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We construct models for the level-by-level equivalence between strong compactness and supercompactness containing failures of the Generalized Continuum Hypothesis (GCH) at inaccessible cardinals. In one of these models, no cardinal is supercompact up to an inaccessible cardinal, and for every inaccessible cardinal , . In another of these models, no cardinal is supercompact up to an inaccessible cardinal, and the only inaccessible cardinals at which GCH holds are also measurable. These results extend and generalize earlier work of the author.
In bilateral systems for classical logic, assertion and denial occur as primitive signs on formulas. Such systems lend themselves to an inferentialist story about how truth-conditional content of connectives can be determined by inference rules. In particular, for classical logic there is a bilateral proof system which has a property that Carnap in 1943 called categoricity. We show that categorical systems can be given for any finite many-valued logic using -sided sequent calculus. These systems are understood as a further development of bilateralism—call it multilateralism. The overarching idea is that multilateral proof systems can incorporate the logic of a variety of denial speech acts. So against Frege we say that denial is not the negation of assertion and, with Mark Twain, that denial is more than a river in Egypt.
Typical applications of Hintikka’s game-theoretical semantics (GTS) give rise to semantic attributes—truth, falsity—expressible in the -fragment of second-order logic. Actually a much more general notion of semantic attribute is motivated by strategic considerations. When identifying such a generalization, the notion of classical negation plays a crucial role. We study two languages, and , in both of which two negation signs are available: and . The latter is the usual GTS negation which transposes the players’ roles, while the former will be interpreted via the notion of mode. Logic extends independence-friendly (IF) logic; behaves as classical negation in . Logic extends , and it is shown to capture the -fragment of third-order logic. Consequently the classical negation remains inexpressible in .
Since Aristotle and the Stoa, there has been a clash, worsened by modern predicate logic, between logically defined operator meanings and natural intuitions. Pragmatics has tried to neutralize the clash by an appeal to the Gricean conversational maxims. The present study argues that the pragmatic attempt has been unsuccessful. The “softness” of the Gricean explanation fails to do justice to the robustness of the intuitions concerned, leaving the relation between the principles evoked and the observed facts opaque. Moreover, there are cases where the Gricean maxims fail to apply. A more adequate solution consists in the devising of a sound natural logic, part of the innate cognitive equipment of mankind. This account has proved successful in conjunction with a postulated cognitive mechanism in virtue of which the universe of discourse (Un) is stepwise and recursively restricted, so that the negation selects different complements according to the degree of restrictedness of Un. This mechanism explains not only the discrepancies between natural logical intuitions and known logical systems; it also accounts for certain systematic lexicalization gaps in the languages of the world. Finally, it is shown how stepwise restriction of Un produces the ontogenesis of natural predicate logic, while at the same time resolving the intuitive clashes with established logical systems that the Gricean maxims sought to explain.
This is the first part of a study on cardinals that are characterizable by Scott sentences. Building on previous work of Hjorth, Malitz, and Baumgartner, we study which cardinals are characterizable by a Scott sentence , in the sense that characterizes , if has a model of size but no models of size .
We show that the set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions, and countable products (see Theorems 3.3 and 4.6 and Corollary 4.8). We also prove that if is characterized by a Scott sentence, at least one of , , or is homogeneously characterizable (see Definitions 1.3 and 1.4 and Theorem 3.19). Based on an argument of Shelah, we give counterexamples that characterizable cardinals are not closed under predecessors or cofinalities.
Peter van Inwagen has long claimed that he doesn’t understand substitutional quantification and that the notion is, in fact, meaningless. Van Inwagen identifies the source of his bewilderment as an inability to understand the proposition expressed by a simple sentence like “() ( is a dog),” where “” is the existential quantifier understood substitutionally. I should think that the proposition expressed by this sentence is the same as that expressed by “() ( is a dog).” So what’s the problem? The problem, I suggest, is that van Inwagen takes traditional existential quantification to be ontologically committing and substitutional quantification to be ontologically noncommitting, which requires that the two quantifiers have different meanings—but no different meaning for the substitutional quantifier is forthcoming. What van Inwagen fails to appreciate is that substitutional quantification is directed at a criterion of ontological commitment, namely, W. V. O. Quine’s, which is quite different from van Inwagen’s criterion. Substitutional quantification successfully avoids the commitments Quine’s criterion would engender but has the same commitments as existential quantification given van Inwagen’s criterion. The question, then, is whether the existential quantifier is ontologically committing, as van Inwagen believes. The answer to that question will depend on whether the ordinary language “there is/are,” which is codified by the existential quantifier, is ontologically committing. There are good reasons to doubt that it is.
In this paper we prove that the preordering of provable implication over any recursively enumerable theory containing a modicum of arithmetic is uniformly dense. This means that we can find a recursive extensional density function for . A recursive function is a density function if it computes, for and with , an element such that . The function is extensional if it preserves -provable equivalence. Secondly, we prove a general result that implies that, for extensions of elementary arithmetic, the ordering restricted to -sentences is uniformly dense. In the last section we provide historical notes and background material.