logic

_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…

Richard Pettigrew, currently professor of philosophy at the University of Bristol, will be moving to the University of Oxford, where he will be the new Wykeham Chair of Logic. Professor Pettigrew works on questions across a range of philosophical subfields, including epistemology, formal epistemology, decision theory, logic, philosophy of math, and ethics. He is the author of several books, inclu…

Plato's Divided Line in Republic VI prescribes one line cut into four unequal segments under the proportion whole : larger :: larger : smaller. The geometry permits exactly one ratio: the Golden Ratio Φ. Under the mirror-canonical normalization the four segments are Φ, 1, 1, 1/Φ, summing to Φ³, where the power is unit-dependent and the ratio is the invariant. The two middle segments are conjugate…

Set theory occupies the foundational stratum of modern mathematics, yet the question of whether its machinery reflects structural necessity or descriptive convenience has never been posed from outside the formal tradition itself. Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC) is universally adopted as the axiomatic substrate for virtually all of contemporary mathematics, from analysis…

Prove that no order can be defined in the complex field that turns it into an ordered field. My question is what exactly does it mean to "define an order?". Here is my proof, which I think ...

Let $L_0$ be an axiomatic proof system for propositional logic which has the following axiom schema: A1. $(\alpha\to(\beta\to\alpha))$ A2. ...

This paper establishes the formal structure of La Profilée. Reading P000 — the canonical derivation — against the requirements of formal axiomatics reveals a precise result: LP has one genuine primitive (the definition of state as minimal distinguishability) and one genuine axiom (real transformation). Everything else — M1, M3, F/M/K, IR ≤ 1, structural time, and the regime architecture — is deri…

This paper argues against the standard mathematical and linguistic treatment of zero, nullity, absence, empty collections, default falsity, and related symbolic devices. The critique is not merely that the glyph $\Zglyph$ is historically convenient, pedagogically overused, or technically dangerous in certain operations. The deeper criticism is that the entire zero-family performs a forbidden ling…

We first offer a minimal version of PRA where the subscript on any function symbol f_i allows us to recover the axiom defining the function symbol. An algorithm is then constructed where the relevant subscript guides the generation of a "canonical proof" of f_i(i,n) = m. (Halting occurs when a line is reached with zero occurrences of 'f' to the right of '='.) The resulting algorithm is then shown…

If you switch a lamp on and off an infinite number of times, will the light end up on or off? Somehow math says both

At 25, Kurt Gödel proved there can never be a mathematical “theory of everything.” Columnist Natalie Wolchover explores the implications. The post What Do Gödel’s Incompleteness Theorems Truly Mean? first appeared on Quanta Magazine

The Brouwer-Hilbert dispute is often treated as a conflict between classical and intuitionistic logic. This paper argues that the deeper structural distinction is between formal admissibility and constructive existence. Hilbertian formalism asks whether a symbolic transformation is lawful inside a formal system; Brouwerian intuitionism asks whether the asserted object or proposition has been cons…

Every theory of persistence must answer two questions: does this system continue to exist? And: does the continuing system remain the same system? These are not the same question. La Profilée formalizes the first as the persistence condition IR ≤ 1 and the second as the Frame Continuity Condition (FCC). The Q1/Q2 Separation Theorem establishes that Q2 presupposes Q1, while Q1 does not imply Q2. T…

_Journal of Applied Logics_ 8:89-113. 2021We discuss the strategy of using dyslogistic terms in a novel, laudatory manner, or eulogistic terms pejoratively. By such up- and downgrading of evaluative terms the proponent of a standpoint may attempt to turn the tables in a public controversy: what formerly looked like a bad argument comes to be regarded as a strong one, or vice versa. Is this a lici…

Synthetic domain theory, as its name suggests, is a synthetic axiomatization of domain theory, typically in toposes. The slogan for synthetic domain theory is that “domains are sets”, or more precisely, “domains are certain intuitionistic sets” in that one works in an intuitionistic set theory with certain anti-classical axioms and defines a notion of domain internally. These synthetic domains ac…

Written: Aug. 14, 2007. Theorems are nice, but they are usually deadends. A lemma may be "trivial", or easy, to prove, once stated, but if it is good, its value far surpasses even the deepest theorems. Wiki has a fairly long list of lemmas, including the powerful Schur's Lemma, Lovasz's Local Lemma, and many others. But my absolute favorite, that lead to at least two Fields medals (so far), is Sz…

Bearers track closure classes; closure classes do not track bearers. This inversion is the substantive claim the present paper defends, and its grounds are structural: the framework operates at the admissibility stratum, characterizing what closure formation can produce; bearers are realization-stratum products of closure formation; the partition between class members and non-members is a consequ…

An apartness relation is a binary relation that, instead of saying when two things are the same (as an equivalence relation), states when two things are different – an inequality relation. Apartness relations are most used in constructive mathematics; in classical mathematics, equivalence relations can take their place (mediated by negation). The apartness relations that we discuss here are somet…

constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism In constructive mathematics, we often do algebra by equipping an algebra with a tight apartness (and requiring the algebraic operations to be strongly extensional), in the sense of ring with tight apartness. In this context, it is convenient to replace subalgebras with an…