mathematical-logic

Who was the first major female logician in the history of logic who made more major contributions to the development of logic than to other related fields (like mathematics or computing)? Wikipedia's ...

The question is in the title. How would you formalize the statement :"all generalizations are false, except this one"? in predicate calculus?

Wikipedia gives the following proof that the Cauchy-completeness of the real numbers implies Dedekind-completeness: Let $S$ be a nonempty set of real numbers. If $S$ has exactly one element, then its ...

Does there exist a turing degree $A$, such that $A \not \geq_T \emptyset''$ and $A\oplus \emptyset' \geq_T \emptyset''$? (edit: yes, by Friedberg's inversion theorem) I asked this to Claude AI which ...

indiscernible sequence? Morley sequence? Ramsey theorem? Erdos-Rado theorem? Ehrenfeucht-Fraïssé games (back-and-forth games) Hrushovski construction? generic predicate? Given a collection of finite(ly-generated) first-order structures which formally resemble collections of finite(ly-generated) substructures found inside an infinite structure, there are additional assumptions on such collections …

Reverse mathematics refers mainly to a program introduced by Harvey Friedman and Stephen Simpson, which aims to establish for many theorems of classical analysis exactly which set existence principles they rely on. They consider the formalization of analysis based on Polish spaces in the language of second order arithmetic (hence the need to focus on separable spaces). One then tries to show that…

_Mathematical Structures in Computer Science_ 18 (1):145-164. 2008This paper proposes a new relevant logic B+⊓⊔, which is obtained by adding two binary connectives, intensional conjunction ⊓ and intensional disjunction ⊔, to Meyer–Routley minimal positive relevant logic B+, where ⊓ and ⊔ are weaker than fusion ◦ and fission +, respectively. We give Kripke-style semantics for B+⊓⊔, with →, ⊓ and ⊔…

_Bulletin of the Section of Logic_. forthcomingThis article aims to study, proof-theoretically and semantically, Gentzen-style sequent calculi (including possibly Cut-free and Identity-free systems), containing combinations of canonical and cocanonical rules, i.e. Gentzen systems for sequents, with well-behaved forms of left and right introduction and elimination rules for logical expressions. Ou…

I am reading through Errett Bishops "Foundations of Constructive Analysis" because I got interested in constructive as well as intuitionistic mathematics, and have a confusion in regards to ...

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…