Related papers: Logic is Algebra
People usually regard algorithms as more abstract than the programs that implement them. The natural way to formalize this idea is that algorithms are equivalence classes of programs with respect to a suitable equivalence relation. We argue…
In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a…
This work is divided between two main areas: in the theory of multialgebras, we focus mostly on a new definition of what a freely generated object should be in their category, and on how this category is equivalent to another with partially…
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…
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…
There is no mysterious link between mathematics and physics, because both of them are human inventions designed to study the world.
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…
We show that a class of algebras is closed under the taking of homomorphic images and direct products if and only if the class consists of all algebras that satisfy a set of (generally simultaneous) equations. For classes of regular…
Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary "propositional" logic should in general be the logic of subsets of a given universe set. Partitions on…
In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone's representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof…
The purpose of this survey article is to introduce the reader to a connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive type theory of Martin-L\"of into…
We consider three modern roles for logic in artificial intelligence, which are based on the theory of tractable Boolean circuits: (1) logic as a basis for computation, (2) logic for learning from a combination of data and knowledge, and (3)…
We establish a novel connection between two research areas in non-classical logics which have been developed independently of each other so far: on the one hand, input/output logic, introduced within a research program developing logical…
Matching logic is a logical framework for specifying and reasoning about programs using pattern matching semantics. A pattern is made up of a number of structural components and constraints. Structural components are syntactically matched,…
Plausible reasoning concerns situations whose inherent lack of precision is not quantified; that is, there are no degrees or levels of precision, and hence no use of numbers like probabilities. A hopefully comprehensive set of principles…
We introduce ologisms. They generate from ologs by extending their logical expressivity, from the possibility of considering constraints of equational nature only to the possibility of considering constraints of syllogistic nature, in…
The article demonstrates that logic is not necessarily singleton and does not always have the standard interpretation of negation. Appropriate generalizations of logic are suggested. Positive logic and multivalued negation operations are…
Type-free systems of logic are designed to consistently handle significant instances of self-reference. Some consistent type-free systems also have the feature of allowing the sort of general abstraction or comprehension principle that…
Leibniz algebras are certain generalization of Lie algebras. In this paper we survey the important results in Leibniz algebras which are analogs of corresponding results in Lie algebras. In particular we highlight the differences between…
The Boolean logic of subsets, usually presented as `propositional logic,' is considered as being "classical" while intuitionistic logic and the many sublogics and off-shoots are "non-classical." But there is another mathematical logic, the…