Page 362 - Encyclopedia of Philosophy of Language

Formal Semantics
B is evaluated as true on every assignment on which all the premises A , , . . . , An come out true. Finally, the notion of provability is linked with that of validity by demonstrating that every WFF that is provable is valid ('soundness') and that every WFFthat is valid is provable ('completeness').
These resources are not adequate for prepositional modal logic, since the modal operators are not truth- functional. In the case of modal logic, the intuitive semantic idea—which goes back to Leibniz—is that necessary truth is truth in all possible worlds. The intuitive idea was turned into model-theoretic sem- antics for several modal systems by Kripke (1963). A modal model involves not just one assignment of truth-values to sentence letters, but a whole set of assignments: each assignment can be thought of as describing a possible world. The simplest way to develop the Leibnizian idea of necessity would then be to say that the truth of D A on any one assignment requires the truth of A on every assignment in the set. However, Kripke's model theory introduces a further element: a relation of accessibility between assign- ments. The truth of DA on any one assignment requires the truth of A on every asignment that is accessible from that one.
In the formal development of the semantics, a modal model is a triple <W, R, V>, where W is a set (the set of possible worlds), R is a binary relation on W (the accessibility relation), and V is a function from ordered pairs of sentence letters and worlds to truth- values (the valuation). The valuation V thus deter- mines an assignment of truth-values to sentence letters corresponding to each world w in the set W.
The valuation V is then extended—by induction on the complexity of WFFS—to an evaluation of each WFF as true or false at each world. For example, a negation
~AistrueataworldwifandonlyifAisnottrueat w; a conjunction A&B is true at a world w if and only if each conjunct A and B is true at w; and so on for the other truth-functional connectives. Finally, DA is true at a world w if and only if A is true at every world that is R-related to w, that is, every world w' such that R(w ,w').
A WFF of prepositional modal logic is then 'valid' if it is true at every world in every model. Similarly, an argument in prepositional modal logic is valid if, whenever all the premises are true at a world in a model, the conclusion is also true at that world in that model. In addition, a WFF is said to be 'valid in a model' if it is true at every world in that model; and a WFF is said to be 'valid in a class of models' if it is valid in every model in the class.
If every theorem of a given modal system is valid in a class of models, then the system is said to be 'sound' with respect to that class. If every WFF that is valid in a class of models is a theorem of a given modal system, then the system is said to be 'complete' with respect to that class.
3.1 Soundness Results
The system K is sound with respect to the class of all modal models. This is to say that every theorem of the system K is valid; that is, valid in every model. In order to see that this is so, one needs to consider the three components of the system K. First, it is clear that any WFF that is a substitution instance of a tautology of nonmodal prepositional logic will be evaluated as true at every world in every model. Fur- thermore, the rule of modus ponens preserves validity. Second, one has to establish the validity of every instance of the schema
K. D(Ar>B)z>(nA3DB).
If D(A o B)and DA are both true at a world w,then A =>B and A are both true at every world w' accessible from w. But then, by the truth-table for' = ,' B is also true at every such w', and so DB is true at w, as required. Third, it is necessary to check that the rule of Necessitation preserves validity: if A is valid, then D A is also valid. Assume that A is true at every world in every model, and suppose that for some world w, in some model, DA is not true at w. Then there is some world w' accessible from w in the model, such that A is not true at w'. But this contradicts the assumption that A is true at every world in every model. So, D A is after all true at every world in every model.
The system T is not sound with respect to the class of all modal models. It is a straightforward matter to construct a model in which some instance of the sch- ema T is false at some world. For example, suppose that W = {W|,w2), that w, is R-related to w2 and to no other world, and that for some sentence letter p, V(p,w,) = False while V(p,w2)=True. Then Dp is true at w, while p is false at w,. So, Dp ^ P is false at w,inthismodel.
The system T is, however, sound with respect to the class of all models in which the R-relation is 'reflexive'; that is, in which, for every world w, R(w, w). (It is a routine matter to check this.) For short, T is said to be sound with respect to the class of all reflexive models.
There are similar soundness results for the systems S4, B, and S3. The system S4 is sound with respect to the class of all models in which R is both reflexive and transitive. (R is transitive if, whenever R(u,v) and R(v, w), one also has R(u, w).) The system B is sound with respect to the class of all models in which R is both reflexive and symmetric. (R is symmetric if, whenever (u, v), one also has R(v, u).) The system S5 is sound with respect to the class of models in which R is reflexive, transitive, and symmetric; that is, in which R is an 'equivalence' relation.
Furthermore, the system K4 is sound with respect to the class of models in which R is transitive. There is also a soundness result for the system G; but the crucial property of the R-relation is more complex. G
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