Quantized Linear Logic,Involutive Quantales and Strong Negation

A new logic, quantized intuitionistic linear logic (QILL), is introduced, and is closely related to the logic which corresponds to Mulvey and Pelletier's (commutative) involutive quantales. Some cut-free sequent calculi with a new property quantization principle and some complete semantics such as an involutive quantale model and a quantale model are obtained for QILL. The relationship between QILL and Wansing's extended intuitionistic linear logic with strong negation is also observed using such syntactical and semantical frameworks.     

A Uniform Approach for Characterizing the Provably Total Number-Theoretic Functions of KPM and (Some of) its Subsystems

In this article we show how to extract with the use of the Buchholz-Cichon-Weiermann approach to subrecursive hierarchies from Rathjen's 1991 ordinal analysis of KPM a characterization of the provably total number-theoretic functions of KPM and some of its (most prominent) subsystems in a uniform and direct way.     

On the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics.     

Algorithmic Proof Methods and Cut Elimination for Implicational Logics Part I: Modal Implication

In this work we develop goal-directed deduction methods for the implicational fragment of several modal logics. We give sound and complete procedures for strict implication of K, T, K4, S4, K5, K45, KB, KTB, S5, G and for some intuitionistic variants. In order to achieve a uniform and concise presentation, we first develop our methods in the framework of Labelled Deductive Systems [Gabbay 96]. The proof systems we present are strongly analytical and satisfy a basic property of cut admissibility. We then show that for most of the systems under consideration the labelling mechanism can be avoided by choosing an appropriate way of structuring theories. One peculiar feature of our proof systems is the use of restart rules which allow to re-ask the original goal of a deduction. In case of K, K4, S4 and G, we can eliminate such a rule, without loosing completeness. In all the other cases, by dropping such a rule, we get an intuitionistic variant of each system. The present results are part of a larger project of a goal directed proof theory for non-classical logics; the purpose of this project is to show that most implicational logics stem from slight variations of a unique deduction method, and from different ways of structuring theories. Moreover, the proof systems we present follow the logic programming style of deduction and seem promising for proof search [Gabbay and Reyle 84, Miller et al. 91].     

Slot Mereology Revised

The paper suggests two revisions of K. Bennett's system of slot mereology. The revisions do not touch on the philosophical rationale for this system, but are focused on certain logical deficiencies in her formalisation.     

Philosophy and Science Dialogue: Mental Causation

This paper is a dialogue between Thalia Wheatley and Terence Horgan. Horgan maintains that philosophy is a broadly empirical discipline, and that philosophical theorizing about how concepts work treats certain intuitions about proper concept-usage as empirical data. He holds that the possibility of strong multiple realizability undermines the psychophysical identity theory. He holds that the concept of causation is governed by implicit contextual parameters, and that this dissolves Kim’s problem of “causal exclusion.” He holds that the concept of free will is governed by implicit contextual parameters, and that free-will attributions are often true, in typical contexts, even if determinism is true. Thalia Wheatley holds that the concept of multiple realizability hinges on the level of abstraction discussed and that neuroscientific data does not yet support multiple realizability of mental states from specific, high resolution brain states. She also holds that compatibilism redefines the concept of free will in ways that bear little resemblance to the common understanding―that of being free to choose otherwise in the moment. She maintains that this folk understanding is incompatible with the brain as a physical system and is not rescued by concepts of context and capacity.     

Logics with the Qualitative Probability Operator

The paper presents several strongly complete axiomatizationsof qualitative probability within the framework of probabilisticlogic. We show that in the proposed semantics qualitative probabilitiesare characterized by probability functions, so they also arecomparative probabilities.     

BETWEEN PHYSICALISM AND MENTALISM: PHILIP CLAYTON ON MIND AND EMERGENCE

Abstract. Philip Clayton's work on emergence is a valuable contribution to the fields of religion, science, and philosophy. I focus on three narrow but extremely important areas of Clayton's work. First, Clayton deems that Terrence Deacon's emergence theory is difficult to accept because it is constructed from thermodynamics, thereby rendering it unable to address phenomenological issues. I examine Deacon's theory and show that development from a physics base is warranted. Furthermore, Clayton does not convincingly demonstrate that such a constructive approach is necessarily incapable of attending to mental phenomena or offer an alternative that explains the causal power of a physically nonconstructible mental realm. Second, I argue that Clayton's notion of emergentist supervenience for comprehending the mental/physical relation is unnecessarily redundant and problematic in relation to causal power. Third, I explore Clayton's alternative use of agent causation to make sense of mental properties having causal power in the world. His effort to resolve emergence difficulties by appealing to phenomenology receives primary attention. Clayton's use of emergence theory is an important contribution to the religion‐and‐science community, and I encourage further dialogue on the exchange that Clayton commences.     

Proof Systems Combining Classical and Paraconsistent Negations

New propositional and first-order paraconsistent logics (called L _ω and FL _ω , respectively) are introduced as Gentzen-type sequent calculi with classical and paraconsistent negations. The embedding theorems of L _ω and FL _ω into propositional (first-order, respectively) classical logic are shown, and the completeness theorems with respect to simple semantics for L _ω and FL _ω are proved. The cut-elimination theorems for L _ω and FL _ω are shown using both syntactical ways via the embedding theorems and semantical ways via the completeness theorems. Presented by Yaroslav Shramko and Heinrich Wansing     

Critical thinking in North America: A new theory of knowledge,learning, and literacy

The pace of change in the world is accelerating, yet educational institutions have not kept pace. Indeed, schools have historically been the most static of social institutions, uncritically passing down from generation to generation outmoded didactic, lecture-and-drill-based, models of instruction. Predictable results follow. Students, on the whole, do not learn how to work by, or think for, themselves. They do not learn how to gather, analyze, synthesize and assess information. They do not learn how to analyze the diverse logic of the questions and problems they face and hence how to adjust their thinking to those problems. They do not learn how to enter sympathetically into the thinking of others, nor how to deal rationally with conflicting points of view. They do not learn to become critical readers, writers, speakers and listeners. They do not learn how to use their native languages clearly, precisely, or persuasively. They do not, therefore, become literate, in the proper sense of the word. Neither do they gain much in the way of genuine knowledge since, for the most part, they could not explain the basis for what they believe. They would be hard pressed to explain, for example, which of their beliefs were based on rational assent and which on simple conformity to what they have been told. They have little sense as to how they might critically analyze their own experience, or identify national or group bias in their own thinking. They are much more apt to learn on the basis of irrational than rational modes of thought. They lack the traits of mind of a genuinely educated person: intellectual humility, courage, integrity, perseverance, and faith in reason.Happily, there is a movement in education today striving to address these problems in a global way, with strategies and materials for the modification of instruction at all levels of education. At its foundation is an emerging new theory of knowledge, learning, and literacy, one which recognizes the centrality of independent critical thinking to all substantial learning, one which recognizes that higher-order, multilogical thinking is as important to childhood as to adult learning, and as important to foundational learning in monological as in multilogical disciplines. This educational reform movement is not proposing an educational miracle cure, for its leading proponents recognize that many social and historical forces must come together before the ideals of the critical thinking movement will become a full academic reality. Schools do not exist in a social vacuum. To the extent that the broader society is uncritical so, on the whole, will be society's schools. Nevertheless, the social conditions necessary for fundamental changes in schooling are increasingly apparent. The pressure for fundamental change is growing. Whether and to what extent these needed basic changes will be delayed or side-tracked, thus requiring new periodic resurgences of this movement, with new, more elaborate articulations of its ideals, goals, and methods — only time will tell.