Contents 1 Introduction to type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 1.1 Early versions of type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 1.2 Type theory with -notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969 1.3 The Axiom of Choice and Skolemization . . . . . . . . . . . . . . . . . . . . . . 973 1.4 The expressiveness of type theory . . . . . . . . . . . . . . . . . . . . . . . . . 975 1.5 Set theory as an alternative to type theory . . . . . . . . . . . . . . . . . . . . 976 2 Metatheoretical foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977 2.1 The Unifying Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977 2.2 Expansion proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978 2.3 Proof translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982 2.4 Higher-order uni cation . . . . . . . . . . . .

Abstract. The HOL system is an LCF-style mechanized proof-assistant for conducting proofs in higher order logic. This paper discusses a proposal to extend the primitive basis of the logic underlying the HOL system with a very simple form of quantification over types. It is shown how certain practical problems with using the definitional mechanisms of HOL would be solved by the additional expressive power gained by making this extension.

Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re

Abstract. Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens. Many non-classical logics are, at the propositional level, funny toys which work quite good, but when one wants to extend them to higher levels to get a real logic that would enable one to do mathematics or other more sophisticated reasonings, sometimes dramatic troubles appear.

The systems K of transnite cumulative types up to are extended to systems K 1 that include a natural innitary inference rule, the so-called limit rule. For countable a semantic completeness theorem for K 1 is proved by the method of reduction trees, and it is shown that every model of K 1 is equivalent to a cumulative hierarchy of sets. This is used to show that several axiomatic rst-order set theories can be interpreted in K 1 , for suitable . Keywords: cumulative types, innitary inference rule, logical foundations of set theory. MSC: 03B15 03B30 03E30 03F25 1 Introduction The idea of founding mathematics on a theory of types was rst proposed by Russell [20] (foreshadowed already in [19]), and subsequently implemented by Whitehead and Russell [26]. The formal systems presented in these works were later simplied and cast into their modern shape by Ramsey [18]. Godel [9] and Tarski [25] were the rst to restrict the type structure to types of unary predi...

Abstract. A classical higher-order logic PFsub of partial functions is defined. The logic extends a version of Farmer’s logic PF by enriching the type system of the logic with subset types and dependent types. Validity in PFsub is then reduced to validity in PF by a translation. 1

. In this paper, we present methods for eliminating higher-order quantifiers in proof goals arising in the verification of digital circuits. For the description of the circuits, a subset of higher-order logic called hardware formulae is used which is sufficient for describing hardware specifications and implementations at register transfer level. Real circuits can be dealt with as well as abstract (generic) circuits. In case of real circuits, it is formally proved, that the presented transformations result in decidable formulae, such that full automation is achieved for them. Verification goals of abstract circuits can be transformed by the presented methods into goals of logics weaker than higher-order logic, e.g. (temporal) propositional logic. The presented transformations are also capable of dealing with hierarchy and have been implemented in HOL90. 1 Introduction Higher-order logic is well suited for hardware verification [Gord86, Joyc91], but unfortunately this logic is neither ...

This paper deals with the problem of a program that is essentially the same over any of several types but which, in the older imperative languages must be rewritten for each separate type. For example, a sort routine may be written with essentially the same code except for the types for integers, booleans, and strings. It is clearly desirable to have a method of writing a piece of code that can accept the specific type as an argument. Milner developed his ideas in terms of type assignment to lambda-terms. It is based on a result due originally to Curry (Curry 1969) and Hindley (Hindley 1969) known as the principal type-scheme theorem, which says that (assuming that the typing assumptions are sufficiently wellbehaved) every term has a principal type-scheme, which is a type-scheme such that every other type-scheme which can be proved for the given term is obtained by a substitution of types for type variables. This use of type schemes allows a kind of generality over all types, which is known as polymorphism.

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