Modal logic(s)

Encoding modality linguisticallyAuxiliary (modal) verbscan, should, may, must, could, ought to, ...Adverbspossibly, perhaps, allegedly, ...Adjectivesuseful, possible, inflammable, edible, ...Many languages are much richer

Modal-based ambiguity in NLJohn can sing.Fred would take Mary to the movies. The dog just ran away.Dave will discard the newspaper.Jack may come to the party.

Propositional logic (review)Used to represent properties of propositions Formal properties, allows for wide range of applications, usable crosslinguisticallyHas three parts: vocabulary, syntax, semantics

Propositional logic (1)Vocabulary:Atoms representing whole propositions: p, q, r, s, Logic connectives: &, V, , ,Parentheses and brackets: (, ), [, ]ExamplesJohn is hungry.: pJohn eats Cheerios.: qp qp q

Propositional logic (2)Syntax (well-formed formulas, wffs):Any atomic proposition is a wff.If is a wff, then is a wff.If and are wffs, then ( & ), ( v ), ( ), and ( ) are wffs.Nothing else is a wff.Examples & pq is not a wff((pq) & (pr)) is a wff(p v q) s is a wff((((p & q) v r) s) t) is a wff

Propositional logic (3)Semantics:V() = 1 iff V() = 0.V( & ) = 1 iff V() = 1 and V() = 1.V( v ) = 1 iff V() = 1 or V() = 1.V( ) = 1 iff V() = 0 or V() = 1.V( ) = 1 iff V() = V().The valuation function V is all-important for semantic computations.

Logical inferencesModus Ponens: p q p -------- qModus Tollens: p q q --------- pHypothetical syllogism: p q q r -------- p rDisjunctive syllogism: p v q p -------- q

Formal logic and inferencesDeMorgans Laws( v ) ( & )( & ) ( v )Conditional Laws( ) ( v )( ) ( )( ) ( & )Biconditional Laws( ) ( ) & ( )( ) ( & ) v ( & )

Lexical items and predicationsneezed x.(sneeze(x)) saw y.x.(see(x,y)) laughed and is not a woman x.(laugh(x) & woman(x)) respects himself x.respect(x,x)respects and is respected by y.x.[respect(x,y) & respect(y,x)]

The function of lambdasLambdas fill open predicates variables with contentJohn sneezed. John, x.(sneeze(x)) x.(sneeze(x)) (John) x.(sneeze(x)) (John) sneeze(John)

The basic op: -conversionIn an expression (x.W)(z), replace all occurrences of the variable x in the expression W with z.(x.hungry(x))(John) hungry(John) (x.[married(x) & male(x) & adult(x)])(John) married(John) & male(John) & adult(John)

Contingency and truth

Two necessary ingredientsBackground: premises from which conclusions are drawnRelation: force of the conclusionJohn may be the murderer.John must be the murderer.

Model-theoretic valuationM = whereU is domain of individualsV is a valuation functionFor example,U = {mary, bill, pc23}V (likes) = {, }V (hungry) = {mary, bill}V (is broken) = {pc23}V (is French) =

Model-theoretic valuation[[Mary is hungry]]M = [[is hungry]]([[Mary]]) = [V(hungry)](mary) is true iff mary V(hungry) = 1[[my computer likes Mary]]M = 1 iff [[likes]] iff V(likes) = 0So far, have only used constants BUT variables are also possiblefunction g assigns to any variable an element from U

Possible worldsVariants, miniscule or drastic, from the actual context (world)W is the set of all possible worlds w, w, w, ...Ordering can be induced on the set of all possible worldsThe ordering is reflexive and transitiveModal logic: evaluates truth value of p w/rt each of the possible worlds in W

Modal logicBuild up a useful system from propositional logicAdd two operators: : It is possible that ...: It is necessary that ...K Logic: propositional logic plus:If A is a theorem, then so is A(AB) (A B)

Semantics of operatorsIf = , then [[]]M,w,g=1 iff wW, [[]]M,w,g=1.If = , then [[]]M,w,g=1 iff there exists at least one wW such that [[]]M,w,g=1.

Notes on KObvious equivalencies: A = AOperators behave very much like quantifiers in predicate calculusK is too weak, so add to it: M: A A

The result is called the T logic.

Notes on TStill too weak, so: (4) A A(5) A A

Logic S4: adding (4) to TLogic S5: adding (5) to T

S5Not adequate for all types of modalityHowever, it is commonly used for database work

O say what is (modal) truth?Let M = be a model with mapping I, and V be a valuation in the model; then:M,w v iff I()(w) = trueIf R(t1,...,tk) is atomic, M,w v R(t1...tk) iff V(R)(w)M,w v iff M,w v M,w v & iff M,w v and M,w v M,w v (x) iff M,w v [x/u] for all u UM,w v iff M,w v for all w WM,w v [x.(x)](t) if M,w v [x/u] where u = g(t,w)

Human necessity is a human necessity iff it is true in all worlds closest to the idealIf W is the modal base, wW there exists wW such that:w w, and wW , if w w then is true in w is a human possibility iff is not a human necessity

Backgrounds (Kratzer)Realistic: for each w, set of ps that are trueTotally realistic: set of ps that uniquely define wEpistemic: ps that are established knowledge in wStereotypical: ps in the normal course of wDeontic: ps that are commanded in wTeleological: ps that are related to aims in wBuletic: ps that are wished/desirable in wEmpty: the empty set of ps in any w

Related notionsConditionalsCounterfactualsGenericsTenseIntensionalityDoxastics (belief models)

The Fitting paperApplies modal logic to databasesmodel-theoretic, S5, formulas tableau methods for proofs, derived rulesOperator that associates, combines semantic items compositionallyPredicates, entitiesVariables

The Fitting paperdb records: possible worldsaccess: ordering on possible worldstwo types of axioms:constraint axiomsinstance axiomsQueries: modal logic expressionsProofs and derivations: tableau methods (several rules)