Bachelor's Thesis: Studies in Higher-Order Equational Logic

Mark Kaminski, Betreuer: Gert Smolka

Motivation

Higher-order logic, also known as type theory, has been introduced in 1908 by Bertrand Russell [33] as a formal basis for mathematical reasoning, based on a functional view of logic originally developed by Gottlob Frege [13]. In its modern form, type theory is based on Alonzo Church’s simply typed lambda-calculus [8] and the formulations by Leon Henkin [22] and Peter Andrews [4]. Over the years type theory has become an integral part of every subject of study that is in some way concerned with the relationship between computation and logical reasoning. In computer science, higher-order logic has lots of applications, including proof assistant systems like e.g. Isabelle [28] or PVS [29].
Classical formulations of type theory employ rules of inference depending on some dedicated logical constants. Consider, for instance, the well-known rule "Modus ponens", commonly formulated as:
From A and A->B, infer B
The rule involves the constant -> and is therefore specific to logical systems where such a constant is built in.

Abstract

We show that higher-order logic (HOL) can be axiomatized in S, the simply typed lambda-calculus with equational deduction. Unlike traditional formulations of HOL, S does not rely on pre-defined semantics of logical constants.
First we show how deduction in traditional HOL can be simulated within S, thus proving S to be a general-purpose higher-order logical system. Afterwards we prove the completeness of S for first-order axioms.
An important task of the thesis is to investigate in how far the usual logical constants and semantic structures can be axiomatized within S. We start by considering Boolean algebras, i.e. systems generated by Boolean axioms and show how they can be axiomatically extended by quantification. We define the identity test and show some important properties of identity in S. We axiomatize in S the usual semantic structure of HOL, thus showing that the semantic expressiveness of S matches that of traditional higher-order formalisms. Finally we analyze the deductive power of S in more detail and obtain interesting incompleteness results for specific instances of the system.

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