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 logic. Presented by Heinrich Wansing     

Constructive Logic with Strong Negation is a Substructural Logic. I

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 . In this paper, it is shown 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. This answers a longstanding question of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL _ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL _ew . Presented by Heinrich Wansing     

Priestley Duality for Paraconsistent Nelson’s Logic

The variety of N4^{^}{{\bf N4}^\perp}-lattices provides an algebraic semantics for the logic N4^{^}{{\bf N4}^\perp} , a version of Nelson’s logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of N4^{^}{{\bf N4}^\perp}-lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson algebras constructed by R. Cignoli and A. Sendlewski.     

Equivalential logics (I)

The class of equivalential logics comprises all implicative logics in the sense of Rasiowa [9], Suszko's logicSCI and many Others. Roughly speaking, a logic is equivalential iff the greatest strict congruences in its matrices (models) are determined by polynomials. The present paper is the first part of the survey in which systematic investigations into this class of logics are undertaken. Using results given in [3] and general theorems from the theory of quasi-varieties of models [5] we give a characterization of all simpleC-matrices for any equivalential logicC (Theorem I.14). In corollaries we give necessary and sufficient conditions for the class of all simple models for a given equivalential logic to be closed under free products (Theorem I.18). Theorem I.17 can be generalized as follows:For any equivalential logic C, clauses (i), (iii)and (v),formulated in Th.I.17,are equivalent.     

A Negationless Interpretation of Intuitionistic Theories. I

The present work contains an axiomatic treatment of some parts of the restricted version of intuitionistic mathematics advocated by G. F. C. Griss, also known as negationless intuitionistic mathematics. Formal systems NPC, NA, and FIM ^N for negationless predicate logic, arithmetic, and analysis are proposed. Our Theorem 4 in Section 2 asserts the translatability of Heyting's arithmetic HAinto NA. The result can in fact be extended to a large class of intuitionistic theories based on HAand their negationless counterparts. For instance, in Section 3 this is shown for Kleene's system of intuitionistic analysis FIMand our FIM ^N . This revised version was published online in June 2006 with corrections to the Cover Date.     

The effects of news report valence and linguistic labels on prejudice against social minorities

ABSTRACT

Combating prejudice against social minorities is a challenging task in current multicultural societies. Mass media can decisively shape prejudice, because it often represents the main source of information about social minorities. In 3 studies in the Czech Republic (N = 445) and Switzerland (N = 362; N = 220), we investigated how prejudice against negatively and positively perceived minorities (the Roma in Study 1, Kosovo Albanians in Study 2, Italians in Study 3) is influenced by a single exposure to a print news report, by manipulating the valence of reports about minority members (positive vs. negative vs. mixed) and linguistic forms for minorities’ ethnicity (nouns vs. adjectives). Positive and negative reports shaped prejudice in the respective directions; the effect of mixed reports mostly did not differ from positive reports. Labeling ethnicity with nouns (e.g., a male Roma) resulted in more prejudice than adjectives (e.g., a Roma man), independent of report valence. Report valence influenced the affective part of prejudice (i.e., feelings toward a minority), whereas language consistently shaped the behavioral part of prejudice (i.e., preferred social distance from a minority).     

Undecidability and intuitionistic incompleteness

Let S be a deductive system such that S-derivability (s) is arithmetic and sound with respect to structures of class K. From simple conditions on K and s, it follows constructively that the K-completeness of s implies MP(S), a form of Markov's Principle. If s is undecidable then MP(S) is independent of first-order Heyting arithmetic. Also, if s is undecidable and the S proof relation is decidable, then MP(S) is independent of second-order Heyting arithmetic, HAS. Lastly, when s is many-one complete, MP(S) implies the usual Markov's Principle MP.An immediate corollary is that the Tarski, Beth and Kripke weak completeness theorems for the negative fragment of intuitionistic predicate logic are unobtainable in HAS. Second, each of these: weak completeness for classical predicate logic, weak completeness for the negative fragment of intuitionistic predicate logic and strong completeness for sentential logic implics MP. Beth and Kripke completeness for intuitionistic predicate or sentential logic also entail MP.These results give extensions of the theorem of Gödel and Kreisel (in [4]) that completeness for pure intuitionistic predicate logic requires MP. The assumptions of Gödel and Kreisel's original proof included the Axiom of Dependent Choice and Herbrand's Theorem, no use of which is explicit in the present article.     

The Tense Logic for Master Argument in Prior’s Reconstruction

In this paper we examine Prior’s reconstruction of Master Argument [4] in some modal-tense logic. This logic consists of a purely tense part and Diodorean definitions of modal alethic operators. Next we study this tense logic in the pure tense language. It is the logic K _t 4 plus a new axiom (P): ‘p Λ G p ⊃ P G p’. This formula was used by Prior in his original analysis of Master Argument. (P) is usually added as an extra axiom to an axiomatization of the logic of linear time. In that case the set of moments is a total order and must be left-discrete without the least moment. However, the logic of Master Argument does not require linear time. We show what properties of the set of moments are exactly forced by (P) in the reconstruction of Prior. We make also some philosophical remarks on the analyzed reconstruction. Presented by Jacek Malinowski     

Functional Monadic Bounded Algebras

The variety MBA of monadic bounded algebras consists of Boolean algebras with a distinguished element E, thought of as an existence predicate, and an operator ${\exists} reflecting the properties of the existential quantifier in free logic. This variety is generated by a certain class FMBA of algebras isomorphic to ones whose elements are propositional functions.     

Rough Sets and 3-Valued Logics

In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix). With the strong semantics, where only the value t is treated as designated, the above logic is a “common denominator” for Kleene and Łukasiewicz 3-valued logics, which represent its two different “determinizations”. In turn, the weak semantics—where both t and u are treated as designated—represents such a “common denominator” for two major 3-valued paraconsistent logics. We give sound and complete, cut-free sequent calculi for both versions of the logic generated by the rough set Nmatrix. Then we derive from these calculi sequent calculi with the same properties for the various “determinizations” of those two versions of the logic (including Łukasiewicz 3-valued logic). Finally, we show how to embed the four above-mentioned determinizations in extensions of the basic rough set logics obtained by adding to those logics a special two-valued “definedness” or “crispness” operator.     

Perfectionism and achievement goals in athletes: Relations with approach and avoidance orientations in mastery and performance goals

ObjectivesChallenging views that perfectionism is a maladaptive factor in sport and that it is related to a preoccupation with performance goals and a neglect of mastery goals, the present article argues that perfectionism in athletes is not generally maladaptive, but shows differential relationships with mastery and performance goals depending on which facets of perfectionism are regarded.MethodGoing beyond the dichotomous achievement goal framework, two studies with N=204 high school athletes and N=147 university student are presented investigating how two facets of perfectionism—striving for perfection and negative reactions to imperfection [Stoeber, J., Otto, K., Pescheck, E., Becker, C., & Stoll, O. (2007). Perfectionism and competitive anxiety in athletes: Differentiating striving for perfection and negative reactions to imperfection. Personality and Individual Differences, 42, 959–969.]—relate to athletes’ achievement goals.DesignStudy 1 employed a cross-sectional correlational design, Study 2 a longitudinal correlational design.ResultsFollowing the trichotomous achievement goal framework, Study 1 found striving for perfection to be positively related to mastery and performance-approach goals, whereas negative reactions to imperfection were positively related to performance-approach and performance-avoidance goals and inversely to mastery goals. Following the 2×2 framework, Study 2 found striving for perfection to be positively related to mastery-approach and performance-approach goals whereas negative reactions to imperfection were positively related to mastery-avoidance, performance-approach, and performance-avoidance goals. Moreover, negative reactions to imperfection predicted residual increases in mastery-avoidance goals over 3 months.ConclusionsIt is concluded that striving for perfection in athletes is associated with an adaptive pattern of achievement goals whereas negative reactions to imperfection are associated with a maladaptive pattern. Thus, striving for perfection in sport may be adaptive in athletes who do not experience strong negative reactions when performance is less than perfect.     

Reliable measurement in sport psychology: The case of performance outcome measures

ObjectivesThe present research addresses a neglected aspect within the current Zeitgeist of improving methodological standards in (sport)psychology: reliable measurement. We discuss and highlight the importance of reliable measurement from different perspectives and empirically assess reliability of three commonly used performance outcome measures in order to give guidelines to researchers on how to increase reliability of measurements of performance outcomes.MethodIn three studies we estimate 5 different reliability coefficients for three performance outcome measures based on 14 golf putts (study 1; N = 100), 14 dart throws (study 2; N = 200; 100 sports students; 100 non-sports students) and 14 free throws in basketball (study 3; N = 192; 100 non-basketball players; 92 basketball players).ResultsThe highest reliability was the odd-even reliability for darts for the whole sample (0.888), followed by golf putts (0.714 for distance from the hole, 0.614 for successful putts) and free throws (0.504 non-basketball players; 0.62 for basketball players; and 0.826 for whole sample).ConclusionsBased on theoretical considerations and our empirical findings we give practical guidelines to improve reliability for performance outcome measures in sport psychology.     

AI,ME AND LEWIS (ABELIAN IMPLICATION,MATERIAL EQUIVALENCE AND C I LEWIS 1920)

C I Lewis showed up Down Under in 2005, in e-mails initiated by Allen Hazen of Melbourne. Their topic was the system Hazen called FL (a Funny Logic), axiomatized in passing in Lewis 1921. I show that FL is the system MEN of material equivalence with negation. But negation plays no special role in MEN. Symbolizing equivalence with → and defining ∼A inferentially as A→f, the theorems of MEN are just those of the underlying theory ME of pure material equivalence. This accords with the treatment of negation in the Abelian l-group logic A of Meyer and Slaney (Abelian logic. Abstract, Journal of Symbolic Logic 46, 425–426, 1981), which also defines ∼A inferentially with no special conditions on f. The paper then concentrates on the pure implicational part AI of A, the simple logic of Abelian groups. The integers Z were known to be characteristic for AI, with every non-theorem B refutable mod some Zn for finite n. Noted here is that AI is pre-tabular, having the Scroggs property that every proper extension SI of AI, closed under substitution and detachment, has some finite Zn as its characteristic matrix. In particular FL is the extension for which n = 2 (Lewis, The structure of logic and its relation to other systems. The Journal of Philosophy 18, 505–516, 1921; Meyer and Slaney, Abelian logic. Abstract. Journal of Symbolic Logic 46, 425–426, 1981; This is an abstract of the much longer paper finally published in 1989 in G. G. Priest, R. Routley and J. Norman, eds., Paraconsistent logic: essays on the inconsistent, Philosophica Verlag, Munich, pp. 245–288, 1989). Meyer was supported in this work as a Visiting Fellow in the College of Engineering and Computer Science, ANU.     

Guilty,But Not Ashamed: “True” Self‐Conceptions Influence Affective Responses to Personal Shortcomings

The current research examined how true self‐conceptions (who a person believes he or she truly is) influence negative self‐relevant emotions in response to shortcomings. In Study 1 (N = 83), an Internet sample of adults completed a measure of authenticity, reflected on a shortcoming or positive life event, and completed state shame and guilt measures. In Study 2 (N = 49), undergraduates focused on true versus other determined self‐attributes, received negative performance feedback, and completed state shame and guilt measures. In Study 3 (N = 138), undergraduates focused on self‐determined versus other determined self‐aspects, reflected on a shortcoming or neutral event, and completed state shame, guilt, and self‐esteem measures. In Study 4 (N = 75), undergraduates thought about true self‐attributes, an achievement, or an ordinary event; received positive or negative performance feedback; and completed state shame and guilt measures. In Study 1, differences in true self‐expression positively predicted shame‐free guilt (but not guilt‐free shame) following reminders of a shortcoming. Studies 2–4 found that experimental activation of true self‐conceptions increased shame‐free guilt and generally decreased guilt‐free shame in response to negative evaluative experiences. The findings offer novel insights into true self‐conceptions by revealing their impact on negative self‐conscious emotions.     

An Incomplete Relevant Modal Logic

The relevant modal logic G is a simple extension of the logic RT, the relevant counterpart of the familiar classically based system T. Using the Routley–Meyer semantics for relevant modal logics, this paper proves three main results regarding G: (i) G is semantically complete, but only with a non-standard interpretation of necessity. From this, however, other nice properties follow. (ii) With a standard interpretation of necessity, G is semantically incomplete; there is no class of frames that characterizes G. (iii) The class of frames for G characterizes the classically based logic T.     

The Incompleteness of RGL

RGLis a version of the modal logic GLbased on the relevant logic R. It is shown that the class of RKframes that verify all theorems of RGLalso verify a scheme that we call (!). If RGLhas (!) as a theorem, however, it is not a relevant logic. I go on to show that not all instances of (!) are theorems of RGL, hence this logic is not complete over any class of RKframes.     

On analysing relevance constructively

This paper lays out a program for analysing relevance constructively. It begins with a summary of results concerning the system C of Pottinger [197a] which has entailment, relevant implication, S4 strict implication, and intuitionist implication among its connectives. A full working out of the motivation for C will require formal analysis of informal concepts derived from the usual explanation of the meanings of the constants of intuitionist propositional logic. Formal machinery which should be adequate for the proof theoretic side of the analysis of the nonmodal part of C is described in detail, and the direction in which semantical results are to be sought is indicated.     

Louis Couturat,modern logic,and the international auxiliary language

ABSTRACT

To some extent, the early twentieth century revival of universal languages was the work of logicians and mathematicians. Pioneers of modern logic such as Frege, Russell and Peano wanted to overcome the diversity and deficiencies of natural languages. Through the rigour of formal logic, they aimed at providing scientific thinking with a reliable medium free from the ambiguity and inconsistencies of ordinary language. This article shows some interconnections between modern logic and the search for a common tongue that would unite scientists and people of all nations. The French mathematician and philosopher Louis Couturat is a key figure in understanding the interplay between these two movements. Through his work in composing the Ido language as an alternative to Esperanto, Couturat gave a new life to the Leibnizian idea of a universal characteristics. In addition, his multifaceted work provides a valuable insight into some political implications of early analytic philosophy.