As a historian of logic, I am frequently puzzled by the things which people have to say about the relationship between mathematical logic and some other kind of logic which is variously described as ‘intentional’ and ‘traditional.’ Part of my puzzlement arises from my failure to understand precisely what kind of system is being offered under the guise of intentional logic. I have always taken it that logic is concerned with valid inferences, with showing us how we may legitimately derive (...) a conclusion from a set of premisses; yet the validation of inferences seems to be the least of the concerns of the intentional logician. He says that it can be done, but he does not bother to show us how. My purpose in this paper is to list some of the sources of my puzzlement in the hope that an exponent of intentional logic will show me how they can be resolved, and how their resolution will contribute to the building of a system (however informal) in which different types of argument can be validated. (shrink)

The paper provides an alternative interpretation of ‘pair points’, discussed in Beall et al., "On the ternary relation and conditionality", J. of Philosophical Logic 41(3), 595-612. Pair points are seen as points viewed from two different ‘perspectives’ and the latter are explicated in terms of two independent valuations. The interpretation is developed into a semantics using pairs of Kripke models (‘pair models’). It is demonstrated that, if certain conditions are fulfilled, pair models are validity-preserving copies of positive substructural models. This (...) yields a general soundness and completeness result for a variety of (positive) substructural logics with respect to multimodal Kripke frames with binary accessibility relations. In addition, an epistemic interpretation of pair models is provided. (shrink)

The operators of first-order logic, including negation, operate on whole formulae. This makes it unsuitable as a tool for the formal analysis of reasoning with non-sentential forms of negation such as predicate term negation. We extend its language with negation operators whose scope is more narrow than an atomic formula. Exploiting the usefulness of subatomic proof-theoretic considerations for the study of subatomic inferential structure, we define intuitionistic subatomic natural deduction systems which have several subatomic operators and an additional operator for (...) formula negation at their disposal. We establish normalization and subexpression property results for the systems. The normalization results allow us to formulate a proof-theoretic semantics for formulae composed of the subatomic operators. We illustrate the systems with applications to reasoning with combinations of sentential negation, predicate term negation, subject term negation, and antonymy. (shrink)

The essay revisits the puzzle of the ‘passage’ of time in relation to EPR-type measurements and asks what philosophical consequences can be drawn from them. Some argue that the lack of invariance of temporal order in the measurement of a space-like related EPR pair, under relativistic motion, casts serious doubts on the ‘reality’ of the lapse of time. Others argue thatcertain features of quantum mechanics establisha tensed theory of time – understood here as Possibilism or the growing block universe. The (...) paper analyzes the employment of frame-invariant entropic clocks in a relativistic setting and argues that tenselessness does not imply timelessness. But this conclusion does not support a tensed theory of time, which requires a preferred foliation. It is argued that the only reliable inference from the EPR example and the use of entropic clocks is an inference not just to a Leibnizian order of the succession of events but a frame-invariant order according to some selected clocks. (shrink)

In this essay, I discuss some observations by Peirce which suggest he had some idea of the substantive metalogical differences between logics which permit both quantifiers and relations, and those which do not. Peirce thus seems to have had arguments?which even De Morgan and Frege lacked?that show the superior expressiveness of relational logics.

In the predicate calculus, variables provide a flexible indexing service which selects the actual arguments to a predicate letter from among possible arguments that precede the predicate letter (in the parse of the formula). In the process of selection, the possible arguments can be permuted, repeated (used more than once), and skipped. If this service is withheld, so that arguments must be the immediately preceding ones, taken in the order in which they occur, the formula is said to be fluted. (...) Quine showed that if a fluted formula contains only homogeneous conjunction (conjoins only subformulas of equal arity), then the satisfiability of the formula is decidable. It remained an open question whether the satisfiability of a fluted formula without this restriction is decidable. This paper answers that question. (shrink)

Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th (...) century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$. (shrink)

Summary This paper will attempt to integrate (1) some new reflections on the implications for ontology of Monistic interpretations of formulae in quantification theory, with (2) a review of earlier material that I have published on such implications, and with (3) a sketch of several points made by others which bear on related issues.

Few philosophers believe in the existence of so-called negative properties. Indeed, many find it mind-boggling just to imagine such entities. By contrast, I believe not only that negative properties are quite conceivable, but also that there are good reasons for thinking that some such properties actually exist. In this paper, I would like to explicate a concept of negative properties which I think avoids the logical absurdities commonly believed to frustrate theories of negative existences. To do this, I shall deploy (...) a conceptual approach to ontology: I first evaluate the ontological commitments of ordinary negative statements, before I argue that, from the perspective of conceptual analysis, negative properties are no more suspicious than positive properties. In the second part of this paper, I probe the extent to which one can explain the distinction between positive and negative properties in terms of facts about logico-semantical entailment. (shrink)

Les travaux de Fred Sommers dans le domaine de l'ontologie ont été l'objet, jusqu'à présent, de beaucoup moins d'attention critique qu'ils ne sont aptes à en susciter. C'est au cours des années soixante que Sommers a élaboré sa théorie et qu'il l'a rendue publique par le moyen d'une série d'articles et de conférences. Bien que ses travaux aient alors fait l'objet de quelques critiques ou commentaires, ils ont, depuis, généralement été passés sous silence. Récemment, la revue Monist a fait paraître (...) un numéro spécial entièrement consacré au thème des catégories. Considérant le thème, l'importance et la qualité des articles regroupés dans ce numéro, il est surprenant de constater que les idées hautement originales et suggestives de Sommers ne soient mentionnées par aucun des auteurs. Evidemment, une des raisons pour lesquelles les idées de Sommers suscitent présentement si peu d'intérêt est le fait que, depuis la fin des années soixante, il a lui-même laissé de côté l'ontologie pour se tourner vers la logique—un domaine où il a fait et fait toujours couler beaucoup d'encre. (shrink)

Fred Sommers passed away in October of 2014 in his 92nd year. Having begun his teaching at Columbia University, he eventually became the Harry A. Wolfson Chair in Philosophy at Brandeis University, where he taught from 1963 to 1993. During his long and productive career, Sommers authored or co-authored over 50 books, articles, reviews, etc., presenting his ideas on numerous occasions throughout North America and Europe. His work was characterized by a commitment to the preservation and application of historical insights (...) and to the value of a well-articulated, coherent logical system. He was recognized for his independence and refusal to accept any view on the basis of authority alone. This made him a formidable critic but accounted in part for his many innovative and original ideas. In spite of his general contrariness in logic, Sommers earned the respect of the majority of his contemporaries, including Russell, Quine, van Benthem, Hacking, Suppes, and Strawson. In 2005, he was the subjec... (shrink)

Traditionally one main emphasis of the quantum mechanical measurement theory is on the question how the pure state of the compound system 'measured system + measuring apparatus' is transformed into the 'mixture' of all possible results of that measurement, weighted with their probability: the so-called “disappearance of the interference terms”. It is argued in this note that in reality there is no such transformation, so that there is no need to account for such a transformation theoretically. _German_ Gewöhnlich liegt ein (...) Hauptgewicht der quantenmechanischen Messtheorie auf der Frage des Übergangs vom,reinen Fall' des Gesamtsystems,gemessenes System + Messgerät' in das,Gemenge' aus allen möglichen Messergebnissen, gewichtet mit ihren Wahrscheinlichkeiten: Das sog.,,Verschwinden der Interferenzterme“. In dieser Notiz wird die Meinung vertreten, dass es einen solchen Übergang faktisch nicht gibt, so dass es nicht notwendig ist, eine theoretische Erklärung für den Übergang zu geben. (shrink)

I use van Heijenoort’s published writings and manuscript materials to provide a comprehensive overview of his conception of modern logic as a first-order functional calculus and of the historical developments which led to this conception of mathematical logic, its defining characteristics, and in particular to provide an integral account, from his most important publications as well as his unpublished notes and scattered shorter historico-philosophical articles, of how and why the mathematical logic, whose he traced to Frege and the culmination of (...) its formative period in the incompleteness results of Gödel, became modern logic, as distinct from the traditional logic of Aristotle, and why and how the logistic tradition that led from Frege through Russell, rather than the algebraic tradition that led from De Morgan and Boole through Peirce and Schröder, came, in his view, to define modern logic. (shrink)

Peter Ludlow shows how word meanings are much more dynamic than we might have supposed, and explores how they are modulated even during everyday conversation. The resulting view is radical, and has far-reaching consequences for our political and legal discourse, and for enduring puzzles in the foundations of semantics, epistemology, and logic.

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it (...) is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.(This work is completed in 2017 -5- 2, it is in vixra in 2017-5-14, presented in Unilog 2018, Vichy). (shrink)

Aristotle's General Definition of the Syllogism may be taken as consisting of two parts: the Inferential Conditions and the Final Clause. Although this distinction is well known, traditional interpretations neglect the Final Clause and its influence on syllogistic. Instead, the aforementioned tradition focuses on the Inferential Conditions only. We intend to show that this neglect has severe consequences not just on syllogistic but on the whole exegesis of Aristotle's Prior Analytics I. Due to these consequences, our objective is to analyse (...) the General Definition's Final Clause and its consequences on syllogistic. We propose a reading of the Final Clause as an additional criterion for distinguishing some arguments as properly syllogistic ones and as a main theme which connects all parts of the Prior Analytics I into one coherent piece of work. (shrink)