Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that the domain of interpretation is not empty, every name denotes exactly one object in the domain and the quantifiers have existential import. Free logics usually reject the claim that names need to denote in, and of the systems considered in this paper, the positive free logic concedes that (...) some atomic formulas containing non-denoting names are true, while negative free logic rejects even the latter claim. Inclusive logics, which reject, are likewise considered. These logics have complex and varied axiomatizations and semantics, and the goal of this paper is to present an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization, using sequent calculi which possess all the desired structural properties of a good proof system, including admissibility of contraction and cut, while streamlining free logics in a way no other approach has. We then present a simple and unified system of abstract semantics, which allows for a straightforward demonstration of the meta-theoretical properties, and offers insights into the relationship between different logics. The final part of this paper is dedicated to extending the system with modalities by using a labeled sequent calculus, and here we are again able to map out the different approaches and their mutual relations using the same framework. (shrink)

What an intuitionist may refer to with respect to a given epistemic state depends not only on that epistemic state itself but on whether it is viewed concurrently from within, in the hindsight of some later state, or ideally from a standpoint “beyond” all epistemic states (though the latter perspective is no longer strictly intuitionistic). Each of these three perspectives has a different—and, in the last two cases, a novel—logic and semantics. This paper explains these logics and their semantics and (...) provides soundness and completeness proofs. It provides, moreover, a critique of some common versions of Kripke semantics for intuitionistic logic and suggests ways of modifying them to take account of the perspective-relativity of reference. (shrink)

The search for the extensions of sentences can be guided by Frege’s “principle of compositionality of extension”, according to which the extension of a composed expression depends only on its logical form and the extensions of its parts capable of having extensions. By means of this principle, a strict criterion for the admissibility of objects as extensions of sentences can be derived: every object is admissible as the extension of a sentence that is preserved under the substitution of co-extensional expressions. (...) The question is: what are the extensions of elementary sentences containing empty singular terms, like ‘Vulcan rotates’. It can be demonstrated that in such sentences, states of affairs as structured objects (but not truth-values) are preserved under the substitution of co-extensional expressions. Hence, such states of affairs are admissible (while truth-values are not) as extensions of elementary sentences containing empty singular terms. (shrink)

I’ve often noticed how debates within the SSE community sometimes parallel debates in the political arena, perhaps especially with respect to the passion they elicit and the intolerance and condescension sometimes lavished on members of the “opposition.” Occasionally, of course, the debates in the SSE are nearly indistinguishable from those in the political arena—say, over the evidence for human-caused climate change. But what I find most striking is how the passion, intolerance, etc.—perhaps most often displayed by those defending whatever the (...) “received” view happens to be—betrays either a surprising ignorance or else a seemingly convenient lapse of memory, one that probably wouldn’t appear in less emotionally charged contexts. What impassioned partisans tend to ignore or forget concerns (a) the tentative nature of both scientific pronouncements and knowledge claims generally (including matters ostensibly much more secure than those under debate), as well as (b) the extensive network of assumptions on which every knowledge claim rests. So I’d like to offer what I hope will be a perspective-enhancer, concerning how even our allegedly most secure and fundamental pieces of a priori knowledge are themselves open to reasonable debate. A widespread, but naïve, view of logic is that no rational person could doubt its elementary laws. But that bit of popular “wisdom” is demonstrably false. And if that’s the case, then so much the worse for the degree of certitude we can expect in more controversial arenas. Let me illustrate with a few examples.1. (shrink)