Algebraic aspects of deduction theorems

The first known statements of the deduction theorems for the first-order predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary set P of formulas of I built up with at most two distinct sentential variables p and q is called a uniform deduction theorem scheme for C if it satisfies the following condition: for every set X of formulas of I and for any formulas and , C(X{{a}}) iff P(, ) AC(X). [P(, ) denotes the set of formulas which result by the simultaneous substitution of for p and for q in all formulas in P]. The above definition encompasses many particular formulations of theorems considered in the literature to be deduction theorems. Theorem 1.3 gives necessary and sufficient conditions for a logic to have a uniform deduction theorem scheme. Then, given a sentential logic C with a uniform deduction theorem scheme, the lattices of deductive filters on the algebras A similar to the language of C are investigated. It is shown that the join-semilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian.A part of this paper was presented in abstracted form in Bulletin of the Section of Logic, Vol. 12, No. 3 (1983), pp. 111–116, and in The Journal of Symbolic Logic.     

Unrealistic Optimism and Positive-Negative Asymmetry: A Conceptual and Cross-cultural Study of Interrelationships between Optimism,Pessimism, and Realism

Using a variant of Weinstein's (1980) technique for the measurement of unrealistic optimism, subjects were classified as optimistic, realistic, or pessimistic about their chances of being confronted in the future with problems such as divorce, nervous breakdown, etc. On the basis of previous theory and research on cognition and affect, it was hypothesized that, across problems, subjects would alternate optimism with realism rather than with pessimism. Cross-cultural data obtained from 19 samples of Belgian, Moroccan, and Polish subjects not only confirmed the hypothesis but also showed high agreement about the nature of the problems that were selectively associated with optimism, realism and pessimism.     

Some theorems on structural entailment relations

The classesMatr( \( \subseteq \) ) of all matrices (models) for structural finitistic entailments \( \subseteq \) are investigated. The purpose of the paper is to prove three theorems: Theorem I.7, being the counterpart of the main theorem from Czelakowski [3], and Theorems II.2 and III.2 being the entailment counterparts of Bloom's results [1]. Theorem I.7 states that if a classK of matrices is adequate for \( \subseteq \) , thenMatr( \( \subseteq \) ) is the least class of matrices containingK and closed under the formation of ultraproducts, submatrices, strict homomorphisms and strict homomorphic pre-images. Theorem II.2 in Section II gives sufficient and necessary conditions for a structural entailment to be finitistic. Section III contains theorems which characterize finitely based entailments.     

Social Problem-Solving Skills in Children with Traumatic Brain Injury: Long-Term Outcomes and Prediction of Social Competence

The effects of childhood traumatic brain injury (TBI) on social problem-solving were examined in 35 children with severe TBI, 40 children with moderate TBI, and 46 children with orthopedic injuries (OI). The children were recruited prospectively following injuries that occurred between 6 and 12 years of age. They were followed longitudinally, and ranged from 9 to 18 years of age at the time of the current study, which occurred on average 4 years post injury. They were administered a semi-structured interview used in previous research on social problem-solving to assess the developmental level of their responses to hypothetical dilemmas involving social conflict. Children in the severe TBI group defined the social dilemmas and generated alternative strategies to solve those dilemmas at the same developmental level as did children in the OI group. However, they articulated lower-level strategies as the best way to solve the dilemmas and used lower-level reasoning to evaluate the effectiveness of the strategies. After controlling for group membership, race, socioeconomic status, IQ, and age, children's social problem-solving, and particularly the developmental level of their preferred strategies for resolving conflicts, predicted parents ratings of children's social skills, peer relationships, aggressive behavior, and academic performance. The findings indicate that children with severe TBI demonstrate selective, long-term deficits in their social problem-solving skills that may help to account for their poor social and academic outcomes.     

Deontology of Compound Actions

Studia Logica - This paper, being a companion to the book [2] elaborates the deontology of sequential and compound actions based on relational models and formal constructs borrowed from formal...