logic

_Logique Et Analyse_ 32 (128):241-245. 1989In 1979 Meyer, Routley and Dunn found a new form of Russell's paradox, different from that found by Curry. By means of elementary syntactic transformations it is possible to clarify the relation which these paradoxes have with each other. Also, we will find new relationships with important metatheoric results relative to Peano's arithmetic (PA). We will …

Terence Tao notes AI can now rigorously check mathematical proofs up to 100,000 lines, raising questions about the future of mathematics and its goals.

This book presents a systematic development of Refined Quadruple Neutrosophic (T, I, N, F) Logic, extending classical neutrosophic logic by distinguishing Neutrality (N) from Pure Indeterminacy (I) and by refining each of the four components—Truth, Indeterminacy, Neutrality, and Falsehood—into semantically labeled subcomponents. The resulting n-valued structure provides a multidimensional represe…

Contents (This is a stub) Idea Subtractive logic is an extension of (propositional or first order) intuitionistic logic with a new connective, subtraction, dual to implication, such that each sentence have a “dual” verifying if and only if . Propositional subtractive logic is a conservative extension over propositional intuitionistic logic, but it is not the case for the first order case. Synt…

Are there semantically adequate languages that are authorities of their own truth but have no meta-language, so they are the sole authority of their own truth? A language on top of Tarski's infinite tower would be such a language. This can be proved by demonstration. We will now demonstrate it by building a rigorous paraconsistent model inside a non-well-founded set domain.

constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism topos, homotopy topos type theory, homotopy type theory canonical form, univalence Bishop set, h-set decidable equality, decidable subset, inhabited set, subsingleton A proof assistant or proof management system is a kind of software designed to help with proofs in formal…

constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism topos, homotopy topos type theory, homotopy type theory canonical form, univalence Bishop set, h-set decidable equality, decidable subset, inhabited set, subsingleton The arithmetical hierarchy or arithmetic hierarchy or Kleene–Mostowski hierarchy is a hierarchy used in c…

HIGHLIGHTS ▸ A theorem that identifies a pathology owes a test for identifying it. The Inversion Theorem states what inversion is: a disposition deprived of its substrate does not diminish but persists in form, occupied by the power that emptied it. It does not state how to determine, in a given case, whether inversion has occurred or whether something has merely worn out. Until it does, the theo…

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 This page is to record the reference: William Lawvere, Robert Rosebrugh: Sets for Mathematics Cambridge Universi…

_Journal of Applied Logic – Ifcolog Journal_ 12 (2):189-220. 2025Building on McNamara’s DWE (Doing Well Enough) framework, a substantial generalization and refinement of that framework extends its expressive resources by adding two ordering operators, changing the underlying semantic ordering’s scope, and adding a number of other operators making fuller use of the resulting expanded frames. A sol…

The very architecture of our modern engagement with philosophy encourages logic-chopping and the straw-manning of positions, reducing philosophy to single arguments.

A Boolean topos is a topos that is also a Boolean category. There are several conditions on a topos that are necessary and sufficient to be Boolean: Let be a Boolean pretopos, i.e. a pretopos in which is also a Boolean category, and let be the classifying topos for . Then it does not follow that is Boolean. In fact, this is rarely the case, even if is the classifying pretopos for a theory in clas…

language, signature theory, first-order theory model interpretation structure in model theory elementary embedding type in model theory compactness theorem diagram of a first-order structure definable set (combinatorial) pregeometry ultraproduct, ultraroot, ultrapower indiscernible sequence? Morleyization Morley sequence? Ramsey theorem? Erdos-Rado theorem? Ehrenfeucht-Fraïssé games (back-and-for…

_Zenodo_. 2026In mathematics the order of operations is taught as a convention. Parentheses, exponents, multiplication, division, addition, subtraction. The sequence is enforced but never derived. No account in the standard curriculum or the foundations literature explains why the operations must be performed in that order rather than another. The order of operations is the dependency chain of ar…

Introduction: The Confluence of Data and Logic In the heart of every computational system lies a fundamental duality: data and logic . Both are forms of information , yet they serve distinct roles, their interplay shaping the very essence of computation. To understand this distinction, consider the mechanical process of a CPU. Data, stored as binary patterns in memory, is passive —it waits to be …

Given a vector space $V_\lambda$ which transforms as an irrep $\lambda$ of a subgroup $H$ of a group $G$, we can form the induced representation $\text{Ind}^G_H(\lambda)$ which is $[G:H]$ copies of ...