Related papers: An approach to basic set theory and logic
A re-construction of the fundamentals of programming as a small mathematical theory (PRISM) based on elementary set theory. Highlights: $\bullet$ Zero axioms. No properties are assumed, all are proved (from standard set theory). $\bullet$ A…
This is a detailed and self-contained introduction to the real number system from a categorical perspective. We begin with the categorical definition of the natural numbers, review the Eudoxus theory of ratios as presented in Book V of…
We see how nested sequents, a natural generalisation of hypersequents, allow us to develop a systematic proof theory for modal logics. As opposed to other prominent formalisms, such as the display calculus and labelled sequents, nested…
We develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to…
Those articles (versions) explore a non conventional pregeometrical model encompassing quantics and geometrodynamics (a stochastic graph relational theory). Some key points are : 1) the use of simple boolean atom-like far subquantum…
An origin is often an intriguing issue. It becomes doubly intriguing when the logical form of thinking is considered. In this paper we will investigate exactly that: we will conjecture on the origin of basic instruments of logical thinking.…
The core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analyzed and refined,…
We interpret homogenousness as a second order property and base it on the same principle as nonmonotonic logic: there might be a small set of exceptions. We use this idea to analyse fundamental questions about defeasible inheritance…
This article provides a concise overview of the main mathematical theory of Benford's law in a form accessible to scientists and students who have had first courses in calculus and probability. In particular, one of the main objectives here…
We present several philosophical ideas emerging from the studies of complex systems. We make a brief introduction to the basic concepts of complex systems, for then defining "abstraction levels". These are useful for representing…
Knowing the truth is rarely enough -- we also seek out reasons why the fact is true. While much is known about how we explain contingent truths, we understand less about how we explain facts, such as those in mathematics, that are true as a…
In the realm of light logics deriving from linear logic, a number of variants of exponential rules have been investigated. The profusion of such proof systems induces the need for cut-elimination theorems for each logic, the proof of which…
In this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, unified under the name of tangential extremal principles, combine primal…
A class of models intended to be as minimal and structureless as possible is introduced. Even in cases with simple rules, rich and complex behavior is found to emerge, and striking correspondences to some important core known features of…
Though the truths of logic and pure mathematics are objective and independent of any contingent facts or laws of nature, our knowledge of these truths depends entirely on our knowledge of the laws of physics. Recent progress in the quantum…
The definition is a common form of human expert knowledge, a building block of formal science and mathematics, a foundation for database theory and is supported in various forms in many knowledge representation and formal specification…
My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us…
We investigate here concept learning from incomplete examples. Our first purpose is to discuss to what extent logical learning settings have to be modified in order to cope with data incompleteness. More precisely we are interested in…
Logic reasoning is a significant ability of human intelligence and also an important task in artificial intelligence. The existing logic reasoning methods, quite often, need to design some reasoning patterns beforehand. This has led to an…
In this paper, we present a paradox arising from the acceptance of the Law of Excluded Middle (LEM) within classical mathematics. Specifically, we construct a nonzero analytic function on a connected open subset of the complex plane whose…