The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order predicate logic. In this paper we prove that the lattice of lambda theories is not modular and that the variety generated by the term algebra of a semi-sensible lambda theory is not congruence modular. A...

In this paper, we provide a semantic study of the first-order predicate logic for situations involving uncertainty. We introduce the concepts of uncertain predicate proposition, uncertain predicate formula, uncertain interpretation and degree of truth in the framework of uncertainty theory. Compared with classical predicate formula taking true value in {0, 1}, the degree of truth of uncertain p...

Tarski-Givant’s map calculus is briefly reviewed, and a plan of research is outlined aimed at investigating applications of this ground equational formalism in the theorem-proving field. The main goal is to create synergy between first-order predicate calculus and the map calculus. Techniques for translating isolated sentences, as well as entire theories, from first-order logic into map calculu...

It is well-known that there exist consistent first-order theories that become inconsistent when we add Tarski’s schema T. This is Tarski’s Theorem. To avoid the inconsistency result, one can restrict Tarski’s schema in different ways. In our paper we restrict Tarski’s schema T by only instantiating the schema with a proper subset of the set of all sentences. We prove several results concerning ...

All promiment examples of first-order predicate fuzzy logics are undecidable. This leads to the problem of the arithmetical complexity of their sets of tautologies and satisfiable sentences. This paper is a contribution to the general study of this problem. We propose the classes of first-order core and ∆-core fuzzy logics as a good framework to address these arithmetical complexity issues. We ...

Quantification theory, or first-order predicate logic, can be formulated in terms purely of predicate letters and a few predicate functors which attach t o predicates to form further predicates. Apart from the predicate letters, which are schematic, there are no variables. On this score the plan is reminiscent of the combinatory logic of Schonfinkel and Curry. Theirs, however, had the whole of ...

1. Life and work Kurt Gödel was a solitary genius, whose work influenced all the subsequent developments in mathematics and logic. The striking fundamental results in the decade 1929-1939 that made Gödel famous are the completeness of the first-order predicate logic proof calculus, the incompleteness of axiomatic theories containing arithmetic, and the consistency of the axiom of choice and the...