In this article first-order semantic tableaux and free variable tableaux are extended with sorts. Sorts are sets of unary predicates. They can be attached to variables. Semantically, the domain of a variable is restricted to the intersection of the denotations of the attached predicates. Syntactically, the sort information is exploited by modified γ and δ tableaux rules. The standard unification algorithm of free variable tableaux is replaced by a sorted unification algorithm. The resulting calculi, tableaux with sorts and free variable (...) tableaux with sorts are proved sound, complete and more suitable for mechanization than their standard counterparts without sorts. (shrink)

Condensed detachment is usually regarded as a notation, and defined by example. In this paper it is regarded as a rule of inference, and rigorously defined with the help of the Unification Theorem of J. A. Robinson. Historically, however, the invention of condensed detachment by C. A. Meredith preceded Robinson's studies of unification. It is argued that Meredith's ideas deserve recognition in the history of unification, and the possibility that Meredith was influenced, through ukasiewicz, by ideas of Tarski going back (...) at least to 1939, and possibly to 1930 or earlier, is discussed. It is proved that a term is derivable by substitution and ordinary detachment from given axioms if and only if it is a substitution instance of a term which is derivable from these axioms by condensed detachment, and it is shown how this theorem enables the ideas of ukasiewicz and Tarski mentioned above to be formalized and extended. Finally, it is shown how condensed detachment may be subsumed within the resolution principle of J. A. Robinson, and several computer studies of particular Hilbert-type propositional calculi using programs based on condensed detachment or on resolution are briefly discussed. (shrink)

Theoria , the international Swedish philosophy journal, was founded in 1935. Its contributors in the first 75 years include the major Swedish philosophers from this period and in addition a long list of international philosophers, including A. J. Ayer, C. D. Broad, Ernst Cassirer, Hector Neri Castañeda, Arthur C. Danto, Donald Davidson, Nelson Goodman, R. M. Hare, Carl G. Hempel, Jaakko Hintikka, Saul Kripke, Henry E. Kyburg, Keith Lehrer, Isaac Levi, David Lewis, Gerald MacCallum, Richard Montague, Otto Neurath, Arthur N. (...) Prior, W. V. Quine, Nicholas Rescher, Ernest Sosa, Robert C. Stalnaker, P. F. Strawson, Patrick Suppes, Johan van Benthem, Georg Henrik von Wright and many others. Hempel's confirmation paradoxes, Ross's deontic paradox, Montague's universal grammar and Lindström's theorem are among the contributions to philosophy that were first published in Theoria. (shrink)

We establish a connection between the geometric methods developed in the combinatorial theory of small cancellation and the propositional resolution calculus. We define a precise correspondence between resolution proofs in logic and diagrams in small cancellation theory, and as a consequence, we derive that a resolution proof is a 2-dimensional process. The isoperimetric function defined on diagrams corresponds to the length of resolution proofs.

Van Heijenoort’s account of the historical development of modern logic was composed in 1974 and first published in 1992 with an introduction by his former student. What follows is a new edition with a revised and expanded introduction and additional notes.

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this (...) problem by applying recent methods from sorted logics. This paper presents a tableau calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi. (shrink)

The concept of “necessity of thought” plays a central role in Dag Prawitz’s essay “Logical Consequence from a Constructivist Point of View” (Prawitz 2005). The theme is later developed in various articles devoted to the notion of valid inference (Prawitz, 2009, forthcoming a, forthcoming b). In section 1 I explain how the notion of necessity of thought emerges from Prawitz’s analysis of logical consequence. I try to expound Prawitz’s views concerning the necessity of thought in sections 2, 3 and 4. (...) In sections 5 and 6 I discuss some problems arising with regard to Prawitz’s views. (shrink)