Double-Negation Elimination in Some Propositional Logics

This article answers two questions (posed in the literature), each concerning the guaranteed existence of proofs free of double negation. A proof is free of double negation if none of its deduced steps contains a term of the form n(n(t)) for some term t, where n denotes negation. The first question asks for conditions on the hypotheses that, if satisfied, guarantee the existence of a double-negation-free proof when the conclusion is free of double negation. The second question asks about the existence of an axiom system for classical propositional calculus whose use, for theorems with a conclusion free of double negation, guarantees the existence of a double-negation-free proof. After giving conditions that answer the first question, we answer the second question by focusing on the Lukasiewicz three-axiom system. We then extend our studies to infinite-valued sentential calculus and to intuitionistic logic and generalize the notion of being double-negation free. The double-negation proofs of interest rely exclusively on the inference rule condensed detachment, a rule that combines modus ponens with an appropriately general rule of substitution. The automated reasoning program OTTER played an indispensable role in this study.Comment: 32 pages, no figure

Constructive Logic with Strong Negation is a Substructural Logic. II

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew . The main result of Part I of this series [41] shows that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL ew (namely, a certain variety of FL ew -algebras) are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting of deductive systems to establish the definitional equivalence of the logics N and NFL ew . It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logi

Discriminator logics (Research announcement)

A discriminator logic is the 1-assertional logic of a discriminator variety V having two constant terms 0 and 1 such that V ⊨ 0 1 iff every member of V is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system SBPC, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator logic S can be presented (up to definitional equivalence) as an axiomatic extension of SBPC by a set of extensional logical connectives taken from the language of S. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work

An integration of friendship and social support : relationships with adjustment in college students

A conceptual integration of friendship and social support, explored via factor analysis, was examined in relation to adjustment in 242 undergraduate university men and women. Despite considerable overlap between theoretical components of adult friendship and social support, empirically these two areas have remained quite distinct. The present study sought to consolidate the two important research areas, looking at sex differences and the ways in which interpersonal resources can facilitate adjustment. Subjects were recruited from two local universities for this questionnaire-based study. Participants provided information about their best same-sex friend, their social network as a whole, a romantic relationship (if applicable) and various aspects of adjustment, including depression, self-esteem, quality of life and physical symptoms. Best friend and social network items, respectively, were grouped into subscales representing previously postulated dimensions of friendship and social support. These subscales were entered into exploratory factor analyses, separately for best friend and for network, to determine whether as predicted, friendship and support would combine conceptually. The factors which emerged were entered into hierarchical multiple regressions in order to investigate the connections between these relationship factors, daily hassles and adjustment. The results suggest that relationship factors, particularly those offered by a large, high-quality social network, offer protective benefits for college students. Further, having a trusted, satisfying best friendship and a high-quality romantic relationship appears to enhance certain aspects of adjustment for students as well. Daily hassles were found to detract significantly from the well-being of young men and women. From a theoretical perspective, support was gained for viewing friendship and support not as distinct constructs, but rather, as joint contributors to the phenomena of interpersonal relationships. Practical implications include the need to examine ways of helping college students with small, less than adequate social networks build larger, more beneficial sets of resources

Discriminator logics (Research announcement)

A discriminator logic is the 1-assertional logic of a discriminator variety V having two constant terms 0 and 1 such that V ⊨ 0 1 iff every member of V is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system SBPC, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator logic S can be presented (up to definitional equivalence) as an axiomatic extension of SBPC by a set of extensional logical connectives taken from the language of S. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work