Related papers: Logic is Algebra
We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement's truth value remains the same. This view of mathematics as…
Modern mathematics is known for its rigorous proofs and tight analysis. Math is the paradigm of objectivity for most. We identify the source of that objectivity as our knowledge of the physical world given through our senses. We show in…
Doing research is fighting, what any other thing the human being could do? Fight against powers or to get powers, that depends on us. Science can be a revolution or deadlocked idleness. Still waters, without hitting the stones along their…
Injectives in several classes of structures associated with logic are characterized. Among the classes considered are residuated lattices, MTL-algebras, IMTL-algebras, BL-algebras, NM-algebras and bounded hoops.
A variety is a category of ordered (finitary) algebras presented by inequations between terms. We characterize categories enriched over the category of posets which are equivalent to a variety. This is quite analogous to Lawvere's classical…
An attempt at unifying logic and functional programming is reported. As a starting point, we take the view that "logic programs" are not about logic but constitute inductive definitions of sets and relations. A skeletal language design…
Linear logic was conceived in 1987 by Girard and, in contrast to classical logic, restricts the usage of the structural inference rules of weakening and contraction. With this, atoms of the logic are no longer interpreted as truth, but as…
Logic programming is sometimes described as relational programming: a paradigm in which the programmer specifies and composes n-ary relations using systems of constraints. An advanced logic programming environment will provide tools that…
The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. The purpose of this paper is to lift that framework from universal algebra to the strictly more…
This paper presents a property of propositional theories under the answer sets semantics (called Equilibrium Logic for this general syntax): any theory can always be reexpressed as a strongly equivalent disjunctive logic program, possibly…
Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to…
The paper presents a fresh new comprehensive ideology on Neutrosophic Logic based on contradiction study. It does a survey on Chinese philosophy. The paper aims to solve the chaos of logic and exhibit the potential power of neutrosophy: a…
Predicate Logic with Definitions (PLD or D-logic) is a modification of first-order logic intended mostly for practical formalization of mathematics. The main syntactic constructs of D-logic are terms, formulas and definitions. A definition…
Asymmetric combination of logics is a formal process that develops the characteristic features of a specific logic on top of another one. Typical examples include the development of temporal, hybrid, and probabilistic dimensions over a…
Some notes about quantum physics, an interpretation if one wishes, are put forward, insisting on `closely following the mathematics/formalism, the `nuts and bolts of what quantum physics says'. These, basically well-known, issues seem to…
The concept of a judgment as a logical action which introduces new information into a deductive system is examined. This leads to a way of mathematically representing implication which is distinct from the familiar material implication,…
First-order logic is typically presented as the study of deduction in a setting with elementary quantification. In this paper, we take another vantage point and conceptualize first-order logic as a linear space that encodes "plausibility".…
We define the concept of a logic frame, which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive…
At its core, abstraction is the process of generalizing from specific instances to broader concepts or models, with the primary objective of reducing complexity while preserving properties essential to the intended purpose. It is…
Up to equivalence, a substitution in propositional logic is an endomorphism of its free algebra. On the dual space, this results in a continuous function, and whenever the space carries a natural measure one may ask about the stochastic…