Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions (...) of the connectives in question. (shrink)

We give a new and elementary construction of primitive positive decomposition of higher arity relations into binary relations on finite domains. Such decompositions come up in applications to constraint satisfaction problems, clone theory and relational databases. The construction exploits functional completeness of 2-input functions in many-valued logic by interpreting relations as graphs of partially defined multivalued ‘functions’. The ‘functions’ are then composed from ordinary functions in the usual sense. The construction is computationally effective and relies on well-developed methods of functional (...) decomposition, but reduces relations only to ternary relations. An additional construction then decomposes ternary into binary relations, also effectively, by converting certain disjunctions into existential quantifications. The result gives a uniform proof of Peirce’s reduction thesis on finite domains, and shows that the graph of any Sheffer function composes all relations there. (shrink)

Gentzen’s and Jaśkowski’s formulations of natural deduction are logically equivalent in the normal sense of those words. However, Gentzen’s formulation more straightforwardly lends itself both to a normalization theorem and to a theory of “meaning” for connectives . The present paper investigates cases where Jaskowski’s formulation seems better suited. These cases range from the phenomenology and epistemology of proof construction to the ways to incorporate novel logical connectives into the language. We close with a demonstration of this latter aspect by (...) considering a Sheffer function for intuitionistic logic. (shrink)

This technical and sophisticated book has a misleading title. In only 140 pages it has chapters entitled: Truth Functions, Sentence Logic, Model Theory, Predicate Logic, Recursive Functions, Formalized Arithmetic, Free-Variable Arithmetic, and Axiomatic Set Theory. Goodstein says little about, let alone studies, the historical development of any of these topics. The history consists of a few references to some of the individuals who have played a role in the development of the field; but not always to the major figures. For (...) instance, his chapter on model theory contains no discussion of the development of classical model theory. Only Herbrand is noted as a developer of model theory. However, even if these eight studies in mathematical logic do not tell much about the development of the topics, they still have considerable value. The first chapter on truth functions and the seventh on free-variable arithmetic are the most intriguing because they contain more of what is not usually presented in logic books. The first two chapters on sentential logic are not the standard textbook treatment. This really is a book for other working logicians and mathematicians. Quickly, Goodstein moves into treating truth functions for any number of truth values as functions in modulo arithmetic and investigates the topic of finding Sheffer functions, viz., a function from which all others for a given number of truth values can be obtained. In the second chapter, he is soon presenting Intuitionistic sentential logic and has some remarks about sentential logics between Intuitionistic logic and classical logic. He also presents sequent calculi. Since the first two chapters contain only 36 pages, this suggests that the book would be suitable as a text only for advanced and sophisticated students. (shrink)

The essays in philosophical logic collected in this volume are dedicated to Henry S. Leonard who was one of the first American philosophers to urge the application of modern logic to non-mathematical areas. Leonard also inspired the development of certain areas of contemporary philosophical logic discussed in some of the papers of this volume. This is especially clear in the case of free, or presupposition free, logics which Leonard's early work on a logic of existence inspired. In one essay of (...) this volume Bas C. Van Frassen further develops his work on the semantics of free logic. In another, Milton Fisk relates free logic to modal logic, suggesting a new semantics of strength to deal with various problems in the area and in a third essay, Karel Lambert relates free logic to certain logical puzzles about quantum theory. Leonard was also one of the first to entertain the idea of and discuss a logic of questions and an erotetic logic or logic of interests. The volume contains essays on these topics by Nuel D. Belnap and David Harrah. Leonard's interest in modal logic is reflected in Fisk's article as well as in an article by Richmond Thomason on modal logic and metaphysics. His interest in epistemic logic is seen in an essay on the logic of belief by Jon Vickers and an essay by Hintikka applying the logic of belief and knowledge to problems about the ontological argument. In addition to these essays, there are five others of high quality on diverse topics: R. M. Martin on intentions, Wilfrid Sellars on the metaphysics of the person, R. M. Chisholm on agency, Frederick Fitch on combinatory logic and negative numbers, and H. E. Hendry and G. J. Massey on Sheffer functions. This is a worthy memorial to a quietly influential American philosopher.--R. H. K. (shrink)

Henry M. Sheffer: a bibliography (p. xv-xvi)--Structure: A formulation of the logic of sense and denotation, by A. Church. Notes on the logic of intension, by C.I. Lewis. The logic of terms, by J.W. Miller. Two-valued truth tables for modal functions, by H.S. Leonard. N-valued Boolean algebra, by P. Henle. Triangular matrices determined by two sequences, by L.L. Silverman. The ordered pair in number theory, by W.V. Quine.

Wittgenstein's conception of the general form of a truth function given in thesis 6 can be presented as a sort of a trade-off: the author of the Tractatus is unable to reconcile the simplicity of his original idea of a series of forms with the simplicity of his generalisation of Sheffer's stroke; therefore, he is forced to sacrifice one of them. As we argue in this paper, the choice he makes – to weaken the logical constraints put on (...) the concept of a series of forms, thus effectively metaphorising that concept for the sake of upholding the N-operation's role of generating the series – is unfortunate. An actual expansion of a series of truth functions as defined in 6 would require either making decisions at each step (Anscombe) or outwardly rejecting the concept of a series (Sundholm). However, neither is faithful to Wittgenstein's own fundamental intuitions regarding the nature of logic. For this reason, a different trade-off that prioritises upholding the basic features of a series of forms over the simplicity of the operation that generates that series seems to be much more reasonable. We offer such a trade-off by developing the schema already present in the Tractatus (5.101). The key element of our alternative solution is the construction of an operation that can perform the task of producing all consecutive truth functions of a given collection of atomic propositions as an invariant difference between any base and its successor throughout the series. We define it as a composition of three functions (in an algebraic sense): the first turns a symbol of a given truth function into a binary number, the second increments that number, the third turns the result back into a symbol of another truth function. We also show how the operation can be defined by means of a different notation without invoking binary arithmetic. (shrink)

Why do people share or publicly engage with fake stories? Two possible answers come to mind: (a) people are deeply irrational and believe these stories to be true; or (b) they intend to deceive their audience. Both answers presuppose the idea that people put the stories forward as true. But I argue that in some cases, these outlandish (yet also very popular) stories function as signals of one's group membership. This signaling function can make better sense of why, (...) despite their unusual nature or lack of a factual basis, some of these stories are so widespread. (shrink)