of sentences in an S5 interpretation. But within Cw* R is universal: every world is possible relative to every other. Consequently, in an S5 interpretation, we need not specify a relative possibility relation, and the evaluation rules for B and A need not mention relative possibility; e.g., we can say that B f is true at a world u if there is at least one world v at which f is true. Note that by the characteristics of R, whenever 9 X s in K, T, B, or S4, then 9 X s in S5: the other systems are contained in S5. K is contained in all the systems we have mentioned, while T is contained in B and S4, neither of which is contained in the other. Sentential modal logics give rise to quantified modal logics, of which quantified S5 is the bestknown. Just as, in the sentential case, each world in an interpretation is associated with a valuation of sentence letters as in non-modal sentential logic, so in quantified modal logic, each world is associated with a valuation of the sort familiar in non-modal first-order logic. More specifically, in quantified S5, each world w is assigned a domain Dw – the things that exist at w – such that at least one Dw is non-empty, and each atomic n-place predicate of the language is assigned an extension Extw of n-tuples of objects that satisfy the predicate at w. So even restricting ourselves to just the one first-order extension of a sentential system, S5, various degrees of freedom are already evident. We discuss the following: (a) variability of domains, (b) interpretation of quantifiers, and (c) predication. (a) Should all worlds have the same domain or may the domains of different worlds be different? The latter appears to be the more natural choice; e.g., if neither of of Dw* and Du are subsets of the other, this represents the intuitive idea that some things that exist might not have, and that there could have been things that do not actually exist (though formulating this latter claim requires adding an operator for ‘actually’ to the language). So we should distinguish two versions of S5, one with constant domains, S5C, and the other with variable domains, S5V. (b) Should the truth of (Dn)f at a world w require that f is true at w of some object in Dw or merely of some object in D (D is the domain of all possible objects, 4weWDw)? The former treatment is called the actualist reading of the quantifiers, the latter, the possibilist reading. In S5C there is no real choice, since for any w, D % Dw, but the issue is live in S5V. (c) Should we require that for any n-place atomic predicate F, an n-tuple of objects satisfies F at w only if every member of the n-tuple belongs to Dw, i.e., should we require that atomic predicates be existence-entailing? If we abbreviate (Dy) (y % x) by Ex (for ‘x exists’), then in S5C, A(Ex)AEx is logically valid on the actualist reading of E (%-D-) and on the possibilist. On the former, the formula says that at each world, anything that exists at that world exists at every world, which is true; while on the latter, using the definition of ‘Ex’, it says that at each world, anything that exists at some world or other is such that at every world, it exists at some world or other, which is also true; indeed, the formula stays valid in S5C with possibilist quantifiers even if we make E a primitive logical constant, stipulated to be true at every w of exactly the things that exist at w. But in S5V with actualist quantifiers, A(Ex)AEx is invalid, as is (Ex)AEx – consider an interpretation where for some u, Du is a proper subset of Dw*. However, in S5V with possibilist quantifiers, the status of the formula, if ‘Ex’ is defined, depends on whether identity is existence-entailing. If it is existenceentailing, then A(Ex)AEx is invalid, since an object in D satisfies (Dy)(y % x) at w only if that object exists at w, while if identity is not existence-entailing, the formula is valid.
The interaction of the various options is also evident in the evaluation of two well-known schemata: the Barcan formula, B (Dx)fx P (Dx) B fx; and its converse, (Dx) B fx P B (Dx)fx. In S5C with ‘Ex’ either defined or primitive, both schemata are valid, but in S5V with actualist quantifiers, they both fail. For the latter case, if we substitute -E for f in the converse Barcan formula we get a conditional whose antecedent holds at w* if there is u with Du a proper subset of Dw*, but whose consequent is logically false. The Barcan formula fails when there is a world u with Du not a subset of Dw*, and the condition f is true of some non-actual object at u and not of any actual object there. For then B (Dx)f holds at w* while (Dx) B fx fails there. However, if we require atomic predicates to be existence-entailing, then instances of the converse Barcan formula with f atomic are valid. In S5V with possibilist quantifiers, all instances of both schemata are valid, since the prefixes (Dx) B and B (Dx) correspond to (Dx) (Dw) and (Dw) (Dx), which are equivalent (with actualist quantifiers, the prefixes correspond to (Dx 1 Dw*), and (Dw) (Dx 1 Dw) which are non-equivalent if Dw and Dw* need not be the same set).
Finally in S5V with actualist quantifiers, the standard quantifier introduction and elimination rules must be adjusted. Suppose c is a name for an object that does not actually exist; then – Ec is true but (Dx) – Ex is false. The quantifier rules must be those of free logic: we require Ec & fc before we infer (Dv)fv and Ec P fc, as well as the usual EI restrictions, before we infer (Ev)fv.
See also CONTINGENT , ESSENTIALISM , MATHEMATICAL INTUITIONISM , POSSIBLE WORLDS , SECOND -ORDER LOGI. G.Fo.