Related papers: Logic in Mathematics and Computer Science
Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in…
Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to…
Mathematical reasoning and optimization are fundamental to artificial intelligence and computational problem-solving. Recent advancements in Large Language Models (LLMs) have significantly improved AI-driven mathematical reasoning, theorem…
Mathematical reasoning is essential for problem-solving in education, science, and industry, serving as a crucial benchmark for evaluating artificial intelligence systems. As Large Language Models (LLMs) improve their reasoning…
Every scientific endeavour consists of (at least) two components: A hypothesis on the one hand and data on the other. There is always a more or less abstract level - some theory, a set of concepts, certain relations of ideas - and a…
Contemporary semantic description of logic is based on the ontology of all possible interpretations, an insufficiently clear metaphysical concept. In this article, logic is described as the internal organization of language. Logical…
We study syllogistic reasoning in LLMs from the logical and natural language perspectives. In process, we explore fundamental reasoning capabilities of the LLMs and the direction this research is moving forward. To aid in our studies, we…
The fundamental duality theories relating algebra and geometry that were discovered in the mid-20th century can also be applied to logic via its algebraization under categorical logic. They thereby result in known and new completeness…
Linear logic (LL) is a resource-aware, abstract logic programming language that refines both classical and intuitionistic logic. Linear logic semantics is typically presented in one of two ways: by associating each formula with the set of…
With the continuous advancement of reasoning abilities in Large Language Models (LLMs), their application to scientific reasoning tasks has gained significant research attention. Current research primarily emphasizes boosting LLMs'…
Symbolic mathematical computing systems have served as a canary in the coal mine of software systems for more than sixty years. They have introduced or have been early adopters of programming language ideas such ideas as dynamic memory…
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…
Symbolic calculators like Mathematica are becoming more commonplace among upper level physics students. The presence of such a powerful calculator can couple strongly to the type of mathematical reasoning students employ. It does not merely…
Algebraic characterization of logic programs has received increasing attention in recent years. Researchers attempt to exploit connections between linear algebraic computation and symbolic computation in order to perform logical inference…
Relational descriptions have been used in formalizing diverse computational notions, including, for example, operational semantics, typing, and acceptance by non-deterministic machines. We therefore propose a (restricted) logical theory…
Currently it is widely accepted that the language of science is mathematics. This book explores an alternative idea where the future of science is based on the language of algorithms and programs. How such a language can actually be…
A logic family is a bunch of logics that belong together in some way. First-order logic is one of the examples. Logics organized into a structure occurs in abstract model theory, institution theory and in algebraic logic. Logic families…
Automated theorem provers and formal proof assistants are general reasoning systems that are in theory capable of proving arbitrarily hard theorems, thus solving arbitrary problems reducible to mathematics and logical reasoning. In…
Logical formalisms provide a natural and concise means for specifying and reasoning about preferences. In this paper, we propose lexicographic logic, an extension of classical propositional logic that can express a variety of preferences,…
This report describes the development and use of an online teaching tool giving students exercises in logical modelling, or \emph{formalisation} as it is called in the older literature. The original version of the site, `Logic for Fun',…