Page 364 - Encyclopedia of Philosophy of Language

Formal Semantics
for K4 is transitive, any frame for S4 is reflexive and transitive, any frame for B is reflexive and symmetric, and any frame for S3 is reflexive, symmetric, and transitive. A great deal of work in modal logic is concerned with the characterization of modal systems by classes of frames (Hughes and Cresswell 1984).
3.4 Simpler Semanticsfor S5
The system S5 is characterized by the class of models <W, R, V> (and by the class of all frames <W, R » in which R is an equivalence relation. An equivalence relation divides its domain—the set of worlds W in this case—into equivalence classes: the worlds in one equivalenceclassareR-relatedtoeachotherandto no worlds outside that equivalence class. It is straight- forward to show that S3 is characterized by the class of models in which R is an equivalence relation with only one equivalence class.
The system SS is clearly sound with respect to this class of models. It is also complete with respect to this class; but this cannot be established directly by the method of canonical models. The canonical model for S3 contains more than one equivalence class, since for a sentence letter p both the set {Dp} and the set {~ p} are consistent. Therefore, in the canonical model, there will be a world w, at which Dp is true, and a world w2 at which ~ p is true. But then w, and w2 must be in different equivalence classes, since the truth of Dp at w, requires the truth of p at every world that is R-related to w,.
However, suppose that £ is a set of WFFS that is consistent in S3. Then there is a world w in the canoni- cal model for S3 at which all the WFFS in £ are evalu- ated as true. The evaluation of WFFS at w depends only upon the evaluation of WFFSat other worlds that are accessible from w. Consequently, it would make no difference to the evaluation of WFFS at w if one were to remove from the canonical model all those worlds that lie outside the equivalence class to which w belongs. If one removed all those worlds, then one would be left with a model with only one equivalence class, containing a world w at which all the WFFS in £ are evaluated as true. Consequently, for every con- sistent set of WFFS E, there is a model with only one equivalence class, containing a world w that satisfies I.
If a model contains only one equivalence class, then every world is accessible from every other world, and so there is no need to specify the accessibility relation separately. Consequently, a 'simple S3 model' is a pair <W, V>, where W is a set of worlds, and V is a function from ordered pairs of sentence letters and worlds to truth-values. The valuation V is extended to an evalu- ation of each WFF as true or false at each world just as before, save that DA is true at a world w if and only if A is true at every world in W. Thus, in the case of S3, one returns to the original Leibnizian idea that necessary truth is truth in all possible worlds.
It has already been noted that the axioms of S3 are plausibly valid for the notion of metaphysical, or broadly logical, necessity. Simple S3 models are typically used as the semantic basis for philosophical discussions of necessity (Plantinga 1974; Forbes 1985). However, these discussions cannot proceed very far without the resources of quantified modal logic.
4. Quantified Modal Logic
Suppose that modal operators are added to the language of predicate logic, while still allowing 'D ' the same privileges of occurrence as ' ~ . ' Then, not only sentences of the form Q(3x)F(x) are permitted, wherethe'D'hasacompletesentencewithinitsscope, but also sentences of the form (3x)DF(x), where the 'G ' has only an open sentence within its scope.
Quine (1960; 1976) points out that this—the third 'grade of modal involvement'—is only very dubiously intelligible if the modal operator is properly conceived as expressing a property of sentences. If the property of sentences is expressed by a predicate 'Nee,' then the problem can be seen very clearly by considering an expression of the form (Bx)Nec('F(x)'). For here, the used quantifier does not bind the mentioned variable, and so is vacuous.
The only circumstances in which this problem can clearly be overcome are those in which each object in the domain of quantification has a uniquely canonical name. In those circumstances, a sentence of the form (3x)QF(x) can be interpreted as saying that there is an object z such that the property expressed by 'Nee' applies to the sentence of the form F(n) that results by replacing the variable with the canonical name of z. One case in which these requirements are met is provided by the language of arithmetic, where each natural number has a numeral as its canonical name. Consequently, the use of modal logic in the study of provability can proceed from the prepositional modal system G to the corresponding quantified modal system, without any risk from Quinean objections.
Where there are not uniquely canonical names, quantification into the scope of a modal operator can- not be rendered intelligible in this way, and so the conception of the modal operator as expressing a property of sentences must be given up. Quine's objec- tion to this is that it leads to 'Aristotelian essentialism': the doctrine that an object has some of its properties essentially, and others only contingently. For a sen- tence of the form (Bx)DF(x) says that there is an object that necessarily has such-and-such a property. Quine (1976: 175) also notes that in quantified modal logic one will almost inevitably obtain the theorem
(Vx)(Vy)(x=yz>n(x=y))
which says that 'identity holds necessarily if it holds at all.'
These two features of quantified modal logic—that it is committed to the intelligibility of essentialist
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