logic

The Nitescence Theorem constructs a canonical mechanism for navigating Gödelian incompleteness through structural ascension. Given a consistent, Σ1-sound effective theory S, the nitescence operator N(S) = S + Con*(S)* resolves an obstruction determined by S itself rather than arbitrarily chosen among its undecidable propositions. The construction is classical—the Turing–Feferman consistency progr…

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…

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 …

The whole Principia is very large: It is said that the book is famous for taking a thousand pages to prove that 1+1=2. As the preface stresses, the proofs are excruciatingly detailed so to remove the chance of an unstated premise being used in a proof. The goal of Principia was to put forward a set of very basic notions, and show that they and they alone are sufficient for the whole Mathematics. …

Mathematics > Logic Title:A SAT Attack on Tarski's High School Algebra Problem View PDF HTML (experimental) Abstract:Tarski's high school algebra problem asks whether every true identity concerning addition, multiplication, and exponentiation of positive integers follows from a list of 11 elementary identities. Surprisingly, Wilkie showed that the following identity is valid over the …

In Proceedings of Sinn und Bedeutung 29. forthcomingWe submit a new puzzle for logicality. It has been argued that the logicality of language is reflected both in the exclusion of contradictions and tautologies, and in the tendency to maximize information in various interpretive phenomena, such as scalar implicatures. We discuss that these two lines of research take on different and intimately co…

_Kriterion – Journal of Philosophy_. forthcomingTranscendental Logic (TL) offers a mathematisation of the unsayable background-reality (noumenal domain, N-domain) that underlies the logical models (phenomenal domain, P-domain). According to TL, the N-domain is unsayable not because it is unstructured or inaccessible, but because its structure is governed by the laws of orthomodular logic. In cont…

It’s been an interesting few weeks for counterexamples. This post is basically my perspective of what has been going on in the world of formalization, AI tools and, in particular, counterexamples. Unit distance Two months ago today (20th May 2026), … Continue reading →

Motivation I've been playing Clues By Sam. It's a basic logic puzzle type game where you decide whether each person is innocent or a criminal. When I get stuck, I can reliably get unstuck by going ...

In dependent type theory, having a type of all types results in various paradoxes, such as Russell's paradox and Girard's paradox. There are two ways to resolve this issue. One way is to simply add a universe of small types and accept that not all types are small. On the other hand, one can use a hierarchy of universes or universe hierarchy and postulate that every type is an element of universes…