necessity a modal property attributable to a whole proposition (dictum) just when it is not possible that the proposition be false (the proposition being de dicto necessary). Narrowly construed, a proposition P is logically necessary provided P satisfies certain syntactic conditions, namely, that P’s denial is formally self-contradictory. More broadly, P is logically necessary just when P satisfies certain semantic conditions, namely, that P’s denial is false, and P true, in all possible worlds. These semantic conditions were first suggested by Leibniz, refined by Wittgenstein and Carnap, and fully developed as the possible worlds semantics of Kripke, Hintikka, et al., in the 1960s. Previously, philosophers had to rely largely on intuition to determine the acceptability or otherwise of formulas involving the necessity operator, A, and were at a loss as to which of various axiomatic systems for modal logic, as developed in the 1930s by C. I. Lewis, best captured the notion of logical necessity. There was much debate, for instance, over the characteristic (NN) thesis of Lewis’s system S4, namely, AP / A AP (if P is necessary then it is necessarily necessary). But given a Leibnizian account of the truth conditions for a statement of the form Aa namely (R1) that Aa is true provided a is true in all possible worlds, and (R2) that Aa is false provided there is at least one possible world in which a is false, a proof can be constructed by reductio ad absurdum. For suppose that AP / AAP is false in some arbitrarily chosen world W. Then its antecedent will be true in W, and hence (by R1) it follows (a) that P will be true in all possible worlds. But equally its consequent will be false in W, and hence (by R2) AP will be false in at least one possible world, from which (again by R2) it follows (b) that P will be false in at least one possible world, thus contradicting (a). A similar proof can be constructed for the characteristic thesis of S5, namely, -A-P / A-A-P (if P is possibly true then it is necessarily possible). Necessity is also attributable to a property F of an object O provided it is not possible that (there is no possible world in which) O exists and lacks F – F being de re necessary, internal or essential to O. For instance, the non-repeatable (haecceitist) property of being identical to O is de re necessary (essential) to O, and arguably the repeatable property of being extended is de re necessary to all colored objects. See also CONTINGENT , ESSENTIALISM , HAECCEITY , MODAL LOGIC , POSSIBLE WORLD. R.D.B.