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The paper discusses several first-order modal logics that extend the classical predicate calculus. The model theory involves possible worlds with world-variable domains. The logics rely on the philosophical tenet known as serious actualism in that within modal contexts they allow existential generalization from atomic formulas. The language may or may not have a sign of identity, includes no primitive existence predicate, and has individual constants. Some logics correspond to various standard constraints on the accessibility relation, whereas others correspond to various constraints on the domains of the worlds. Soundness and strong completeness are proved in every case; a novel method is used for proving completeness.
In a finitary closure space, irreducible sets behave like two-valued models, with membership playing the role of satisfaction. If f is a function on such a space and the membership of $fx_1 ,\ldots, x_n$ in an irreducible set is determined by the presence or absence of the inputs $x_1 ,\ldots, x_n$ in that set, then f is a kind of truth function. The existence of some of these truth functions is enough to guarantee that every irreducible set is maximally consistent. The closure space is then said to be expressive. This paper identifies the two-valued truth functional conditions that guarantee expressiveness.
We show that, if a suitable intuitionistic metatheory proves that consistency implies satisfiability for subfinite sets of propositional formulas relative either to standard structures or to Kripke models, then that metatheory also proves every negative instance of every classical propositional tautology. Since reasonable intuitionistic set theories such as HAS or IZF do not demonstrate all such negative instances, these theories cannot prove completeness for intuitionistic propositional logic in the present sense.
Hrushovski constructed an $\omega$-categorical stable pseudoplane which refuted Lachlan's conjecture. In this note, we show that an $\omega$-categorical projective plane cannot be constructed by "the Hrushovski method."