We introduce a new formal logic, called equality propositional logic. It has two basic connectives, $boldsymbol{wedge}$ (conjunction) and $equiv$ (equivalence). Moreover, the $Rightarrow$ (implication) connective can be derived as $ARightarrow B:=(Aboldsymbol{wedge}B)equiv A$. We formulate the equality propositional logic and demonstrate that the resulting logic has reasonable properties such a...

The definition of identity in terms of other logical symbols is a recurrent issue in logic. In particular, in First-Order Logic (FOL) there is no way of defining the global relation of identity, while in standard Second-Order Logic (SOL) this definition is not only possible, but widely used. In this paper, the reverse question is posed and affirmatively answered: Can we define with only equalit...

We investigate and compare various ways of transforming equality formulas to propositional formulas, in order to be able to solve satisfiability in equality logic by means of satisfiability in propositional logic. We propose equality substitution as a new approach combining desirable properties of earlier methods, we prove its correctness and show its applicability by experiments.

in this paper we continue development of formal theory of a special class offuzzy logics, called eq-logics. unlike fuzzy logics being extensions of themtl-logic in which the basic connective is implication, the basic connective ineq-logics is equivalence. therefore, a new algebra of truth values calledeq-algebra was developed. this is a lower semilattice with top element endowed with two binary...

We introduce algorithmic logic – an algebraic approach according to [25]. It is done in three stages: propositional calculus, quantifier calculus with equality, and finally proper algorithmic logic. For each stage appropriate signature and theory are defined. Propositional calculus and quantifier calculus with equality are explored according to [24]. A language is introduced with language signa...

In this paper we continue development of formal theory of a special class offuzzy logics, called EQ-logics. Unlike fuzzy logics being extensions of theMTL-logic in which the basic connective is implication, the basic connective inEQ-logics is equivalence. Therefore, a new algebra of truth values calledEQ-algebra was developed. This is a lower semilattice with top element endowed with two binary...

Effectively Propositional Logic (EPR), also known as the Bernays-Schönfinkel class, allows encoding problems that are propositional in nature, but EPR encodings can be exponentially more succinct than purely propositional logic encodings. We recently developed a DPLL-based decision procedure that builds on top of efficient SAT solving techniques to handle the propositional case efficiently whil...

One can add the machinery of relation symbols and terms to a propositional modal logic without adding quantifiers. Ordinarily this is no extension beyond the propositional. But if terms are allowed to be non-rigid, a scoping mechanism (usually written using lambda abstraction) must also be introduced to avoid ambiguity. Since quantifiers are not present, this is not really a first-order logic, ...

A method for encoding formulas in rst-order logic with equality into some equational logics and their associated propositional logics is presented. This yields some new strategies for theorem-proving in rst-order logic with equality. We outline the techniques and give a simple illustration of their use.