The Classification of Propositional Calculi

We discuss Smirnovs problem of finding a common background for classifying implicational logics. We formulate and solve the problem of extending, in an appropriate way, an implicational fragment H of the intuitionistic propositional logic to an implicational fragment TV of the classical propositional logic. As a result we obtain logical constructions having the form of Boolean lattices whose elements are implicational logics. In this way, whole classes of new logics can be obtained. We also consider the transition from implicational logics to full logics. On the base of the lattices constructed, we formulate the main classification principles for propositional logics.     

Game Semantics for the Lambek-Calculus: Capturing Directionality and the Absence of Structural Rules

In this paper, we propose a game semantics for the (associative) Lambek calculus. Compared to the implicational fragment of intuitionistic propositional calculus, the semantics deals with two features of the logic: absence of structural rules, as well as directionality of implication. We investigate the impact of these variations of the logic on its game semantics. Presented by Wojciech Buszkowski     

Identical Twins,Deduction Theorems,and Pattern Functions: Exploring the Implicative BCSK Fragment of S5

We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we show that for the pure logic of one of these implicational connectives two – in general distinct – consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the ‘propositional operations’ associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives.     

Types of \mathsf{I} -Free Hereditary Right Maximal Terms

The implicational fragment of the relevance logic “ticket entailment” is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of T_→ are in normal form and built in all but one case from and only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define FIT_→ – a Fitch-style calculus (related to FT_→) for the implicational fragment of ticket entailment.     

The finite model property for BCK and BCIW

This paper shows that both implicational logicsBCK andBCIW have the finite model property. The proof of the finite model property forBCIW, which is equal to the relevant logicR , was originally given by the first author in his unpublished paper [6] in 1973. The finite model property forBCK can be obtained by modifying the proof of that forBCIW. Here, both of these proofs will be given in a unified form and the difference between them will be clarified. Further discussions will be given in the last section.The first author wishes to thank IIAS-SIS (Fujitsu Laboratories Ltd., Numazu) for 9 months hospitality in Japan, and for facilitating this research.Presented byJan Zygmunt     

On Varieties of Biresiduation Algebras

A biresiduation algebra is a 〈/,\,1〉-subreduct of an integral residuated lattice. These algebras arise as algebraic models of the implicational fragment of the Full Lambek Calculus with weakening. We axiomatize the quasi-variety B of biresiduation algebras using a construction for integral residuated lattices. We define a filter of a biresiduation algebra and show that the lattice of filters is isomorphic to the lattice of B-congruences and that these lattices are distributive. We give a finite basis of terms for generating filters and use this to characterize the subvarieties of B with EDPC and also the discriminator varieties. A variety generated by a finite biresiduation algebra is shown to be a subvariety of B. The lattice of subvarieties of B is investigated; we show that there are precisely three finitely generated covers of the atom. Mathematics Subject Classification (2000): 03G25, 06F35, 06B10, 06B20 Dedicated to the memory of Willem Johannes Blok     

On structural completeness of implicational logics

We consider the notion of structural completeness with respect to arbitrary (finitary and/or infinitary) inferential rules. Our main task is to characterize structurally complete intermediate logics. We prove that the structurally complete extension of any pure implicational in termediate logic C can be given as an extension of C with a certain family of schematically denned infinitary rules; the same rules are used for each C. The cardinality of the family is continuum and, in the case of (the pure implicational fragment of) intuitionistic logic, the family cannot be reduced to a countable one. It means that the structurally complete extension of the intuitionistic logic is not countably axiomatizable by schematic rules.This work was supported by the Polish Academy of Sciences, CPBP 08.15, Struktura logiczna rozumowa niesformalizowanych.     

Conditionals,probability, and non-triviality

We show that the implicational fragment of intuitionism is the weakest logic with a non-trivial probabilistic semantics which satisfies the thesis that the probabilities of conditionals are conditional probabilities. We also show that several logics between intuitionism and classical logic also admit non-trivial probability functions which satisfy that thesis. On the other hand, we also prove that very weak assumptions concerning negation added to the core probability conditions with the restriction that probabilities of conditionals are conditional probabilities are sufficient to trivialize the semantics.     

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.     

Implicational Paradoxes and the Meaning of Logical Constants

I discuss paradoxes of implication in the setting of a proof-conditional theory of meaning for logical constants. I argue that a proper logic of implication should be not only relevant, but also constructive and nonmonotonic. This leads me to select as a plausible candidate LL, a fragment of linear logic that differs from R in that it rejects both contraction and distribution.     

Rosser orderings and free variables

It is shown that for arithmetical interpretations that may include free variables it is not the Guaspari-Solovay system R that is arithmetically complete, but their system R ^–. This result is then applied to obtain the nonvalidity of some rules under arithmetical interpretations including free variables, and to show that some principles concerning Rosser orderings with free variables cannot be decided, even if one restricts oneself to usual proof predicates.     

Ordered sets R and Q as bases of Kripke models

Those formulas which are valid in every Kripke model having constant domain whose base is the ordered set R of real numbers (or, the ordered set Q of rational numbers) are characterized syntactically.     

Deduction theorems for weak implicational logics

The standard deduction theorem or introduction rule for implication, for classical logic is also valid for intuitionistic logic, but just as with predicate logic, other rules of inference have to be restricted if the theorem is to hold for weaker implicational logics.In this paper we look in detail at special cases of the Gentzen rule for and show that various subsets of these in effect constitute deduction theorems determining all the theorems of many well known as well as not well known implicational logics. In particular systems of rules are given which are equivalent to the relevance logics E,R, T, P-W and P-W-I.     

A New Normalization Strategy for the Implicational Fragment of Classical Propositional Logic

The introduction and elimination rules for material implication in natural deduction are not complete with respect to the implicational fragment of classical logic. A natural way to complete the system is through the addition of a new natural deduction rule corresponding to Peirce’s formula (((A → B) → A) → A). E. Zimmermann [6] has shown how to extend Prawitz’ normalization strategy to Peirce’s rule: applications of Peirce’s rule can be restricted to atomic conclusions. The aim of the present paper is to extend Seldin’s normalization strategy to Peirce’s rule by showing that every derivation Π in the implicational fragment can be transformed into a derivation Π′ such that no application of Peirce’s rule in Π′ occurs above applications of →-introduction and →-elimination. As a corollary of Seldin’s normalization strategy we obtain a form of Glivenko’s theorem for the classical {→}-fragment.     

Relevant Robinson's arithmetic

In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in relevant arithmetic.This paper has been greatly influenced by the (largely unpublished) work of E. K. Meyer (cf. [7]) on relevant arithmetic, and I wish to thank him, and also N. D. Belnap, Jr. and D. Cohen for helpful advice. In fairness to Meyer it must be said that he finds my axioms 13 and 13(1) too strong (they are not theorems of his system R# - cf. §4 below). Meyer tells me be finds vindication for his view in my chief theorem of §2. For myself, I find the insights behind Meyer's work on R# to be both stable and fruitful, and if I now had to make a choice, I would follow Meyer in his rejection of my axioms. However, the systems I explore in this paper themselves have a surprising amount of internal consistency of motivation (cf. §5). Let a hundred formal systems bloom.     

A case of extreme simplicity of the core matrix in three-mode principal components analysis

In three-mode Principal Components Analysis, theP ×Q ×R core matrixG can be transformed to simple structure before it is interpreted. It is well-known that, whenP=QR,G can be transformed to the identity matrix, which implies that all elements become equal to values specified a priori. In the present paper it is shown that, whenP=QR − 1,G can be transformed to have nearly all elements equal to values spectified a priori. A cllsed-form solution for this transformation is offered. Theoretical and practical implications of this simple structure transformation ofG are discussed. Constructive comments from anonymous reviewers are gratefully acknowledged.     

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.     

Insomnia symptoms are associated with elevated C-reactive protein in young adults

Abstract

Objective: Insomnia is associated with elevated inflammation; however, studies have not investigated if this relationship is confounded with depression and neuroticism, which are associated with insomnia and inflammation. The current study examined the association of insomnia symptoms with C-reactive protein (CRP) and with interleukin-6 (IL-6), independently and after controlling for depressive symptoms and neuroticism. Design: Fifty-two young adults (mean age?=?25.2?±?3.9 years, 52% female) completed a baseline survey to assess psychological characteristics, followed by a plasma blood draw. Main outcome measures: Plasma CRP and IL-6. Results: When examined alone, insomnia symptoms were significantly associated with elevated CRP (β?=?0.52; R²?=?0.27), as was neuroticism (β?=?0.41, R²?=?0.17), but not depressive symptoms (β?=?0.21, R²?=?0.05). The association between insomnia symptoms and CRP remained significant when depressive symptoms and neuroticism were entered into the model simultaneously; this model did not explain more variance than the model with insomnia symptoms alone. No variables were associated with IL-6. Conclusions: Results suggest that insomnia symptoms are independently associated with elevated CRP in young adults, even after controlling for presumed overlapping psychological constructs. Findings highlight the potential importance of treating insomnia to reduce systemic inflammation.     

Relationships among motor skill,perceived self-competence,fitness, and physical activity in young adults

PurposeThis large-scale, exploratory study sought to examine the associations between motor skill proficiency (MSP), perceived self-competence (PC), physical fitness (PF), and self-reported physical activity (PA) to assess the assumption that relationships among these variables found in childhood and adolescence persist into young-adulthood. Specifically, predictors of PF and PA were assessed. Additionally, the contributions body mass index (BMI) and gender made to these relationships, as well as that of process- versus product-oriented motor skills data, were studied. Finally, how these relationships differed by gender was explored.MethodsParticipants’ (n = 448) MSP was assessed using select measures from the TGMD-2 and the BOT-2, while PF was evaluated using the FitnessGram. Participants also completed surveys to assess PC (SPPCS), PA (past-week MAQ), and demographic information. Height and weight were collected to calculate BMI.ResultsModels predicting physical fitness had good fit to the observed data, with perceived athletic competence, locomotor skill, upper-limb coordination, BMI, and gender accounting for about 75% (R² = 0.748) of aerobic PF variance and about 48% (R² = 0.476) of variance in push-up performances. About 18% (R² = 0.178) of variance in curl-up performances was predicted by perceived athletic competence, locomotor skill, upper-limb coordination, and PA. The PA model demonstrated weaker predictive power, with only about 10% (R² = 0.097) of variance explained. When considering male and female models separately, female models demonstrated better fit when predicting all PF and PA outcome variables. BMI and gender operated as significant predictors to differing degrees across the various models.ConclusionsThere is compelling evidence to believe relationships found among MSP, PC, PF, and PA in childhood and adolescence also exist in early adulthood. However, differences in the roles of gender and physical fitness versus physical activity are likely to exist.