Term (logic)

Abstract: In a recent paper the author has proposed a definition of the commonly used term “effectively calculable” and has shown on the basis of this definition that the general case of the Entscheidungsproblem is unsolvable in any system of symbolic logic which is adequate to a certain portion of arithmetic and is ω-consistent. The purpose of the present note is to outline an extension of this result to the engere Funktionenkalkul of Hilbert and Ackermann.In the author's cited paper it is pointed out that there can be associated recursively with every well-formed formula a recursive enumeration of the formulas into which it is convertible. This means the existence of a recursively defined function a of two positive integers such that, if y is the Godel representation of a well-formed formula Y then a(x, y) is the Godel representation of the xth formula in the enumeration of the formulas into which Y is convertible.Consider the system L of symbolic logic which arises from the engere Funktionenkalkul by adding to it: as additional undefined symbols, a symbol 1 for the number 1 (regarded as an individual), a symbol = for the propositional function = (equality of individuals), a symbol s for the arithmetic function x+1, a symbol a for the arithmetic function a described in the preceding paragraph, and symbols b1, b2, …, bk for the auxiliary arithmetic functions which are employed in the recursive definition of a; and as additional axioms, the recursion equations for the functions a, b1, b2, …, bk (expressed with free individual variables, the class of individuals being taken as identical with the class of positive integers), and two axioms of equality, x = x, and x = y →[F(x)→F(y)].

Abstract: From the Publisher: This book, the first on the subject in English, gives a readable but rigorous introduction to the theory of term rewriting systems. These are a technique used in computer science, especially functional programming, for abstract data type specification and automatic theorem-proving. The book is self-contained, and begins with a discussion of elementary systems and progresses to the most general cases that involve concepts such as second-order lambda calculus. Exercises are included throughout, and solutions to a selection of them are provided. Complete proofs of results that are often buried in the literature are also given, so researchers in functional programming, theoretical computer science, and logic will find this book useful.

Abstract: Should the title of this paper prompt you to ask, “What is the logic of perception?”, there is an answer at hand. I shall argue here that the logic of our perceptual terms is a branch of modal logic.1 In saying this, I mean by ‘perceptual terms’ both such words as ‘sees’, ‘hears’, ‘feels’, etc., which involve a reference to one particular sense modality, and such words as ‘perceives’, which are neutral in this respect. By modal logic, I mean not only the logic of the terms ‘necessary’ and ‘possible’ but also the logic of all the other terms that can be studied in the same ways as they. Among these terms are most of the words that are usually said to express propositional attitudes, including ‘knows’, ‘believes’, ‘remembers’, ‘hopes’, ‘strives’, etc. What is in common to all the modal notions in this extended sense of the term will be partly explained later.2

Abstract: Automated and semi-automated manipulation of so-called labelled transition systems has become an important means in discovering flaws in software and hardware systems. Process algebra has been developed to express such labelled transition systems algebraically, which enhances the ways of manipulation by means of equational logic and term rewriting.The theory of process algebra has developed rapidly over the last twenty years, and verification tools have been developed on the basis of process algebra, often in cooperation with techniques related to model checking. This textbook gives a thorough introduction into the basics of process algebra and its applications.