Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to (...) express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)

Charles S. Peirce’s pragmatist theory of logic teaches us to take the context of utterances as an indispensable logical notion without which there is no meaning. This is not a spat against compositionality per se , since it is possible to posit extra arguments to the meaning function that composes complex meaning. However, that method would be inappropriate for a realistic notion of the meaning of assertions. To accomplish a realistic notion of meaning (as opposed e.g. to algebraic meaning), Sperber (...) and Wilson’s Relevance Theory (RT) may be applied in the spirit of Peirce’s Pragmatic Maxim (PM): the weighing of information depends on (i) the practical consequences of accommodating the chosen piece of information introduced in communication, and (ii) what will ensue in actually using that piece in further cycles of discourse. Peirce’s unpublished papers suggest a relevance-like approach to meaning. Contextual features influenced his logic of Existential Graphs (EG). Arguments are presented pro and con the view in which EGs endorse non-compositionality of meaning. (shrink)

The theory of existential graphs, which Peirce ultimately divided into four quadrants , is a rich method of analysis in the philosophy of logic. Its $$\upbeta $$ β -part boasts a diagrammatic theory of quantification, which by 1902 Peirce had used in the logical analysis of natural-language expressions such as complex donkey-type anaphora, quantificational patterns describing new mathematical concepts, and cognitive information processing. In the $$\upbeta $$ β -quadrant, he came close to inventing independence-friendly logic, the idea of which he (...) found indispensable in fulfilling the tasks –. (shrink)

Many do not doubt that Peirce's Existential Graphs are diagrammatic, as opposed to symbolic. However, when we are pressured to draw a distinction between the two different forms of representation, we find ourselves at a loss and our intuition quite vague. In this paper, I locate fundamental differences between two logically equivalent systems, Peirce's Alpha system and propositional languages. Suppose we have only two sentential connectives, ¬ and ^. In spite of its truth-functional completeness, we don't want to use this (...) language for the translation of English sentences or as a deductive calculus. We would adopt this language only when we intend to develop logical theories. That is, it is convenient to have fewer connectives for a meta-theory, but not for practical use. So, there seems to be a trade-off between a language with fewer connectives and a language with more connectives. This view has been accepted without question. In this paper, I will argue that this trade-off is limited to linear symbolic systems and that we could have a diagrammatic system with fewer operations but no need to suffer from problems like those of a sentential language with fewer connectives. How is that possible? A comparison between Peirce's Alpha system and a propositional language is presented to answer this question. The case study identifies the following unique property of a non-linear diagrammatic system as a main source of the discrepancy between two different types of representation: One and the same diagram can be read off in more than one way by carving it up in many ways, but without ambiguity. (shrink)

1 INTRODUCTION Above the other titles he might justly have claimed, Charles S. Peirce prized the title 'logician'. He expressed in several places his ...

In 1898 C. S. Peirce declares that the medieval doctrine of consequences had been the starting point of his logical investigations in the 1860s. This paper shows that Peirce studied the scholastic theory of consequentiae as early as 1866–67, that he adopted the scholastics’ terminology, and that that theory constituted a source of logical doctrine that sustained Peirce for a lifetime of creative and original work.

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. (shrink)

BEFORE Frege’s Begriffsschrift, propositional logic was submerged in the often murky theory of the "hypothetical syllogism." With the exception of the Stoa, a handful of astute mediaeval logicians, Leibniz, and Bolzano, one might well obtain the impression from studying the history of logic that Frege created his theory ex nihilo—which is substantially true, since Frege was apparently little influenced by previous work. One might also obtain the impression, especially by reading Frege himself, that very little was being done on propositional (...) logic in the 19th century. This claim would be false. As one might expect from a philosopher who is sometimes grouped with Frege and De Morgan as one of the greatest nineteenth century logicians, Charles S. Peirce dealt extensively with propositional logic, or as he would have called it, with the theory of the hypothetical syllogism. (shrink)

The Tractatus contains twodifferent proofs of the Grundgedanke, or thenonreferentiality of logical constants. In thispaper, I explicate the first proof in TLP 5.4s andreconstruct the less explicitly stated second proof. My explication of the first proof shows it to beelegant but based on an invalid inference. In myreconstruction of the second proof, the main argumentis that the sign of a logical constant does not denotebecause it possesses the punctuation-mark-nature. Andit possesses the punctuation-mark-nature because,given the analyticity thesis in TLP 5, one (...) canestablish for everyday language an adequate symbolismwith N as the sole fundamental operation such that itssign is a bar indicating merely the order and scope ofits application. (shrink)