Even though propositional provability logic is a modal logic with a kind of "necessity" operator, it withstands Quine's (1976) controversial critique of modal notions as unintelligible, already because of its clear and unambiguous arithmetic interpretation.
That kind of arithmetic, an aggressive government interpretation of 1996 immigration laws that has been increasingly invoked in recent years, was rejected by the Supreme Court in a unanimous decision in June.
Ackermann's paper was the starting point of Kreisel's 1951 "no-counterexample" interpretation of arithmetic.
Field's physicalist interpretation of arithmetic and analysis not only undermines the Quine-Putnam indispensability argument.
As a foundational corollary, we obtain that that learning based realizability is a constructive interpretation of Heyting Arithmetic plus excluded middle over Σ01 formulas (for which it was designed) and of all Peano Arithmetic when combined with Gödel's double negation translation.
An Axiom of Infinity would be wanted to ensure that an interpretation of Heyting arithmetic could be embedded in the theory; it might be simplest to provide type 0 with the primitives of HA (just as the earliest versions of TST had the primitives of classical arithmetic provided for type 0).
A paper based on the ideas in the lecture entitled "Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes," was published only in 1958, and the interpretation of Heyting arithmetic into the quantifier free calculus T in it became known as the "Dialectica Interpretation," after the journal in which the article was published (Gödel 1958).
Findings from behavioral studies of normal development of spatial number representations and arithmetic competence contribute to the interpretation of our results.
In the late thirties, at least from 1938 on, he developed as a response to Gentzen's proof his own special interpretation of intuitionistic logic and arithmetic, what came to be known as the Dialectica interpretation.
Figure 7 Geometrical interpretation of the harmonic mean, the arithmetic mean, and the geometric mean of and.
