Inquisitive Logic Ivano Ciardelli www.illc.uva.nl/inquisitive-semantics
Information states A state is a set of valuations. Support Let s be a state. The system InqB 1. s = p iff w s : w(p) = 1 2. s = iff s = 3. s = ϕ ψ iff s = ϕ and s = ψ 4. s = ϕ ψ iff s = ϕ or s = ψ 5. s = ϕ ψ iff t s : if t = ϕ then t = ψ Three features Persistence: if t s and s = ϕ then t = ϕ Classical behaviour of singletons: {w} = ϕ iff w = ϕ Inconsistent state: = ϕ for any ϕ
Propositions and informative content Definition The proposition expressed by ϕ is [ϕ] = {s s = ϕ} The alternatives for ϕ Alt(ϕ) are the maximal states in [ϕ] The informative content of ϕ is info(ϕ) = [ϕ] The truth-set of ϕ is ϕ = {w ϕ is true in w} Proposition (Classical treatment of information) For all ϕ, info(ϕ) = ϕ 11 10 11 10 01 00 [p q] 01 00 p q
Assertions Proposition An assertion ϕ has only one alternative, namely, ϕ. Proposition Any negation ϕ is an assertion. ϕ is an assertion iff ϕ ϕ Any disjunction-free formula is an assertion. 11 10 11 10 11 10 01 00 p p assertion 01 00 p q inquisitive 01 00 (p q) assertion
Definition Inquisitive logic Γ = InqL ϕ for all s, if s = Γ then s = ϕ InqL = {ϕ = InqL ϕ} A combined entailment Inquisitive entailment measures informative and inquisitive content. We have ϕ = InqL ψ in case: 1. ϕ provides at least as much information as ψ 2. ϕ requests at least as much information as ψ And ϕ InqL ϕ is neither informative nor inquisitive. Some particular cases If ϕ is an assertion, then ϕ InqL ϕ CPL. But, e.g.,?p = p p InqL, since it is inquisitive.
Basic properties Disjunction property If ϕ ψ InqL then ϕ InqL or ψ InqL Deduction theorem Γ, ϕ = ψ Γ = ϕ ψ Compactness If Γ = ϕ then Γ 0 = ϕ for some finite Γ 0 Γ Decidability The problem of deciding whether ϕ InqL is decidable.
Support, again Support 1. s = p iff w s : w(p) = 1 2. s = iff s = 3. s = ϕ ψ iff s = ϕ and s = ψ 4. s = ϕ ψ iff s = ϕ or s = ψ 5. s = ϕ ψ iff t s : if t = ϕ then t = ψ
Kripke model for Inquisitive Semantics Definition The Kripke model for IS is M I = ( (ω) { },, V I ) where s V I (p) w(p) = 1 for each w s Proposition (Support amounts to Kripke semantics on M I ) s = ϕ M I, s ϕ {11} {10} {01} {00} {11,01} {10,00} {10,01} {01,00} {11,10} {11,00} {11,01,10} {01,10,00} {11,10,00} {11,01,00} {11,01,10,00}
InqL is a non-substitution closed intermediate logic Corollary IPL InqL CPL Remark InqL is not closed under uniform substitution: p p InqL but (p p) (p p) InqL Definition A weak intermediate logic is a set IPL L CPL which is closed under modus ponens.
Reminder: negative translation of CPL to IPL Theorem There exists a recursively defined map nt such that for all ϕ nt(ϕ) is a negation ϕ CPL nt(ϕ) Theorem (nt is a translation from CPL to IPL) Γ = CPL ϕ nt[γ] = IPL nt(ϕ) Corollary CPL is (isomorphic to) the negative fragment of IPL.
Disjunctive-negative translation It is possible to give a map dnt such that for all ϕ, dnt(ϕ) is a disjunction of negations ϕ InqL dnt(ϕ) Ingredients for dnt 1. intuitionistic logic 2. atomic double negation law: p p for p P 3. for all k N, all substitution instances of the scheme ND k ( χ ξ i ) ( χ ξ i ) i k i k
Theorem (dnt is a translation of InqL to IPL) Γ = InqL ϕ dnt[γ] = IPL dnt(ϕ) Corollary InqL is (isomorphic to) the disjunctive-negative fragment of IPL. [ ] [ ].. [ p] [p p] [ ] [ p p] [ p p] [p p] [ p] [p] [ p] [p] [ ]
dnt+disjunction property uniquely characterize InqL among intermediate logics Theorem 1. If a weak intermediate logic L justifies dnt, then InqL L (L justified dnt in case ϕ L dnt(ϕ) for all ϕ) 2. If additionally L has the disjunction property, then L = InqL. Proof. 1. Suppose ϕ InqL. Then dnt(ϕ) InqL as well. 2. Let dnt(ϕ) = χ 1 χ n. 3. By the disjunction property, χ i InqL for some i. 4. Then, since all w.i.l. agree on negations, χ i L. 5. So, dnt(ϕ) = χ 1 χ n must be in L as well. 6. Finally since L justifies dnt, ϕ L. 7. If L has the d.p., the argument can be reversed.
Axiomatizatizing InqL Theorem (Axiomatization 1: InqL = ND + p p) InqL is axiomatized by a Hilbert-style system having modus ponens as inference rule and the following axioms: axioms for intuitionistic logic all instances of Maksimova s scheme: ND ( ϕ χ i ) ( ϕ χ i ) i k p p where p is an atom i<k
Axiomatizing InqL Theorem (Axiomatization 2: InqL = KP + p p) InqL is axiomatized by a Hilbert-style system having modus ponens as inference rule and the following axioms: axioms for intuitionistic logic every instance of the KP scheme: KP ( ϕ ψ χ) ( ϕ ψ) ( ϕ χ) p p where p is an atom
Schematic fragment of inquisitive logic Definition Given a weak intermediate logic L, the schematic fragment of L is: Sch(L) := {ϕ ϕ L for any substitution instance ϕ of ϕ} Fact Sch(L) is the greatest intermediate logic included in L. Question Sch(InqL)=?
Medvedev s logic of finite problems Definition (Medvedev s logic) ML is the logic of intuitionistic Kripke frames of the form ( (X ) { }, ), with X finite. ML arises from interpreting propositional formulas as problems (Medvedev 62, 66). ML is not finitely axiomatizable (Maksimova et al. 79). It is an open problem whether ML is recursively axiomatizable. Theorem Sch(InqL) = ML Corollary ML is the schematic fragment of a recursively axiomatized logic.
Refining the completeness theorem Theorem For any intermediate logic Λ, Λ + p p = InqL ND Λ ML
Summing up 1. Inquisitive logic InqL takes into account both informative and inquisitive content. 2. InqL is a non-substitution closed intermediate logic. 3. InqL is (isom. to) the disjunctive-negative fragment of IPL. 4. the set of schematic validities of InqL is Mevedev s logic. 5. InqL can be axiomatized as Λ + p p for ND Λ ML.
Some references Ciardelli (2009) Inquisitive semantics and intermediate logics, MSc thesis, University of Amsterdam. Ciardelli and Roelofsen (2011) Inquisitive logic, Journal of Philosophical Logic, 40:55-94. Maksimova (1986) On maximal intermediate logics with the disjunction property, Studia Logica, 45, 69-75. Maksimova, Shetman and Skvorcov (1979) The impossibility of a finite axiomatization of Medvedev s logic of finite problems, Soviet Mathematics Doklady, 20, 394-398. Medvedev (1962) Finite problems, Soviet Mathematics Doklady, 3, 227-230. Medvedev (1966) Interpretation of logical formulas by means of finite problems, Soviet Mathematics Doklady, 7, 857-860.