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Fact verification systems typically assess whether a claim is supported by retrieved evidence, assuming that truthfulness depends solely on what is stated. However, many real-world claims are half-truths, factually correct yet misleading…
This paper presents correspondence between Albert Einstein and the mathematical analyst J. L. B. Cooper on the Einstein-Podolsky-Rosen (EPR) paradox of quantum theory published in 1935. Two letters written by Cooper, and the replies from…
We study a real valued propositional logic with unbounded positive and negative truth values that we call R-valued logic. Such logic slightly extends continuous propositional logic which, in turn, builds on Lukasiewicz many-valued logic.…
Towards better explainability in the field of information retrieval, we present CREDENCE, an interactive tool capable of generating counterfactual explanations for document rankers. Embracing the unique properties of the ranking problem, we…
In 1911, Alfred North Whitehead published a short book "Introduction to Mathematics" (IM) intended for students wanting an explanation of the fundamental ideas of mathematics. Whitehead's IM has enduring value because it was written not…
These are lecture notes for a course I gave in mid-1990s for MSc students at the University of Bath. It presents an algorithm with singly exponential complexity for the existential theory of the reals, in the spirit of J. Renegar. The aim…
This is an exposition of facts about Arithmetic with an approach via mathematical logic. In Section 1 we present Peano Arithmetic, PA, and the complete theory of $\mathbb{N}$, and we show that $\mathbb{N}$ is a prime model of the theory of…
We analyze the informal notion of truth and conclude that it can be formalized in essentially two distinct ways: constructively, in terms of provability, or classically, as a hierarchy of concepts which satisfy Tarski's biconditional in…
The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of…
In 1955, Paul Lorenzen is a mathematician who devotes all his research to foundations of mathematics, on a par with Hans Hermes, but his academic background is algebra in the tradition of Helmut Hasse and Wolfgang Krull. This shift from…
We present a formulation of quantum circuits where the focus is set on whether a given circuit (made of unitary operators and projective measurements with definite outcomes) does reflect an actually realizable physical experiment. In order…
Ordinary and transfinite recursion and induction and ZF set theory are used to construct from a fully interpreted object language and from an extra formula a new language. It is fully interpreted under a suitably defined interpretation.…
Supplementing earlier literature by e.g. Tipler, Clarke, & Ellis (1980), Israel (1987), Thorne, (1994), Earman (1999), Senovilla & Garfinkle (2015), Curiel (2019ab), and Landsman (2021ab), I provide a historical and conceptual analysis of…
In this paper a class of languages which are formal enough for mathematical reasoning is introduced. First-order formal languages containing natural numbers and numerals belong to that class. Its languages are called mathematically…
In 1984, Wim Ruitenburg published a surprising result about periodic sequences in intuitionistic propositional calculus (IPC). The property established by Ruitenburg naturally generalizes local finiteness; recall that intuitionistic logic…
The problem of induction has persisted since Hume exposed the logical gap between repeated observation and universal inference. Traditional attempts to resolve it have oscillated between two extremes: the probabilistic optimism of Laplace…
A true lie is a lie that becomes true when announced. In a logic of announcements, where the announcing agent is not modelled, a true lie is a formula (that is false and) that becomes true when announced. We investigate true lies and other…
Here we consider some well-known facts in syntax from a physics perspective, allowing us to establish equivalences between both fields with many consequences. Mainly, we observe that the operation MERGE, put forward by N. Chomsky in 1995,…
Wright (1928) deals with demand and supply of oils and butter. In Appendix B of this book, Philip Wright made several fundamental contributions to causal inference. He introduced a structural equation model of supply and demand, established…
This draft introduces the technical machinery of a semantic framework for potentialist truthmaking based on our innovation of intentic states, which are structured partial models accounting for our distinction between non-hypothetical and…