Classical Pitowsky correlation polytopes are reviewed with particular emphasis on the Minkowski-Weyl representation theorem. The inequalities representing the faces of polytopes are Boole’s “conditions of possible experience.” Many of these inequalities have been discussed in the context of Bell’s inequalities. We introduce CddIF, a Mathematica package created as an interface between Mathematic...

Optimization can make at least two contributions to boolean logic. Its solution methods can address inference and satisfiability problems, and its style of analysis can reveal tractable classes of boolean problems that might otherwise have gone unnoticed. They key to linking optimization with logic is to provide logical formulas a numerical interpretation or semantics. While syntax concerns the...

P E R H A P S T H E M O S T enduring idea from the early days of AI is that of a declarative system reasoning over explicitly represented knowledge with a general inference engine. Such systems require a formal language to describe the real world; and the real world has things in it. For this rea son, classical AI adopted first-order logic—the mathe matics of objects and relations—as its founda...

Geometry and Meaning is an interesting book about a relationship between geometry and logic defined on certain types of abstract spaces and how that intimate relationship might be exploited when applied in computational linguistics. It is also about an approach to information retrieval, because the analysis of natural language, especially negation, is applied to problems in IR, and indeed illus...

Finding solution values for unknowns in Boolean equations was, along with second-order quantifier elimination, a principal reasoning mode in the Algebra of Logic of the 19th century. Schröder [19] investigated it as Auflösungsproblem (solution problem). It is closely related to the modern notion of Boolean unification. For a given formula that contains unknowns formulas are sought such that aft...

The calculus of relational algebra has its roots in the second half of the last century with the pioneering work on binary relations of G. Boole, A. de Morgan, C.S. Peirce, and E. Schröder. The modern axiomatic development is due to A. Tarski and his co-workers. In the last two decades this formalization has been accepted by many mathematicians and computer scientists as a fruitful base for des...

Lattice theory originated late in the last century along two strands. Many of the structures used in logic have lattices, particularly distributive lattices, associated with them. The work of Boole [4] and Schröder [27] are seminal in this regard. The other strand, which concentrated more on the connections with algebra, was initiated with Richard Dedekind [6] in his studies of ideals in algebr...

This paper addresses the question of how we can understand reasoning in general and mathematical proofs in particular. It argues the need for a high-level understanding of proofs to complement the low-level understanding provided by Logic. It proposes a role for computation in providing this high-level understanding, namely by the association of proof plans with proofs. Proof plans are deened a...

and S are a start state and a start symbol, respectively.) is activated by a reducel or reduce2 action, and this processor reads the terminal symbol (the end mark), by a shift action, then the input string is accepted. If there are no processors performing a shift and a reducel action, the input string is rejected. 4.3 Handling Syntactic Features In order to extend the syntactic analysis method...

On montre ici comment transformer en programmes les démonstrations utilisant l’axiome du choix dépendant et l’existence d’un ultrafiltre non trivial sur N (qu’on peut, de plus, supposer sélectif ). La méthode est une extension de la technique de réalisabilité en logique classique introduite dans [3,4,5], et de la méthode du forcing bien connue en théorie des ensembles. Pour simplifier un peu l’...