Related papers: An Introduction to Mathematical Logic
We define an extension of predicate logic, called Binding Logic, where variables can be bound in terms and in propositions. We introduce a notion of model for this logic and prove a soundness and completeness theorem for it. This theorem is…
Logic rules and inference are fundamental in computer science and have been studied extensively. However, prior semantics of logic languages can have subtle implications and can disagree significantly, on even very simple programs,…
Logical reasoning is essential in a variety of human activities. A representative example of a logical task is mathematics. Recent large-scale models trained on large datasets have been successful in various fields, but their reasoning…
Every definite logic program has as its meaning a least Herbrand model with respect to the program-independent ordering "set-inclusion". In the case of normal logic programs there do not exist least models in general. However, according to…
We consider an extension of first-order logic with a recursion operator that corresponds to allowing formulas to refer to themselves. We investigate the obtained language under two different systems of semantics, thereby obtaining two…
This is a reflection on the author's experience in teaching logic at the graduate level in a computer science department. The main lesson is that model building and the process of modelling must be placed at the centre stage of logic…
This invited paper is a passionate pitch for the significance of logic in scientific education. Logic helps focus on the essential core to identify the foundations of ideas and provides corresponding longevity with the resulting approach to…
First-order logic is the basis for many knowledge representation formalisms and methods. Providing technological support for learning to write first-order formulas for natural language specifications requires methods to test formulas for…
We investigate cut-elimination and cut-simulation in impredicative (higher-order) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic -- in our case a sequent calculus for…
I have argued elsewhere that second order logic provides a foundation for mathematics much in the same way as set theory does, despite the fact that the former is second order and the latter first order, but second order logic is marred by…
The construction of first-order logic and set theory gives rise to apparent circularities of mutual dependence, making it unclear which can act as a self-contained starting point in the foundation of mathematics. In this paper, we carry out…
G\"odel's argument for the First Incompleteness Theorem is, structurally, a proof by contradiction. This article intends to reframe the argument by, first, isolating an additional assumption the argument relies on, and then, second, arguing…
Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as…
We begin the study of categorical logic for continuous model theory. In particular, we 1. introduce the notions of metric logical categories and functors as categorical equivalents of a metric theory and interpretations, 2. prove a…
In this paper, first-order logic is interpreted in the framework of universal algebra, using the clone theory developed in three previous papers. We first define the free clone T(L, C) of terms of a first order language L over a set C of…
Many representation schemes combining first-order logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable…
We present a systematic approach to logical predicates based on universal coalgebra and higher-order abstract GSOS, thus making a first step towards a unifying theory of logical relations. We first observe that logical predicates are…
The emphasis is made on the juxtaposition of (quantum~theorem) proving versus quantum (theorem~proving). The logical contents of verification of the statements concerning quantum systems is outlined. The Zittereingang (trembling input)…
This is the first volume of a textbook for a two-semester course in mathematical analysis. This first volume is about analysis of functions of a single variable. The topics covered include completeness axiom, Archimedean property,…
This paper proposes an alternative to standard first-order logic that seeks greater naturalness, generality, and semantic self-containment. The system removes the first-order restriction, avoids type hierarchies, and dispenses with external…