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The present text discusses whether there is a tension between aphorisms 6.1-6.13 of the Tractatus and the Church-Turing theorem about the decidability of predicate logic. We attempt to establish the following points: (i) Aphorisms 6.1-6.13 are not consistent with the Church-Turing theorem. (ii) The logical symbolism of the Tractatus, built from the N-operator, can (and should) be interpreted as expressively complete with respect to first-order formulas. (iii) Wittgenstein's reasons for believing that Logic is decidable were purely philosophical and the undecidability result shows that there are aspects of his criticisms of Frege and Russell that become unjustified in light of these results. [ABSTRACT FROM AUTHOR]

From the mid-1600s to the beginning of the eighteenth century, there were two main circles of German scholars which focused extensively on diagrammatic reasoning and representation in logic. The first circle was formed around Erhard Weigel in Jena and consists primarily of Johann Christoph Sturm and Gottfried Wilhelm Leibniz; the second circle developed around Christian Weise in Zittau, with the support of his students, particularly Samuel Grosser and Johann Christian Lange. Each of these scholars developed an original form of using geometric diagrams in logic. In this paper, I will trace the historical notes of John Venn and other modern logicians back to the original works published in the Weigel and Weise circles and describe the development of using geometric figures for logical reasoning and representation in that period of time. [ABSTRACT FROM PUBLISHER]

This paper examines the contemporary philosophical and cognitive relevance of Charles Peirce's diagrammatic logic of existential graphs (EGs), the 'moving pictures of thought'. The first part brings to the fore some hitherto unknown details about the reception of EGs in the early 1900s that took place amidst the emergence of modern conceptions of symbolic logic. In the second part, philosophical aspects of EGs and their contributions to contemporary logical theory are pointed out, including the relationship between iconic logic and images, the problem of the meaning of logical constants, the cognitive economy of iconic logic, the failure of the Frege-Russell thesis, and the failure of the Language of Thought hypothesis. [ABSTRACT FROM AUTHOR]

In this article, the author studies some central concepts in Avicenna's and Tusī's modal logics as presented in Avicenna's Al-Isharat wa'l Tanbīhat (Pointers and Reminders) and in Tusī's commentary. In this work, Avicenna introduces some remarkable distinctions in order to interpret Aristotle's modal syllogistic in the Prior Analytics. The author outlines a new interpretation of absolute sentences as temporally indefinite sentences and argues on the basis of this that Avicenna seems to subscribe to the Principle of Plenitude. He also shows that he has no valid proof of the modal conversion rules and that he uses some rather ad hoc distinctions to show that Aristotle's modal syllogistic is correct. The author also notes some interesting differences between Avicenna's and Tusī's approaches to modal logic. [ABSTRACT FROM AUTHOR]

I attribute an 'intensional reading' of the second incompleteness theorem to its author, Kurt Godel. My argument builds partially on an analysis of intensional and extensional conceptions of meta-mathematics and partially on the context in which Godel drew two familiar inferences from his theorem. Those inferences, and in particular the way that they appear in Godel's writing, are so dubious on the extensional conception that one must doubt that Godel could have understood his theorem extensionally. However, on the intensional conception, the inferences are straightforward. For that reason I conclude that Godel had an intensional understanding of his theorem. Since this conclusion is in tension with the generally accepted view of Godel's understanding of mathematical truth, I explain how to reconcile that view with the intensional reading of the theorem that I attribute to Godel. The result is a more detailed account of Godel's conception of meta-mathematics than is currently available. [ABSTRACT FROM AUTHOR]

This article presents an historical and conceptual overview on different approaches to logical abstraction. Two main trends concerning abstraction in the history of logic are highlighted, starting from the logical notions of concept and function. This analysis strictly relates to the philosophical discussion on the nature of abstract objects. I develop this issue further with respect to the procedure of abstraction involved by (typed) λ-systems, focusing on the crucial change about meaning and predicability. In particular, the analysis of the nature of logical types in the context of Constructive Type Theory allows elucidation of the role of the previously introduced notions. Finally, the connection to the analysis of abstraction in computer science is drawn, and the methodological contribution provided by the notion of information is considered, showing its conceptual and technical relevance. Future research shall focus on the notion of information in distributed systems, analysing the paradigm of information hiding in dependent type theories. [ABSTRACT FROM AUTHOR]

John Venn and Charles L. Dodgson (Lewis Carroll) created systems of logic diagrams capable of representing classes (sets) and their relations in the form of propositions. Each is a proof method for syllogisms, and Carroll's is a sound and complete system. For a large number of sets, Carroll diagrams are easier to draw because of their self-similarity and algorithmic construction. This regularity makes it easier to locate and thereby to erase cells corresponding with classes destroyed by the premises of an argument, a particularly difficult task in Venn diagrams for more than four sets. Carroll diagrams can represent existential propositions easily, so they are capable of clearly representing more complex problems than Venn's system can. Finally, both Carroll and Venn diagrams are maximal, in the sense that no additional logic information like inclusive disjunctions is able to be represented by them. Carroll's logic diagrams and logic trees constitute his visual logic system. [ABSTRACT FROM AUTHOR]

Explores Charles Sanders Peirce's general insights about the logical status of diagrams and its limit. Peirce's belief that the same logical status can be given to diagrams as to symbolic systems; Peirce's Existential Graphs; Correction of important defects of Venn diagrams; Discrepancy between Peirce's and others' estimates of his contribution to logical diagrams.