Completeness in the theory of types

Abstract

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.For the functional calculus of second order, in which predicate variables may be bound, a very different kind of result is known: no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem. This follows from results of Gödel concerning systems containing a theory of natural numbers, because a finite categorical set of axioms for the positive integers can be formulated within a second order calculus to which a functional constant has been added.By a valid formula of the second order calculus is meant one which expresses a true proposition whenever the individual variables are interpreted as ranging over an (arbitrary) domain of elements while the functional variables of degree n range over all sets of ordered n-tuples of individuals. Under this definition of validity, we must conclude from Gödel's results that the calculus is essentially incomplete.It happens, however, that there is a wider class of models which furnish an interpretation for the symbolism of the calculus consistent with the usual axioms and formal rules of inference. Roughly, these models consist of an arbitrary domain of individuals, as before, but now an arbitrary class of sets of ordered n-tuples of individuals as the range for functional variables of degree n. If we redefine the notion of valid formula to mean one which expresses a true proposition with respect to every one of these models, we can then prove that the usual axiom system for the second order calculus is complete: a formula is valid if and only if it is a formal theorem.

Free Choice in Deontic Inquisitive Semantics (DIS)

“…Fitting (2002) and Benzmüller et al (2004) are two of them, but since both of these papers interpret the central machinery of type logic in some nonstandard way, 2 the logic used here will be the ITL of Muskens (2007). In this logic all operators have standard interpretations and in fact the interpretation of the logic is a rather straightforward generalisation of that of Henkin (1950), making (2) invalid but retaining all classical rules for logical operators. The following somewhat impressionistic description mainly highlights ITL's minor differences with standard simple type theory.…”

Syntax and Semantics of the logic L_omega omega^lambda

“…Classical second order logic is complete with respect to models which are called nowadays Henkin models, see [10]. Combining Henkin's proof and the standard proof of Heyting valued completeness for first order intuitionistic logic one shows that our logic L λ ωω (but in fact, full intuitionistic second order logic) is complete with respect to Heyting valued Henkin models.…”

Borrowing interpolation

“…Higher order logic with Henkin semantics has been introduced and studied in [12,40], a recent book on the topic being [3]. Here, in order to simplify the presentation and the illustration of our interpolation borrowing method, we consider an simplified variant close to the 'higher order algebra' of [51] which does not consider λ-abstraction and choice functions.…”