Higher-order semantics and extensionality

Abstract

Abstract.In this paper we re-examine the semantics of classical higher-order logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higher-order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we develop a methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of (machine-oriented) higher-order calculi with respect to these model classes.

Free Choice in Deontic Inquisitive Semantics (DIS)

“…1 There are now several approaches to type theory that manage to avoid making (2) valid. 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.…”

Higher-order aspects and context in SUMO

“…More recently extensionality and equality reasoning in HOL has been studied [43,44,45,46]. The TPS system [47,48], which is based on a higher-order mating calculus, is a pioneering ATP system for HOL.…”

“…For a detailed discussion of functional and Boolean extensionality in classical higher-order logic we refer to Benzmüller, Brown and Kohlhase [45].…”

“…The semantics of HOL is well understood and thoroughly documented in the literature [67,68,45,69] THF0 encodings obey the convention that the types of constant symbols and variable symbols have to be declared before their first use. Type declarations for constant symbols are typically provided in a type signature part at the beginning of each THF0 file while types of variable symbols are provided in their binding positions.…”

10 Higher order modal logic

“…Higher order logic (often also called type theory or the Theory of Types) began with Frege, was formalized in Russell [46] and Whitehead and Russell [52] early in the previous century, and received its canonical formulation in Church [14]. 1 While classical type theory has since long been overshadowed by set theory as a foundation of mathematics, recent decades have shown remarkable comebacks in the fields of mechanized reasoning (see, e.g., Benzmüller et al [9] and references therein) and linguistics. Since the late 1960's philosophers and logicians, for various reasons which we will dwell upon, have started to combine higher order logic with modal operators (Montague [35,37,38], Bressan [11], Gallin [22], Fitting [19]).…”