Three paralogisms, one logic of utterances

Logic, Rationality, and Interaction

The semantic paradoxes and the paradoxes of vagueness ('soritical paradoxes') display remarkable family resemblances. In particular, the same nonclassical logics have been (independently) applied to both kinds of paradoxes. These facts have been taken by some authors to suggest that truth and vagueness require a uni ed logical framework (see e.g. [5,3]). Some authors go further, and argue that truth is itself a vague or indeterminate concept (see e.g. [7,4]). Importantly, however, there currently is no identi cation of what the common features of semantic and soritical paradoxes exactly consist in. This is what we aim to do in this work: we analyze semantic and soritical paradoxes, and develop our analysis into a theory of paradoxicality. The uni cation of the paradoxes of truth and vagueness we propose here has a wide scope, but for the sake of concreteness we focus on four three-valued logics.

Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic is, the authors write in the preface, intended" to give philosophy students a basic grounding in philosophical logic, in a way that connects with the motivations they derive elsewhere from philosophy"(p. ix). In providing a uniform framework within which one can study the behaviour of, and interactions between, modal, many-valued, intuitionistic, and paraconsistent logic, the text provides a useful contribution along these lines.

This article introduces, studies, and applies a new system of logic which is called 'HYPE'. In HYPE, formulas are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionis-tic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE's logical connectives. Furthermore, HYPE's first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work.

Recently, it has become a custom to treat questions (or, better, questioning) as a game between two subjects. Unfortunately, one rarely goes beyond the scheme of Questioner-Scientist and Answerer-Nature, although the Interlocutor so conceived displays some undesirable features. This paper argues for the idea that logic of questions can be build as a logic of the game between “knowledge resources” persons or theories, rather than errant Scientist and omniscient Nature. To this end the concept of epistemically-possible worlds is discussed, which is conceived as analogous to that of possible worlds in modal logic. And, furthermore, the concepts of relation of epistemic alternativeness and of epistemically-alternative worlds are introduced. On this basis a version of semantics for propositional, three-valued logic of questions is offered and semantic proofs of some theses are given.

This is the first draft of an article introducing, studying, and applying a new system of logic which is called 'HYPE'. In HYPE, formulas are evaluated at states that may be " gappy " (partial) or " glutty " (overdetermined). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE's logical connectives. HYPE's first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when in-tuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyper-intensional contexts, though the details of this application need to be left to follow-up work.

It is proposed that "This sentence is not true" may be true under some interpretations and false under others. The revenge challenge given by "This sentence is false in at least one interpretation" is handled through the observation that a sentence is arguable true under all interpretations if and only if it is provable. This makes the revenge challenge equivalent to the one given by "This sentence is not provable." Apart from arguing a link with Gödel's first incompleteness theorem we also argue that the way a Liar sentence may be true under one interpretation and false under another is related to the way a "many-valued function" in mathematics may have more than one value. We set up a mathematical framework where sentences may be given many different interpretations. Examples are also given of how Liar sentences may be given many interpretations in practice depending on the intentions of speakers and how they are understood, and our approach to dealing with Liar sentences is compared to those of others.

Truth Meets Vagueness. Unifying the Semantic and the Soritical Paradoxes, 2023

Semantic and soritical paradoxes display remarkable family resemblances. For one thing, several non-classical logics have been independently applied to both kinds of paradoxes. For another, revenge paradoxes and higher-order vagueness-among the most serious problems targeting solutions to semantic and soritical paradoxes-exhibit a rather similar dynamics. Some authors have taken these facts to suggest that truth and vagueness require a unified logical framework, or perhaps that the truth predicate is itself vague. However, a common core of semantic and soritical paradoxes has not been identified yet, and no explanation of their relationships has been provided. Here we aim at filling this lacuna, in the framework of many-valued logics. We provide a unified diagnosis of semantic and soritical paradoxes, identifying their source in a general form of indiscernibility. We then develop our diagnosis into a theory of paradoxicality, which formalizes both semantic and soritical paradoxes as arguments involving specific instances of our generalized indiscernibility principle, and correctly predicts which logics can non-trivially solve them.

2016

The traditional representation of an epistemic scenario as a model covers only complete descriptions that specify truth values of all assertions. However, many, perhaps most, epistemic scenarios are not complete and allow partial or asymmetric knowledge. Syntactic Epistemic Logic, SEL, suggests viewing an epistemic situation as a set of syntactic conditions rather than as a model, thus also capturing incomplete descriptions. This helps to extend the scope of Epistemic Game Theory. In addition, SEL closes the conceptual and technical gap, identified by R. Aumann, between the syntactic character of game descriptions and the semantic method of analyzing games.

A many-valued logic in 8 truth-values based upon Classical logic termed ‘Universal Logic’, denoted U8, provides a correspondence to the ‘if-then’ implication meaning of natural language. Truth tables of implication and equivalence for U8 will be given, expanding the definition of validity. Accordingly, a new analysis of the ‘paradoxes of material implication’ will be undertaken. Material implication will be found to be identical with Universal logic when Boolean assignments are employed signifying that Classical is a subset of U8 logic. However, when 2 truth-values {true, false} are employed, denoted U2, implication resembles material equivalence and so validity is amended. As illustration of this approach five major ‘paradoxes of material implication’ will be analysed in terms of U2 validity. Remarkable results will be elucidated showing two ‘paradoxes’ to be affirmed while three were denied providing evidence that ‘Universal Logic’ offers an intuitive inductive logic.

We introduce the reader to game-theoretic semantics (GTS) and to chart some of its current directions in formal epistemology. GTS was originally developed by Jaakko Hintikka in the 1960s and became one of the main approaches in logical and linguistic semantics. I place games in a wider historical and systematic perspective within the overall development of logic, and explore some of the recent advances.