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claims and to the necessity of identity—are regarded by Quine as unattractive. However, in philosophical use of quantified modal logic (particularly following Kripke 1980), they are accepted as foundational.
4.1 Model-theoretic Semanticsfor Quantified Modal Logic
The key idea in the semantics for quantified modal logic is once again that of a possible world. In the case of prepositional modal logic, each world was associated with an assignment of truth-values to sen- tence letters and then—by an induction on the com- plexity of WFFS—with an evaluation of every WFF as true or false. In the case of quantified modal logic, each world must be associated with a domain of objects and an interpretation of each constant and atomic predicate in that domain. In effect, each world determines a structure that is a model for the cor- responding nonmodal predicate calculus language. Given that structure, a complex WFF can be evaluated as true or false (relative to a sequence of objects assigned to free variables). If the modal system is a quantified version of S5, then QA will be true at a world if A is true at every world. (For a brief dis- cussion of the axiomatization and model theory for quantified versions of S5 and other modal systems, see Hughes and Cresswell 1984: ch. 9).
A version of the semantics for quantified S5 can be sketched by first making two simplifying assumptions. First, one assumes that every object in the domain has at least one name in the predicate calculus language to which the modal operator 'Q ' is added. Second, one ignores the fact that which objects exist is itself a contingent matter. That is, one ignores the fact that different worlds should have different domains of objects associated with them, and assumes that the domain is the same for all worlds in the set W.
With these assumptions in place, a quantified S5 model (with fixed domain) is said to be a triple <W, D, I>, where W is a set of worlds, D is a domain of objects, and I is an interpretation function. The function I maps constants (names) to objects in D, and maps each ordered pair of a closed atomic WFF and a world to a truth-value. Thus, I determines an extension for each atomic predicate at each world. The extension of I to an evaluation of each closed WFF as true or false at each world is then straightforward. The truth-functional connectives and the modal oper- ator are treated just as in the case of the simple sem- antics for prepositional S5. The quantifiers are treated just as in the case of nonmodal predicate calculus when every object in the domain has a name. Thus, for example, (3x)C(x) is true at a world w if and only if some substitution instance C(m) is true at w. Because of the second assumption, all instances of the schema
(Vx)DF(x)=>n(Vx)F(x) (13) (the Barcan formula; Marcus 1962) and the schema
D(Vx)F(x)=(Vx)DF(x) (14)
(the converse Barcan formula) are valid.
Once the first assumption is removed, the notion of
truth needs to be replaced with that of satisfaction by sequences, or truth relative to an assignment of objects to free variables, just as in the case of nonmodal predi- cate calculus.
More importantly, once the second assumption is removed, one is faced with several choices. Suppose that a quantified S5 model (with varying domain) is said to be a triple <W, d, I> where d is now a function from worlds in w to sets of objects. Thus, d(w) is the domain of the world w; intuitively, the set of objects thatexistatw.OneletsD=(j (d(w):weW}(andsim- plifies the description of the choices by retaining the first assumption: every object in D has a name in the language).
The first choice to be faced concerns atomic predi- cates. The question is whether the interpretation func- tion I should assign to each atomic predicate, at world w, an extension in the set D, or an extension that lies wholly within the domain of w, d(w). The standard presentations of model-theoretic semantics for quant- ified modal logic (Kripke 1963; Fine 1978) do not impose the restriction that the extension of an atomic predicate at world w should lie within the domain of w. Thus they allow that, so far as the model theory is concerned, an atomic sentence of the form Fm may be true at a world even though the object named by m does not exist at that world. The opposite choice would correspond to the requirement that an atomic sentence should be false at a world if any of the objects named in it fails to exist at that world. This require- ment is called the 'falsehood principle.'
The second choice to be faced concerns the quan- tifiers. In the evaluation of WFFS as true or false at a world w, the question is whether the quantifiers should be interpreted as ranging over the domain of that world, d(w), or as ranging over the larger set D. The standard presentations have the quantifiers ranging only over d(w). This is sometimes called the 'actualist' interpretation of the quantifiers. The opposite choice is the 'possibilist' interpretation (see Forbes (1988) for detailed discussion).
Given the actualist interpretation of the quantifiers, the Barcan formula at (13) above is not generally valid. Intuitively, the reason is as follows. It may be that all the objects that exist at world w have a certain property P at every world, but that at some world w' there are other objects that lack the property P.
The third choice to be faced concerns the modal operators. In the evaluation of modal WFFS as true or false at a world, the question is whether the truth of DA should require the truth of A at every world or only the truth of A at every world in which the objects named in A exist. In standard presentations, the for- mer option is taken, so that 'D' expresses strong necessity. On the latter option, 'D ' expresses weak
Modal Logic
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