Related papers: Foundations
This chapter presents probability logic as a rationality framework for human reasoning under uncertainty. Selected formal-normative aspects of probability logic are discussed in the light of experimental evidence. Specifically, probability…
There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole…
We offer the proofs that complete our article introducing the propositional calculus called semi-intuitionistic logic with strong negation.
Mathematical proofs should be paired with formal proofs, whenever feasible.
This paper shows that, even at the most basic level, the parallel, countable branching and uncountable branching recurrences of Computability Logic (see http://www.cis.upenn.edu/~giorgi/cl.html) validate different principles.
When a decision, such as the approval or denial of a bank loan, is delegated to a computer, an explanation of that decision ought to be given with it. This ethical need to explain the decisions leads to the search for a formal definition of…
The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are…
Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational…
Environments for systematic construction of logic programs are needed in the academy as well as in the industry. Such environments should support well defined construction methods and should be able to be extended and interact with other…
In the words of the esteemed mathematician Paul Erd\"os, the mathematician's task is to \emph{prove and conjecture}. These two processes form the bedrock of all mathematical endeavours, and in the recent years, the mathematical community…
Computability logic is a formal theory of (interactive) computability in the same sense as classical logic is a formal theory of truth. This approach was initiated very recently in "Introduction to computability logic" (Annals of Pure and…
This letter introduces a new, substantially simplified version of the branching recurrence operation of computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html), and proves its equivalence to the old, "canonical" version.
A semantical embedding of input/output logic in classical higher-order logic is presented. This embedding enables the mechanisation and automation of reasoning tasks in input/output logic with off-the-shelf higher-order theorem provers and…
Logic has its origins in basic questions about the nature of the real world and how we describe it. This article seeks to bring out the physical and epistemological relevance of some of the more recent technical work in logic and…
The study of propositional logic -- fundamental to the theory of computing -- is a cornerstone of the undergraduate computer science curriculum. Learning to solve logical proofs requires repeated guided practice, but undergraduate students…
Cirquent calculus is a proof system with inherent ability to account for sharing subcomponents in logical expressions. Within its framework, this article constructs an axiomatization CL18 of the basic propositional fragment of computability…
The overall goal of this paper is to investigate the theoretical foundations of algorithmic verification techniques for first order linear logic specifications. The fragment of linear logic we consider in this paper is based on the linear…
These informal notes are based on the author's lecture at the National Academies of Science, Engineering, and Mathematics workshop on "AI to Assist Mathematical Reasoning" in June 2023. The goal is to think through a path by which we might…
We discuss the idea that computers might soon help mathematicians to prove theorems in areas where they have not previously been useful. Furthermore we argue that these same computer tools will also help us in the communication and teaching…
A introduction to the syntax and Semantics of Answer Set Programming intended as an handout to [under]graduate students taking Artificial Intlligence or Logic Programming classes.