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Abstract. Undefined terms are commonplace in mathematics, particularly in calculus. The traditional approach to undefinedness in mathematical practice is to treat undefined terms as legitimate, nondenoting terms that can be components of meaningful statements. The traditional approach enables statements about partial functions and undefined terms to be stated very concisely. Unfortunately, the traditional approach cannot be easily employed in a standard logic in which all functions are total and all terms are defined, but it can be directly formalized in a standard logic if the logic is modified slightly to admit undefined terms and statements about definedness. This paper demonstrates this by defining a version of simple type theory called Simple Type Theory with Undefinedness (sttwu) and then formalizing in sttwu examples of undefinedness arising in calculus. The examples are taken from M. Spivak’s well-known textbook Calculus. 1

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.

This paper presents an extended version of Church's simple type theory called Basic Extended Simple Type Theory (bestt). By adding type variables and support for reasoning with tuples, lists, and sets to simple type theory, it is intended to be a practical logic for formalized mathematics. 1

abstract. We consider simply typed lambda terms obtained with a single base type B and two constants ⊥ and →, where B is interpreted as the set of the two truth values, ⊥ as falsity, and → as implication. We show that every value of the full set-theoretic type hierarchy can be described by a closed term and that every valid equation can be derived from three axioms with β and η. In contrast to the established approach, we employ a pure lambda calculus where constants appear as a derived notion. 1

We give a first order formulation of Church's type theory in which types are mere sets. This formulation is obtained by replacing -calculus by a language of combinators (skolemized comprehension schemes), introducing a distinction between propositions and their contents, relativizing quantifiers and at last replacing typing predicates by membership to some sets. The theory obtained this way has both a type theoretical flavor and a set theoretical one. Like set theory, it is a first order theory, and it uses only one notion of collection. Like type theory, it gives an explicit notation for objects, a primitive notion of function and propositions are objects.

Peter Andrews has proposed, in 1971, the problem of finding an analog of the Skolem theorem for Simple Type Theory. A first idea lead to a naive rule that worked only for Simple Type Theory with the axiom of choice and the general case has only been solved, more than ten years later, by Dale Miller [9, 10]. More recently, we have proposed with Thérèse Hardin and Claude Kirchner [7] a new way to prove analogs of the Miller theorem for different, but equivalent, formulations of Simple Type Theory. In this paper, that does not contain new technical results, I try to show that the history of the skolemization problem and of its various solutions is an illustration of a tension between two points of view on Simple Type Theory: the logical and the theoretical points of view.

Q0 is an elegant version of Church’s type theory formulated and extensively studied by Peter B. Andrews. Like other traditional logics, Q0 does not admit undefined terms. The traditional approach to undefinedness in mathematical practice is to treat undefined terms as legitimate, nondenoting terms that can be components of meaningful statements. Q u 0 is a modification of Andrews’ type theory Q0 that directly formalizes the traditional approach to undefinedness. This paper presents Q u 0 and proves that the proof system of Q u 0 is sound and complete with respect to its semantics which is based on Henkin-style general models. The paper’s development of Q u 0 closely follows Andrews’ development of Q0 to clearly delineate the differences between the two systems.