logic

Paradoxes have long been treated as flaws in logical systems, with solutions confined to technical patches—setting prohibitions, restricting self-reference, introducing axioms. This essay offers no such patch. Instead, it steps back to the moment before a paradox is even perceived as a contradiction: the original cognitive structure within which logic operates. Reexamining classic paradoxes—the L…

Part 1 — Ten coins in five lines of four From Enigma — the big book of brain-teasers and games of logic (2008), Fabrice Mazza, Sylvain Lhullier, p. 90: Arrange these ten coins into five lines, each ...

basic constructions: strong axioms further There are many systems of formal logic. By “classical logic” one broadly refers to those such systems which reflect the kind of logic as understood, quite literally, by the classics, say starting with Aristotle, Metaphysics 1011b24. If you have never heard of any alternative system of logic, then classical logic is just the kind of logic that you have he…

This book shows that the assumption that classical logic is essentially a bivalent (i.e., 2-valued) logic, a logic of truth and falsity, is an incorrect and harmful conception. Classical logic is certainly a Boolean logic, and the smallest (non-degenerate) Boolean algebra is the 2-element Boolean algebra; however, there are numerous Boolean algebras that have more than 2 elements, such as the 4-,…

By encoding mathematical statements into numbers, mathematician Kurt Gödel used ordinary arithmetic to check whether a statement can be proved

This paper introduces a multi-layered mathematical model designed to quantify the epistemic validity of competing theoretical frameworks. Unlike traditional qualitative heuristics, this framework operationalizes a theory’s overall value by balancing its raw explanatory power against its structural complexity and the psychological bias of the evaluating agent. By combining the linear summation of …

his book introduces Type-k Neutrosophic Sets as a recursive extension of classical neutrosophic logic. While standard neutrosophic sets represent each proposition through three independent components—truth, indeterminacy, and falsity—Type-k Neutrosophic Sets allow each of these components to be recursively characterized by further neutrosophic triplets. This creates a hierarchical epistemic struc…

_Journal of Logic and Computation_ 35 (7):1-27. 2025What happens if we drop the axiom (K) and the necessitation rule from the usual axiomatic presentation of modal logic T? This system was first introduced by Ivlev (1988, Bull. Sect. Log., 17, 114–121). We show that this logic is a syntactical variant of the well-known paraconsistent logic BK (also known as mbCciw), which belongs to a large famil…

t of maximal greatness but from the more fundamental premise that God is the ground of all things. From this premise, the paper derives the ontological omnipresence of God across all possible worlds as the condition of their possibility. Combining this with the incoherence of pure divine absence (via the logic of substitution), it concludes that God exists necessarily in every possible world. The…

basic constructions: strong axioms further Predicative mathematics is a way of doing mathematics without allowing impredicative definitions. Informally, a definition is impredicative if it refers to a totality which includes the thing being defined. For example, the definition of a particular real number as the least upper bound of a given set is impredicative, because it characterizes as a parti…

_Philosophy of Science_. forthcomingMathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for a particular derivation to ''correspond'' to a particular proof. Mere existence of a formalization is not enough, and a sub…

In Diagrammatic Representation and Inference 14th International Conference, Diagrams 2024, Münster, Germany, September 27 – October 1, 2024, Proceedings. Cham: Springer. pp. 476-479. 2024The Region Connection Calculus (RCC) is a qualitative spatial reasoning formalism, developed in knowledge representation and geographical information systems. We argue that RCC can be viewed as a more fine-graine…

_Analysis_. forthcomingThis paper strengthens a triviality result by Santorio (2022, Analysis) for the probability of counterfactuals. The best response may be to drop the idea that the probability of counterfactuals behaves classically.

This paper develops a logic of essence (HLE) in the framework of higher-order logic. The theory aims to provide a general framework for theorizing about the essences of objects, properties, propositions, and logical operations like conjunction, negation, quantification, etc. The first part of the paper presents the formal language and axiom system of HLE. After that, some theorems of the system a…

This paper is the twelfth core treatise of the Evolutionary Manifestation and Ontological Retrieval (hereinafter referred to as EMOR) Philosophy series. It inherits the ultimate conclusion of the eleventh treatise The Ontological Distinction Between Truth and Truth-value: the local closure of truth-value is the endogenous root of the generation of logical contradictions and paradoxes. Based on th…

nLab weak limited principle of omniscience Context Foundations foundations The basis of it all mathematical logic deduction system , natural deduction , sequent calculus , lambda-calculus , judgment type theory , simple type theory , dependent type theory collection , object , type , term , set , element equality , judgmental equality , typal equality universe , size issues higher-order logic Set…

nLab lesser limited principle of omniscience Context Foundations foundations The basis of it all mathematical logic deduction system , natural deduction , sequent calculus , lambda-calculus , judgment type theory , simple type theory , dependent type theory collection , object , type , term , set , element equality , judgmental equality , typal equality universe , size issues higher-order logic S…

A statement can be true or false. But as Kurt Gödel demonstrated, there will always be mathematical assumptions that can neither be proven nor disproven

_Revue Roumaine de Philosophie_ 70 (I):153-168. 2026While the Sorites paradox is often presented in a propositional conditional form, it can be recast using disjunctive, conjunctive, or modus tollens syllogism. This paper challenges Kit Fine’s claim that only the conjunctive form is motivated. All four propositional forms are effective because they express three deeper, first-order principles: (1…

_History and Philosophy of Logic_ 47 (2):201-216. 2026Some of the more prominent contributions to the last fifty years of scholarship on Aristotle’s syllogistic suggest a conceptual framework under which the syllogistic is a logic, a system of inferential reasoning, only if it is not a theory, a system concerned with ontology or general facts. I argue that this a misleading interpretative framewo…