In this article we revisit a number of disputes regarding significance logics---i.e., inferential frameworks capable of handling meaningless, although grammatical, sentences---that took place in a series of articles most of which appeared in the Australasian Journal of Philosophy between 1966 and 1978. These debates concern (i) the way in which logical consequence ought to be approached in the context of a significance logic, and (ii) the way in which the logical vocabulary has to be modified (either by restricting some notions, (...) or by adding some vocabulary) to keep as much of Classical Logic as possible. Our aim is to show that the divisions arising from these disputes can be dissolved in the context of a novel and intuitive proposal that we put forward. (shrink)

The basis of the paper is a logic of analytical consequential implication, CI.0, which is known to be equivalent to the well-known modal system KT thanks to the definition A → B = df A ⥽ B ∧ Ξ (Α, Β), Ξ (Α, Β) being a symbol for what is called here Equimodality Property: (□A ≡ □B) ∧ (◊A ≡ ◊B). Extending CI.0 (=KT) with axioms and rules for the so-called circumstantial operator symbolized by *, one obtains a system CI.0*Eq (...) in whose language one can define an operator ↠ suitable to formalize context-dependent conditionals (so to counterfactual conditionals) via the definition (Def ↠) A ↠ B = df *A ⥽ B ∧ Ξ (Α, Β).. The central problem of the paper is to identify inside CI.0*Eq + Def ↠ a set of axioms yielding the fragment consisting of all and only theorems in which only truth-functional operators and the two operators → and ↠ occur. This system, here called CI.0, is introduced in §3. In view of the intended purpose, it is introduced a complete and tableau-decidable system KTw, which is an extension of KT with axioms for the so-called quasi-variables w(A), w(B)… Three translation functions among the three languages used in the paper are then introduced. The first, Tr, maps every wff of form *A into wffs of form w(A) ∧ A and translates theorems of CI.0*Eq into theorems of KTw. The second, t°, translates theorems of CI.0 into theorems of CI.0*Eq by applying Def↠. A third one, t, translates theorems of CI.0 into theorems of KTw. As a consequence, it follows that t°A = TrtA. In §4 it is proved that CI.0 is complete w.r.t. the class of CI.0-models of form where f is a selection function and R an access relation. It is then proved (i) that CI.0 models may be converted into KTw-models; (ii) that the truth-value of a proposition in a world of a CI.0-model is preserved in the same world of the derived KTw-model; (iii) that A is a CI.0-thesis iff its translation tA is KTw-thesis. It follows that, if A is not a CI.0-thesis, tA is not a KTw-thesis, so also that TrtA is not a CI.0*Eq-thesis. Since t°A = TrtA and t°A is a function built on Def↠, this proves that every t°-translation of a CI.0-wff that is a CI.0*Eq-theorem is a CI.0-theorem. Since the converse proposition is proved in §2, their conjunction establishes the desired result. In the final section, it is proved that CI.0 is tableau-decidable, that ↠ is not a trivial operator and that ↠ enjoys the positive and negative properties required in §1 for context-dependent conditionals. Some final remarks suggest that studying the relations between CI.0 and systems of classical conditional logic may be a promising line of investigation. (shrink)

Here, we discuss logical, philosophical and technical problems associated to relating logic and relating semantics. To do so, we proceed in three steps. The first step is devoted to providing an introduction to both relating logic and relating semantics. We discuss this problem on the example of different languages. Second, we address some of the main research directions and their philosophical applications to non-classical logics, particularly to connexive logics. Third, we discuss some technical problems related to relating semantics, and its (...) application to philosophy of science, language and pragmatics. (shrink)

Logics of significance have been proposed in an attempt to overcome the shortcomings of classical logic as a model of reasoning in the presence of nonsignificant (e.g., meaningless, ill-formed, unverifiable) sentences. Many-valued logicians have addressed this problem by introducing logics with infectious truth values. Cases in point are the weak Kleene logics B3 (paracomplete weak Kleene logic) and PWK (paraconsistent weak Kleene logic). Over time, it has become clear that the valid entailments of these significance logics obey variable inclusion patterns (...) that link them to other, usually better known, logics—such patterns, however, allow for disturbing exceptions. Logics of pure (left or right) variable inclusion have been introduced with an eye to removing these exceptions. In this paper, we consider the pure left variable inclusion companion of classical logic and give a complete description of its subclassical extensions. We also provide relative axiomatizations and characteristic (sets of) matrices for each one of these extensions, as well as syntactic descriptions (in terms of variable inclusion criteria) for the valid entailments of some of them, and determine in each case the algebra reducts of the Suszko-reduced matrix models. (shrink)