The traditional model theory of first-order logic assumes that the interpretation of a formula can be given without reference to its deductive context. This paper investigates an interpretation which depends on a formula's location within a derivation. The key step is to drop the assumption that all quantified variables must have the same range and to require only that the ranges of variables in a derivation must be related in such way as to preserve the soundness of the inference rules. (...) With each (consistent) derivation there is associated a Buridan-Volpin (orBV) structure [M, {r(x)}] which is simply a Tarski structureM for the language and a map giving the ranger(x) of each variablex in the derivation. IfLK* is (approximately) the classical sequent calculusLK of Gentzen from which the structural contraction rules have been dropped, then our main result reads: If a set of first-ordered formulas has a Tarski modelM, then from any normal derivationD inLK* of can be constructed aBV modelM D=[M, {r(x)}] of where each ranger(x) is finite. (shrink)

If the collection of models for the axioms 21 of elementary number theory is enlarged to include not just the " natural numbers " or their non-standard infinitistic extensions but also what are here called "primitive recursive notations", questions arise about the reliability of first-order derivations from 21. In this enlarged set of "models" some derivations usually accepted as "reliable" may be problematic. This paper criticizes two of these derivations which claim, respectively, to establish the totality of exponentiation and to (...) prove Euclid's theorem about the infinity of primes. (shrink)