A three-valued propositional logic is presented, within which the three values are read as ?true?, ?false? and ?nonsense?. A three-valued extended functional calculus, unrestricted by the theory of types, is then developed. Within the latter system, Bochvar analyzes the Russell paradox and the Grelling-Weyl paradox, formally demonstrating the meaninglessness of both.

Kleene’s well-known strong three-valued logic is shown to be one of a family of logics with similar mathematical properties. These logics are produced by an intuitively natural construction. The resulting logics have direct relationships with bilattices. In addition they possess mathematical features that lend themselves well to semantical constructions based on fixpoint procedures, as in logic programming.

Stereotypically, computation involves intrinsic changes to the medium of representation: writing new symbols, erasing old symbols, turning gears, flipping switches, sliding abacus beads. Perspectival computation leaves the original inscriptions untouched. The problem solver obtains the output by merely alters his orientation toward the input. There is no rewriting or copying of the input inscriptions; the output inscriptions are numerically identical to the input inscriptions. This suggests a loophole through some of the computational limits apparently imposed by physics. There can be (...) symbol manipulation without inscription manipulation because symbols are complex objects that have manipulatable elements besides their inscriptions. Since a written symbol is an ordered pair of consisting of a shape and the reader's orientation to that inscription, the symbol can be changed by changing the orientation rather than inscription. Although there are the usual physical limits associated with reading the answer, the computation is itself instantaneous. This is true even when the sub-calculations are algorithmically complex, exponentially increasing or even infinite. (shrink)

While Kripke's original paper on the theory of truth used a three-valued logic, we believe a four-valued version is more natural. Its use allows for possible inconsistencies in information about the world, yet contains Kripke's development within it. Moreover, using a four-valued logic makes it possible to work with complete lattices rather than complete semi-lattices, and thus the mathematics is somewhat simplified. But more strikingly, the four-valued version has a wide, natural generalization to the family of interlaced bilattices. Thus, with (...) little more work, the theory is extended to a broad class of settings. Indeed, a result like Theorem 6.2 would not even be possible to state without the interlaced bilattice machinery. We hope the notion of interlaced bilattice will make apparent further such connections. (shrink)

Frank Ramsey was the greatest of the remarkable generation of Cambridge philosophers and logicians which included G. E. Moore, Bertrand Russell, Ludwig Wittgenstein and Maynard Keynes. Before his tragically early death in 1930 at the age of twenty-six, he had done seminal work in mathematics and economics as well as in logic and philosophy. This volume, with a new and extensive introduction by D. H. Mellor, contains all Ramsey's previously published writings on philosophy and the foundations of mathematics. The latter (...) gives the definitive form and defence of the reduction of mathematics to logic undertaken in Russell and Whitehead's Principia Mathematica; the former includes the most profound and original studies of universals, truth, meaning, probability, knowledge, law and causation, all of which are still constantly referred to, and still essential reading for all serious students of these subjects. (shrink)

I LOGIC IN PHILOSOPHY— PHILOSOPHY OF LOGIC i. On the relation of logic to philosophy I n this book, the consequences of certain logical insights for ...

This book offers a unique synthesis of past and current work on the structure, meaning, and use of negation and negative expressions, a topic that has engaged thinkers from Aristotle and the Buddha to Freud and Chomsky. Horn's masterful study melds a review of scholarship in philosophy, psychology, and linguistics with original research, providing a full picture of negation in natural language and thought; this new edition adds a comprehensive preface and bibliography, surveying research since the book's original publication.

In the general theory of logic built up by Whitehead and Russell to furnish a basis for all mathematics there is a certain subtheory which is unique in its simplicity and precision; and though all other portions of the work have their roots in this subtheory, it itself is completely independent of them. Whereas the complete theory requires for the enunciation of its propositions real and apparent variables, which represent both individuals and propositional functions of different kinds, and as a (...) result necessitates- the introduction of the cumbersome theory of types, this subtheory uses only real variables, and these real variables represent but one kind of entity—which the authors have chosen to call elementary propositions. The most general statements are formed by merely combining these variables by means of the two primitive propositional functions of propositions Negation and Disjunction; and the entire theory is concerned with the process of asserting those combinations which it regards as true propositions, employing for this purpose a few general rules which tell how to assert new combinations from old, and a certain number of primitive assertions from which to begin. (shrink)