A structural characterization of extended correctness-completeness in classical logic.(first order logic; axiomatic systems)

SUMMARY: In this paper I deal with first order logic and axiomatic systems. I present the metalogical results that show the property of satisfying Modus Ponens as a necessary and sufficient condition for the extended completeness of the system, and to the Deduction Metatheorem as a necessary and sufficient condition for the extended correctness of the system. Both supposing that the system satisfies the corresponding restricted properties. These results show that the choice of that rule of inference and of that metatheorem, for any particular axiomatic system, are not a matter of personal liking or of practical convenience, but they playa fundamental role for the extended correctness-completeness properties of the axiomatic system. As a matter of fact, they can be considered as structural properties that characterize the fulfilling of the Extended Correctness and Completeness theorem for the axiomatic system.

KEY WORDS: logical consequence, axiomatic system, compactness, semantics

1. Introduction

In what follows I will only work with formal first order languages with equality. I will deal with the concept of first order classical deductive logic, particularly with the concept of an axiomatic system of Hilbert type that is founded in the concepts of axiom, rule of inference and definition of formal derivation. As comes next I give the definition of what I will understand by an axiomatic system of Hilbert type.

Definition. An axiomatic system S is given by the following:

a) A finite or infinite decidable (1) set A, of formulas. The formulas of [DELTA] are called the axioms of S.

b) A finite set RI of decidable (2) rules of inference of S.