Aristotelian diagrams for semantic and syntactic consequence

Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the well-known square of opposition for the categorical statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system’s soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation.

     

Teaching Legal Theory with Venn Diagrams

Venn diagrams, which are widely used in introductory logic courses, provide a convenient and illuminating way of presenting the various theories concerning the nature of law. When combined with the Aristotelian square of opposition, these diagrams show not only how the theories are related to one another, logically, which is essential to understanding them, but also which theories are compossible. One surprising result of this approach is that it shows the substantive compatibility of the theories of law set forth by H. L. A. Hart and Ronald Dworkin, who are usually pitted against one another. I show, through quotation and visual representation, that there is no essential disagreement between these jurisprudential stalwarts concerning the relation of law and morality.     

存在图的逻辑地位——批评与回应

存在图是继欧拉图、文恩图和皮尔士一文恩图之后的又一种逻辑图。逻辑图是指用于表示命题和推理的二维图形。然而由于存在图之前的逻辑图在表达能力上的缺陷以及现代数理逻辑的成功,长期以来存在图并未引起人们的关注。直到近来计算机表示的图示推理发展起来以后人们才认识到它的重要性,并把它作为一种图式逻辑纳入了哲学逻辑的范畴。人们对存在图逻辑地位的认可经历了一个复杂的过程。存在图在被认可道路上遇到了逻辑系统身份、推理效率和阐述的精确性三个问题,随着这三个问题的解决,人们逐渐认可了存在图的逻辑地位。     

The Broadest Necessity

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language (using truth functional connectives and quantifiers). This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given.     

Empirical Propositions and the Change of Language-Games

People unavoidably provide reasons for their words and deeds when reasoning in a language-game.Wittgenstein thinks that when people in different language-games argue with one other,they insist on adopting a doubtful attitude toward the reasons provided by the other side.His use of the term "language-game" here is a metaphor,and implies that people in different cultures can scarcely reason with one another.Indeed,according to Wittgenstein's consideration of concepts of logic in On Certainty,language-games are incompatible with one another because their internal logic and reasons are different from each other.However,in his discussion of empirical propositions Wittgenstein has also shown us the possibility that the intemal reasons of one language-game can transmit beyond its own borders and be valid in another language-game.     

Between Square and Hexagon in Oresme's Livre du Ciel et du Monde

In logic, Aristotelian diagrams are almost always assumed to be closed under negation, and are thus highly symmetric in nature. In linguistics, by contrast, these diagrams are used to study lexicalization, which is notoriously not closed under negation, thus yielding more asymmetric diagrams. This paper studies the interplay between logical symmetry and linguistic asymmetry in Aristotelian diagrams. I discuss two major symmetric Aristotelian diagrams, viz. the square and the hexagon of opposition, and show how linguistic considerations yield various asymmetric versions of these diagrams. I then discuss a pentagon of opposition, which occupies an uneasy position between the square and the hexagon. Although this pentagon belongs neither to the symmetric realm of logic nor to the asymmetric realm of linguistics, it occurs several times in the literature. The oldest known occurrence can be found in the cosmological work of the 14th-century author Nicole Oresme.     

A multi-purpose logic module for behavioral experiments

A notational system has been developed that can be used to describe most reinforcement schedules or other sequential procedures by appropriately interconnecting the basic notational units. The notational language has been applied in the past to develop a user-oriented computer program for behavioral experiments. The present paper describes a digital logic module designed to be isomorphic with the basic unit of the notational language, thus permitting rapid programming of new experiments directly from the notational system. The major advantage of the new module is that it requires no electronic sophistication on the part of the user, since the one module serves as the basic unit of all experiments.     

On S

The sentential logic S extends classical logic by an implication-like connective. The logic was first presented by Chellas as the smallest system modelled by contraining the Stalnaker-Lewis semantics for counterfactual conditionals such that the conditional is effectively evaluated as in the ternary relations semantics for relevant logics. The resulting logic occupies a key position among modal and substructural logics. We prove completeness results and study conditions for proceeding from one family of logics to another.We are grateful to Peter Apostoli, Kosta Doen, and anonymous referees for their comments on an earlier version of this paper. A.F.'s work has been supported by a grant from the Volkswagen-Stiftung.Presented byJan Zygmunt     

A cirquent calculus system with clustering and ranking

Cirquent calculus is a new proof-theoretic and semantic approach introduced by G. Japaridze for the needs of his theory of computability logic (CoL). The earlier article “From formulas to cirquents in computability logic” by Japaridze generalized formulas in CoL to circuit-style structures termed cirquents. It showed that, through cirquents with what are termed clustering and ranking, one can capture, refine and generalize independence-friendly (IF) logic. Specifically, the approach allows us to account for independence from propositional connectives in the same spirit as IF logic accounts for independence from quantifiers. Japaridze's treatment of IF logic, however, was purely semantical, and no deductive system was proposed. The present paper syntactically constructs a cirquent calculus system with clustering and ranking, sound and complete w.r.t. the propositional fragment of cirquent-based semantics. Such a system captures the propositional version of what is called extended IF logic, thus being an axiomatization of a nontrivial fragment of that logic.     

Does Science Influence the Logic we Ought to Use: A Reflection on the Quantum Logic Controversy

In this article I argue that there is a sense in which logic is empirical, and hence open to influence from science. One of the roles of logic is the modelling and extending of natural language reasoning. It does so by providing a formal system which succeeds in modelling the structure of a paradigmatic set of our natural language inferences and which then permits us to extend this structure to novel cases with relative ease. In choosing the best system of those that succeed in this, we seek certain virtues of such structures such as simplicity and naturalness (which will be explained). Science can influence logic by bringing us, as in the case of quantum mechanics, to make natural language inferences about new kinds of systems and thereby extend the set of paradigmatic cases that our formal logic ought to model as simply and naturally as possible. This can alter which structures ought to be used to provide semantics for such models. I show why such a revolution could have led us to reject one logic for another through explaining why complex claims about quantum mechanical systems failed to lead us to reject classical logic for quantum logic.     

A Lewisian Semantics for S2

This paper sets out a semantics for C.I. Lewis's logic S2 based on the ontology of his 1923 paper ‘Facts, Systems, and the Unity of the World’. In that article, worlds are taken to be maximal consistent systems. A system, moreover, is a collection of facts that is closed under logical entailment and conjunction. In this paper, instead of defining systems in terms of logical entailment, I use certain ideas in Lewis's epistemology and philosophy of logic to define a class of models in which systems are taken to be primitive elements but bear certain relations to one another. I prove soundness and completeness for S2 over this class of models and argue that this semantics makes sense of at least a substantial fragment of Lewis's logical theory.     

Contingent dissociation between recognition and fragment completion: the method of triangulation

Two experiments conforming to the logic of the method of triangulation were conducted. Following the study of a list of words, the first of two successive tests (recognition) was identical for two groups of subjects, but the second one, in which the same word-fragment cues were presented to both groups, differed with respect to retrieval instructions. Subjects in one group engaged in cued recall of study-list words, whereas those in the second group completed the fragments with the first word that came to mind. Both experiments yielded the same result: The dependency between the first and second tests, indexed by Yule's Q statistic, was greater for recognition and cued recall than it was for recognition and fragment completion. These results speak to the controversial issue of the usefulness of contingency analyses of data from successive memory tests. The results are interpreted in a theoretical framework consisting of an integration of the idea of a hypothetical quasi-memory system with the transfer-appropriate procedural approach.     

Hintikka on the Foundations of Mathematics: IF Logic and Uniformity Concepts

The initial goal of the present paper is to reveal a mistake committed by Hintikka in a recent paper on the foundations of mathematics. His claim that independence-friendly logic (IFL) is the real logic of mathematics is supported in that article by an argument relying on uniformity concepts taken from real analysis. I show that the central point of his argument is a simple logical mistake. Second and more generally, I conclude, based on the previous remarks and on another standard fact of IFL, that first-order logic (FOL) can adequately express uniformity concepts in real analysis, whereas IFL (understood as a non-trivial extension of FOL) cannot. This not only radically contradicts Hintikka’s particular claim in that article, but also undermines his whole enterprise of founding mathematics on his logic system.     

逻辑图表的发展

长期以来,逻辑图表被逻辑学家们仅仅视为教学辅助工具;但逻辑图表从Euler图到Venn图再到存在图表和概念图表的发展表明,它已经超越出教学工具这一狭隘界限,从而获得了更为深刻,广泛的意义,实际上逻辑图表是与我们标准数理逻辑记法相区别的另一种有效的逻辑记法,而且有着更强的直观性和可应用性。     

Logic and Ontology

A brief review of the historicalrelation between logic and ontologyand of the opposition between the viewsof logic as language and logic as calculusis given. We argue that predication is morefundamental than membership and that differenttheories of predication are based on differenttheories of universals, the three most importantbeing nominalism, conceptualism, and realism.These theories can be formulated as formalontologies, each with its own logic, andcompared with one another in terms of theirrespective explanatory powers. After a briefsurvey of such a comparison, we argue that anextended form of conceptual realism provides themost coherent formal ontology and, as such, canbe used to defend the view of logic as language.     

Implicit attitude generalization occurs immediately; explicit attitude generalization takes time

People are able to explicitly resist using knowledge about one person to evaluate another person from the same group. After learning about positive and negative behaviors performed by one individual from each of two different groups, participants were introduced briefly to new individuals from the groups. Implicit evaluations of the original individuals readily generalized to the new individuals; explicitly, participants resisted such generalization. Days later, both implicit and explicit evaluations of the original individuals generalized to the new individuals. The results suggest that associative links (e.g., shared group membership) are sufficient for implicit attitude generalization, but deliberative logic (e.g., individual group members are not necessarily the same) can reduce explicit generalization by association. When knowledge distinguishing who did what is unavailable, such as after forgetting, associative knowledge provides the basis of explicit evaluation. We conclude that a simple association linking one individual to another can produce implicit attitude generalization immediately and explicit attitude generalization eventually.     

Dynamic Deontic Logic and its Paradoxes

In Meyer’s promising account [7] deontic logic is reduced to a dynamic logic. Meyer claims that with his account “we get rid of most (if not all) of the nasty paradoxes that have plagued traditional deontic logic.” But as was shown by van der Meyden in [4], Meyer’s logic also contains a paradoxical formula. In this paper we will show that another paradox can be proven, one which also effects Meyer’s “solution” to contrary to duty obligations and his logic in general. Presented by Hannes Leitgeb     

Translating Toulmin Diagrams: Theory Neutrality in Argument Representation

The Toulmin diagram layout is very familiar and widely used, particularly in the teaching of critical thinking skills. The conventional box-and-arrow diagram is equally familiar and widespread. Translation between the two throws up a number of interesting challenges. Some of these challenges (such as the relationship between Toulmin warrants and their counterparts in traditional diagrams) represent slightly different ways of looking at old and deep theoretical questions. Others (such as how to allow Toulmin diagrams to be recursive) are diagrammatic versions of questions that have already been addressed in artificial intelligence models of argument. But there are further questions (such as the relationships between refutations, rebuttals and undercutters, and the roles of multiple warrants) that are posed as a specific result of examining the diagram inter-translation problem. These three classes of problems are discussed. To the first class are addressed solutions based on engineering pragmatism; to the second class, are addressed solutions drawn from the appropriate literature; and to the third class, fuller exploration is offered justifying the approaches taken in developing solutions that offer both pragmatic utility and theoretical interest. Finally, these solutions are explored briefly in the context of the Araucaria system, showing the ways in which analysts can tackle arguments either using one diagrammatic style or another, or even a combination of the two.     

The use of diagrams in analogical problem solving

In four experiments, we examined the impact of perceptual properties on the effectiveness of diagrams in analogical problem solving, using variants of convergence diagrams as source analogues for the radiation problem. Static diagrams representing the initial problematic state (one large line directed at a target) and the final state for a convergence solution (multiple converging lines) were not accessed spontaneously but were often used successfully once a hint to consider the diagram had been provided. The inaccessibility of static diagrams was not alleviated by adding additional diagrams to represent intermediate states (Experiment 1), but spontaneous access was improved by augmenting static diagrams with a verbal statement of the convergence principle (Experiment 3). Spontaneous retrieval and noticing were increased markedly by animating displays representing converging forces and thereby encouraging encoding of the lines as indicating motion toward a target (Experiments 3 and 4). However, neither static nor animated diagrams were effective when the arrows were reversed to imply divergence rather than convergence (Experiment 2). The results indicate that when animation encourages the interpretation of a diagram as a helpful source analogue, it can greatly enhance analogical transfer.