Related papers: Aletheia, double negation and negation
In this essay we examine some aspects of the classical theory of definition as codified in Aristotle's \emph{Topics} and Porphyry's \emph{Eisagog\^e} in the light of the way definition is carried out in modern mathematical practice. Our…
In a 1985 commentary to his collected works, Kolmogorov remarked that his 1932 paper "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of…
The impossibility of eliminating hallucination, understood here as incorrect definite answers, in sufficiently expressive yes-or-no formal domains is an immediate consequence of classical undecidability theorems. This note does not revisit…
Classical logic has a serious limitation in that it cannot cope with the issues of vagueness and uncertainty into which fall most modes of human reasoning. In order to provide a foundation for human knowledge representation and reasoning in…
The failure of distributivity in quantum logic is motivated by the principle of quantum superposition. However, this principle can be encoded differently, i.e., in different logico-algebraic objects. As a result, the logic of experimental…
This paper establishes grounds for deeper exploration into the question of dual nature of mathematics as an abstract discipline and as a concrete science. It is argued, as one of the consequences of the discussion, that the division into…
This paper describes a formalism that subsumes Peterson's intermediate quantifier syllogistic system, and extends the ideas by van Eijck on Aristotle's logic. Syllogisms are expressed in a concise form making use of and extending the…
Five algebraic notions of termination are formalised, analysed and compared: wellfoundedness or Noetherity, L\"ob's formula, absence of infinite iteration, absence of divergence and normalisation. The study is based on modal semirings,…
The Doomsday argument and anthropic reasoning are two puzzling examples of probabilistic confirmation. In both cases, a lack of knowledge apparently yields surprising conclusions. Since they are formulated within a Bayesian framework, they…
The concept of emergence, or synergy in its simplest form, is widely used but lacks a rigorous definition. Our work connects information and set theory to uncover the mathematical nature of synergy as the failure of distributivity. It…
We translate the argumentation networks ${\cal A}=(S, R)$ into a theory $D$ of intuitionistic logic, retaining $S$ as the domain and using intuitionistic negation to model the attack $R$ in ${\cal A}$: the attack $xRy$ is translated to…
Extensional higher-order logic programming has been introduced as a generalization of classical logic programming. An important characteristic of this paradigm is that it preserves all the well-known properties of traditional logic…
The notion of a partition on a set is mathematically dual to the notion of a subset of a set, so there is a logic of partitions dual to Boole's logic of subsets (Boolean subset logic is usually mis-specified as the special case of…
Abstraction is key to human and artificial intelligence as it allows one to see common structure in otherwise distinct objects or situations and as such it is a key element for generality in AI. Anti-unification (or generalization) is…
Axiomatizing mathematical structures and theories is an objective of Mathematical Logic. Some axiomatic systems are nowadays mere definitions, such as the axioms of Group Theory; but some systems are much deeper, such as the axioms of…
This article informally presents a solution to the paradoxes of truth and shows how the solution solves classical paradoxes (such as the original Liar) as well as the paradoxes that were invented as counterarguments for various proposed…
This paper presents a new system of logic, LF, that is intended to be used as the foundation of the formalization of science. That is, deductive validity according to LF is to be used as the criterion for assessing what follows from the…
This work presents the deductive system Double Propositional Logic, LD, along with the semantics of possible worlds that characterize it. LD includes an alternate affirmation operator and an alternate negation operator and is not valid for…
The origins of proof-theoretic semantics lie in the question of what constitutes the meaning of the logical connectives and its response: the rules of inference that govern the use of the connective. However, what if we go a step further…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…