!, ?α := ?fα, :αg Dichotomous inquisitive semantics Notational convention on meta-variables Declaratives Interrogatives Full language Formulas α, β, γ µ, ν; λ '; ; χ Sets of formulas Γ Λ Φ Dichotomous inquisitive semantics Semantics I Usually, the role of semantics is to assign truth-conditions. I However, our language now contains interrogatives as well. I Claim: interrogative meaning = resolution conditions. I We could give a double-face semantics: truth-conditions at worlds for declaratives, resolution conditions at info states for interrogatives. I Instead, we will lift everything to the level of information states. I Our semantics is defined by a relation j= of support between information states and formulas, where: Declaratives: s j= α () α is established in s Interrogatives: s j= µ () µ is resolved in s Dichotomous inquisitive semantics Definition (Models) A model for a set P of atoms is a pair M = hW; Vi where: I W is a set whose elements are called possible worlds I V : W! }(P) is a valuation function Definition (Support) Let M be a model and let s be an information state. 1. M; s j= p () p 2 V(w) for all worlds w 2 s 2. M; s j= ? () s = ; 3. M; s j= ?fα1; : : : ; αng () M; s j= α1 or ::: or M; s j= αn 4. M; s j= ' ^ () M; s j= ' and M; s j= 5. M; s j= ' ! () for any t ⊆ s, if M; t j= ' then M; t j= Dichotomous inquisitive semantics Fact (Perstistence) If M; s j= ' and t ⊆ s then M; t j= '. Fact (Absurd state) M; ; j= ' for any formula ' and model M. Definition (Proposition) The proposition expressed by ' in M is the set of states supporting ': [']M = fs ⊆ W j s j= 'g Fact (Propositions are issues) [']M is an issue for any formula ' and model M. Dichotomous inquisitive semantics Definition (Truth) def M; w j= ' () M; fwg j= ' Definition (Truth-set) j'jM := fw 2 W j M; w j= 'g Fact (Truth and support) S j'jM = [']M Dichotomous inquisitive semantics Fact (Truth-conditions) I M; w j= p () p 2 V(w) I M; w 6j= ? I M; w j= ?fα1; : : : ; αng () M; w j= α1 or ::: or M; w j= αn I M; w j= ' ^ () M; w j= ' and M; w j= I M; w j= ' ! () M; w 6j= ' or M; w j= Dichotomous inquisitive semantics Truth for declaratives I The semantics of a declarative is determined by truth conditions: M; s j= α () for all w 2 s; M; w j= α I That is, we always have [α]M = }(jαjM) I Since truth-conditions are standard, declaratives are classical. 11 10 11 10 11 10 11 10 01 00 01 00 01 00 01 00 p p ^ q p _ q p ! q Dichotomous inquisitive semantics Truth for interrogatives M; w j= µ () w 2 s for some s j= µ 11 10 () w 2 s for some s resolving µ () µ can be truthfully resolved in w 01 00 Definition (Presupposition of an interrogative) I [?fp; qg] π?fα1,...,αn g = α1 _···_ αn I πµ^ν = πµ ^ πν I π'!µ = ' ! πν 11 10 Fact jµjM = jπµjM 01 00 Remark j?fp; qgj For interrogatives, truth-conditions do not fully determine meaning. Ex. consider ?p and ?q. Dichotomous inquisitive semantics Conjunction M; s j= ' ^ () M; s j= ' and M; s j= 11 10 11 10 11 10 01 00 01 00 01 00 p q p ^ q 11 10 11 10 11 10 01 00 01 00 01 00 ?p ?q ?p ^ ?q Dichotomous inquisitive semantics Implication M; s j= ' ! () for any t ⊆ s, if M; t j= ' then M; t j= 11 10 11 10 11 10 11 10 01 00 01 00 01 00 01 00 p q p ! q ?p ! q ≡ q 11 10 11 10 11 10 11 10 01 00 01 00 01 00 01 00 ?p ?q p ! ?q ?p ! ?q Dichotomous inquisitive logic Definition (Entailment) Φ j= () for all M; s, if M; s j= Φ then M; s j= Declarative conclusion Γ; Λ j= α () establishing Γ and ΠΛ implies establishing α. Interrogative conclusion Γ; Λ j= µ () establishing Γ and resolving Λ implies resolving µ. Dichotomous inquisitive logic Example 1 I p $ q ^ r; ?q ^ ?r j= ?p I p $ q ^ r; ?p 6j= ?q ^ ?r Example 2 I ?p ! ?q; ?p j= ?q Dichotomous inquisitive logic Four particular cases I α j= β () α is at least informative as β I α j= µ () α resolves µ I µ j= α () µ presupposes α I µ j= ν () µ is at least as inquisitive as ν Dichotomous inquisitive logic Conjunction Implication Falsum [α] : : : α β α ^ β α ^ β β α α ! β ? α ^ β α β α ! β β - α Double negation axiom ::α ! α Dichotomous inquisitive logic Conjunction Implication Falsum ['] : : : ' ' ^ ' ^ ' ' ! ? ' ^ ' ' ! - ' Double negation axiom ::α ! α Dichotomous inquisitive logic Conjunction Implication Falsum ['] : : : ' ' ^ ' ^ ' ' ! ? ' ^ ' ' ! - ' Interrogative Double negation axiom ::α ! α [α1] [αn ] : : : : : : αi '::: ' ?fα1; : : : ; αn g ?fα1; : : : ; αn g ' Dichotomous inquisitive logic Conjunction Implication Falsum ['] : : : ' ' ^ ' ^ ' ' ! ? ' ^ ' ' ! - ' Interrogative Double negation axiom ::α ! α [α1] [αn ] : : Kreisel-Putnam axiom : : : : α i '::: ' ?fα1; : : : ; αn g (α ! ?fβ1; : : : ; βm g) ! ?fα ! β1; : : : ; α ! βm g ?fα1; : : : ; αn g ' Dichotomous inquisitive logic Definition (Resolutions) To any formula ' we associate a set of declaratives called resolutions. I R(α) = fαg if α is a declarative I R(?fα1; : : : ; αng) = fα1; : : : ; αng I R(µ ^ ν) = fα ^ β j α 2 R(µ) and β 2 R(ν)g V I R(' ! µ) = f α2R(') α ! f(α) j f : R(') !R(µ)g Resolutions of a set Replace each element in the set by one or more resolutions: R( fp; ?q ^ ?rg ) = f fp; q ^ rg fp; q ^ :rg ::: ::: g Dichotomous inquisitive logic Theorem (Resolution theorem) Φ ` () 8Γ 2 R(Φ) 9α 2 R( ) s.t. Γ ` α Corollary There exists an effective procedure that, when given as input: I a proof of Φ ` I a resolution Γ of Φ outputs: I a resolution α of I a proof of Γ ` α Dichotomous inquisitive logic Example If we feed the algorithm I a proof of p $ q ^ r; ?q ^ ?r ` ?p I the resolution p $ q ^ r; q ^ :r It will return I the resolution :p of ?p I a proof of p $ q ^ r; q ^ :r ` :p Dichotomous inquisitive logic Definition (Canonical model) c c c The canonical model for InqDπ is the model M = hW ; V i where: I Wc consists of complete theories of declaratives I V c : Wc ! }(P) is defined by V c (Γ) = fp j p 2 Γg Lemma (Support lemma) For any S ⊆ Wc ; Mc ; S j= ' () T S ` ' Theorem (Completeness) Φ j= () Φ ` Part II Reasoning about entertaining issues: Inquisitive Epistemic Logic Inquisitive epistemic logic Epistemic Logic In standard EL we can reason about facts and (higher-order) information. Inquisitive Epistemic Logic In IEL we can reason about facts, information and issues, including the higher-order cases: I information about information I information about issues I issues about information I issues about issues Inquisitive epistemic logic Standard epistemic models An epistemic model is a triple M = hW; V; fσa (w) j a 2 Agi where: I W is a set of possible worlds I V : W! }(P) is a valuation function I σa : W! }(W) is the epistemic map of agent a, delivering for any w an information state σa (w), in accordance with: Factivity : w 2 σa (w) Introspection : if v 2 σa (w) then σa (v) = σa (w) Inquisitive epistemic logic I We want to add a description of the issues agents entertain.
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