Protoalgebraic Gentzen Systems and the Cut Rule

In this paper we show that, in Gentzen systems, there is a close relation between two of the main characters in algebraic logic and proof theory respectively: protoalgebraicity and the cut rule. We give certain conditions under which a Gentzen system is protoalgebraic if and only if it possesses the cut rule. To obtain this equivalence, we limit our discussion to what we call regular sequent calculi, which are those comprising some of the structural rules and some logical rules, in a sense we make precise. We note that this restricted set of rules includes all the usual rules in the literature. We also stress the difference between the case of two-sided sequents and the case of many-sided sequents, in which more conditions are needed.     

A Note on Formality and Logical Consequence

Logic is formal in the sense that all arguments of the same form as logically valid arguments are also logically valid and hence truth-preserving. However, it is not known whether all arguments that are valid in the usual model-theoretic sense are truth-preserving. Tarski claimed that it could be proved that all arguments that are valid (in the sense of validity he contemplated in his 1936 paper on logical consequence) are truth-preserving. But he did not offer the proof. The question arises whether the usual model-theoretic sense of validity and Tarski's 1936 sense are the same. I argue in this paper that they probably are not, and that the proof Tarski had in mind, although unusable to prove that model-theoretically valid arguments are truth-preserving, can be used to prove that arguments valid in Tarski's 1936 sense are truth-preserving.     

The Logical Style of Confucius’ Analects

Considered from a logical point of view, Confucius’ Analects contain many implicit forms of reasoning and argumentation. This is shown first by analyzing the phenomenon of parallelism: direct parallelism is often a way of hinting at a general assertion, whereas anti-parallelism hides dilemmas, generalizations and modal notions of “moral preference.” The Analects also have various types of conditionals, ranging from material implications, to modalized implications, and counterfactual conditionals, which are the germs of implicit reasoning, concluding with a moral recommendation. Analogies are particularly abundant and a presentation of three examples suggests that, beyond their explicative role, they also involve moral recommendations. The implicit logic of The Analects requires an active, albeit unconscious participation of the reader, which could be an important element in explaining the enduring influence of the text.     

Ist die linguistische Theorie des logischen Apriori obsolet?

The linguistic theory of the logical A Priori: is it obsolete In holistic interpretations, the logical truths are considered as continuous with empirical science: they are revisable, a posteriori, though very near to the centre of our web of belief. In this paper, we consider the merits and demerits of this approach, and we propose that it is necessary to revaluate holistic philosophies of logic. Some arguments are put forward which point in favour of the logical empiricists’ theory of logical truth. We argue (following Hartry Field) that the concept of “correlation between logical facts and logical beliefs” (which is at the heart of the holistic theory) is inconsistent. Finally, we concentrate on the principle of contradiction and argue (following Manley Thompson) that this principle is fundamental for meaning, truth, and thinking. This thesis is derived from considerations on the nature of intentionality. This revised version was published online in August 2006 with corrections to the Cover Date.     

Hilary Putnam's Dialectical Thinking: An Application to Fallacy Theory

In recent and not so recent years, fallacy theory has sustained numerous challenges, challenges which have seen the theory charged with lack of systematicity as well as failure to deliver significant insights into its subject matter. In the following discussion, I argue that these criticisms are subordinate to a more fundamental criticism of fallacy theory, a criticism pertaining to the lack of intelligibility of this theory. The charge of unintelligibility against fallacy theory derives from a similar charge against philosophical theories of truth and rationality developed by Hilary Putnam. I examine how Putnam develops this charge in the case of the conception of rationality pursued by logical positivism. Following that examination, I demonstrate the significance of this charge for how we proceed routinely to analyse one informal fallacy, the fallacy of petitio principii. Specifically, I argue that the significance of this charge lies in its issuance of a rejection of the urge to theorise in fallacy inquiry in general and petitio inquiry in particular. My conclusion takes the form of guidelines for the post-theoretical pursuit of fallacy inquiry.     

Introduction: Special Issue on Spirituality and Adult Development, Part II

This article reviews the purpose of the 3-part Special Issue on Spirituality and Adult Development, described in the Introduction to Special Issue Part I (J. D. Sinnott, 2000). It also outlines the contents of Part II. Developing adults express their belief that spirituality plays a key role in their development, yet developmental psychologists have given comparatively little attention to this factor. Also the concept of spirituality has seldom been considered separately from that of religion. The purpose of this special issue is to begin to remedy this neglect. This introduction is a summary of the 7 empirical and theoretical articles that form Part II of the Special Issue on Spirituality and Adult Development.     

Donald C. Williams’s defence of real metaphysics

In the middle of last century metaphysics was widely criticized, ridiculed, and committed to the flames. During this period a handful of philosophers, against several anti-metaphysical trends, defended metaphysics and articulated novel metaphysical doctrines. Donald C. Williams was one of these philosophers. But while his contributions to metaphysics are well known his defence of metaphysics is not and yet it played a key part in the development and revival of metaphysics. In this paper I present his defence of metaphysics in its historical context. I also show how his defence is relevant in response to recent attacks on metaphysics.     

The Scandal of Deduction

This article provides the first comprehensive reconstruction and analysis of Hintikka’s attempt to obtain a measure of the information yield of deductive inferences. The reconstruction is detailed by necessity due to the originality of Hintikka’s contribution. The analysis will turn out to be destructive. It dismisses Hintikka’s distinction between surface information and depth information as being of any utility towards obtaining a measure of the information yield of deductive inferences. Hintikka is right to identify the failure of canonical information theory to give an account of the information yield of deductions as a scandal, however this article demonstrates that his attempt to provide such an account fails. It fails primarily because it applies to only a restricted set of deductions in the polyadic predicate calculus, and fails to apply at all to the deductions in the monadic predicate calculus and the propositional calculus. Some corollaries of these facts are a number of undesirable and counterintuitive results concerning the proposed relation of linguistic meaning (and hence synonymy) with surface information. Some of these results will be seen to contradict Hintikka’s stated aims, whilst others are seen to be false. The consequence is that the problem of obtaining a measure of the information yield of deductive inferences remains an open one. The failure of Hintikka’s proposal will suggest that a purely syntactic approach to the problem be abandoned in favour of an intrinsically semantic one.     

A Note on “Constant Latent Odds-Ratios Models and the Mantel–Haenszel Null Hypothesis” Hessen, 2005

In a recent paper, Hessen (Psychometrika 70(3):497–516, 2005) introduces the class of constant latent odds-ratios models as an extension of the Rasch model for which the sum score is still the sufficient statistic for ability. In this paper the relation between both the general and the general parametric constant latent odds-ratios model and the Rasch model is considered.