This paper presents a distinctively multiplicative quantificational principle that arguably captures the problematic aspects of Zardini's infinitary rules for a multiplicative quantifier within the context of the semantic paradoxes and the theoretical goal to obtain a (omega)-consistent theory of transparent truth. After showing that the principle is derivable with Zardini's rules and that one obtains through vacuous quantification an inconsistent theory of truth if truth is transparent, the paper presents two results regarding the principle and omega-inconsistency. First, the principle is (...) used to obtain a non-classical variant of McGee's omega-inconsistency result for certain classical theories of truth. Second, it is demonstrated that the conditions for a truth-theoretic variant of Bacon's omega-inconsistency result for certain non-classical theories of transparent truth implies that the principle holds for the paradoxical formula. Finally, the paper argues that the paradoxical reasoning that the principle enables is structurally similar to the kind of infinitary reasoning popularised by Hilbert's Grand Hotel. (shrink)

In order to reason in a non-trivializing way with contradictions, para- consistent logics reject some classically valid inferences. As a way of re- covering some of these inferences, Graham Priest ([Priest, 1991]) proposed to nonmonotonically strengthen the Logic of Paradox by allowing the se- lection of “less inconsistent” models via a comparison of their respective inconsistent parts. This move recaptures a good portion of classical logic in that it does not block, e.g., disjunctive syllogism, unless it is applied to contradictory (...) assumptions. In Priest’s approach the inconsistent parts of models are compared in an extensional way by considering their inconsis- tent objects. This distinguishes his system from the standard format of (inconsistency-)adaptive logics pioneered by Diderik Batens, according to which (atomic) contradictions validated in models form the basis of their comparison. A well-known problem for Priest’s extensional approach is its lack of the Strong Reassurance property, i.e., for specific settings there may be infinitely descending chains of less and less inconsistent models, thus never reaching a minimally inconsistent model. In this paper, we demonstrate that Strong Reassurance holds for the extensional approach under a cardinality-based comparison of the incon- sistent parts of models. Furthermore, we introduce and investigate the metatheory of the class of first-order nonmonotonic inconsistency-tolerant construct over the extensional or quantitative comparisons of their respec- tive models. Core model-theoretic properties for these logics, such as the Löwenheim-Skolem theorems, along with other nonmonotonic properties, are further studied. (shrink)

In order to reason in a non-trivializing way with contradictions, paraconsistent logics reject some classically valid inferences. As a way to recover some of these inferences, Graham Priest proposed to nonmonotonically strengthen the Logic of Paradox by allowing the selection of “less inconsistent” models via a comparison of their respective inconsistent parts. This move recaptures a good portion of classical logic in that it does not block, e.g., disjunctive syllogism, unless it is applied to contradictory assumptions. In Priest’s approach the (...) inconsistent parts of models are compared in an extensional way by consid- ering their inconsistent objects. This distinguishes his system from the standard format of (inconsistency-)adaptive logics pioneered by Diderik Batens, according to which (atomic) contradictions validated in models form the basis of their comparison. A well-known prob- lem for Priest’s extensional approach is its lack of the Strong Reassurance property, i.e., for specific settings there may be infinitely descending chains of less and less inconsistent models, thus never reaching a minimally inconsistent model. In the following paper, we show that Strong Reassurance holds for the extensional ap- proach under a cardinality-based comparison of the inconsistent parts of models. We develop and study the meta-theory of a class of nonmonotonic inconsistency-tolerant logics based on the extensional and the quantitative comparisons of their respective models, including important model-theoretic properties, such as the Löwenheim-Skolem theorems, as well as principles of nonmonotonic inference. (shrink)

This paper develops the model theory of normal modal logics based on partial “possibilities” instead of total “worlds,” following Humberstone [1981] instead of Kripke [1963]. Possibility semantics can be seen as extending to modal logic the semantics for classical logic used in weak forcing in set theory, or as semanticizing a negative translation of classical modal logic into intuitionistic modal logic. Thus, possibility frames are based on posets with accessibility relations, like intuitionistic modal frames, but with the constraint that the (...) interpretation of every formula is a regular open set in the Alexandrov topology on the poset. The standard world frames for modal logic are the special case of possibility frames wherein the poset is discrete. The analogues of classical Kripke frames, i.e., full world frames, are full possibility frames, in which propositional variables may be interpreted as any regular open sets. We develop the beginnings of duality theory, definability/correspondence theory, and completeness theory for possibility frames. The duality theory, relating possibility frames to Boolean algebras with operators (BAOs), shows the way in which full possibility frames are a generalization of Kripke frames. Whereas Thomason [1975a] established a duality between the category of Kripke frames with p-morphisms and the category of complete (C), atomic (A), and completely additive (V) BAOs with complete BAO-homomorphisms (these categories being dually equivalent), we establish a duality between the category of full possibility frames with possibility morphisms and the category of CV-BAOs, i.e., allowing non-atomic BAOs, with complete BAO-homomorphisms (the latter category being dually equivalent to a reflective subcategory of the former). This parallels the connection between forcing posets and Boolean-valued models in set theory, but now with accessibility relations and modal operators involved. Generalizing further, we introduce a class of principal possibility frames that capture the generality of V-BAOs. If we do not require a full or principal frame, then every BAO has an equivalent possibility frame, whose possibilities are proper filters in the BAO. With this filter representation, which does not require the ultrafilter axiom, we are lead to a notion of filter-descriptive possibility frames. Whereas Goldblatt [1974] showed that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of descriptive world frames with p-morphisms, we show that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of filter-descriptive possibility frames with p-morphisms. Applying our duality theory to definability theory, we prove analogues for possibility semantics of theorems of Goldblatt [1974] and Goldblatt and Thomason [1975] characterizing modally definable classes of frames. In addition, we discuss analogues for possibility semantics of first-order correspondence results in the style of Lemmon and Scott [1977], Sahlqvist [1975], and van Benthem [1976a]. Finally, applying our duality theory to completeness theory, we show that there are continuum many normal modal logics that can be characterized by full possibility frames but not by Kripke frames, that all Sahlqvist logics can be characterized by full possibility frames that contain no worlds, and that all normal modal logics can be characterized by filter-descriptive possibility frames. (shrink)

Discussive logic was introduced by Jaskowski as a logic of discussion. In this note we show that some natural translation-based formalizations of discussive logic in modal logic do not yield a paraconsistent logic but rather classical logic. Some alternative modal formalizations of discussive logic that avoid the collapse into classical logic are put forward.

The logic subDL and its quantified extension subDLQ were proposed by Badia and Weber (Dialethism and its Applications, 2019: 155-176) as a basis for developing a version of mathematics in which paradoxes are harmless. In the present paper, subDL as defined in the literature is shown to be too strong to support the theories which motivate it. The crucial point is that contraction is derivable in subDL. It follows that the semantic structure used by Badia and Weber to invalidate contraction (...) is not, in fact, a model of subDL. Here we identify the axioms responsible for contraction in subDL and prove that the logic, weakened by removal of these axioms, is contraction-free and paraconsistent. (shrink)

Connexive expansions of relevant logics tend to prove every negated implication formula. In this paper I discuss why they tend to satisfy this unsavoury property, and discuss avenues by which it can be avoided, providing logics which stand as proofs of concept that these avenues can be made to work.