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By Meir Buzaglo

Scientists and mathematicians often describe the improvement in their box as a approach that incorporates growth of techniques. Logicians commonly deny the potential for conceptual enlargement and the coherence of this description. Meir Buzaglo's cutting edge examine proposes a fashion of increasing common sense to incorporate the stretching of techniques, whereas editing the foundations which it sounds as if block this risk. He deals stimulating discussions of the belief of conceptual enlargement as a normative strategy, and of the relation of the conceptual growth to fact, which means, reference, ontology, and paradox, and analyzes the perspectives of Kant, Wittgenstein, Godel, and others, paying in particular shut realization to Frege. His ebook can be of curiosity to a variety of readers, from philosophers (of good judgment, arithmetic, language, and technological know-how) to logicians, mathematicians, linguists, and cognitive scientists.

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Logic of Concept Expansion

Scientists and mathematicians usually describe the advance in their box as a procedure that incorporates enlargement of options. Logicians normally deny the potential for conceptual growth and the coherence of this description. Meir Buzaglo's leading edge learn proposes a manner of increasing good judgment to incorporate the stretching of options, whereas enhancing the rules which it appears block this probability.

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Here Frege is clearly trying to describe the relation between the old and the new concept as one of subordination. This is, however, an incomplete description of what is involved in non-arbitrary expansions of concepts. ” When we speak about the expansion of a concept we are talking about two concepts such that one of them grew out of the other. The attempt to describe this as subordination is far from capturing the organic-like connection between the concepts. Indeed, in order to capture this connection we must realize that a concept which was defined on some objects is now defined on a broader range of objects, and – what is more important – that the expansion itself is not arbitrary.

This shows that A does not force an internal expansion.  The Logic of Concept Expansion Example Let us take D to be the set of all equations that hold for the rational numbers. We assume that the equations in D are written in a language L = (, +, −, ×,  x ) (interpreted as plus, minus, times, and the power function). Take the expansion of the power function from the rational numbers to the real numbers. This is strongly forced with respect to D.

The functions in N and K agree on their common part. Examples It is easy to check that the examples of expanding the power function to the zero and the negative numbers, and even to the rational and the real numbers, are captured by the definitions above. We only have to supply the appropriate laws S and models M, N. For example: M = The partial model of the field of real numbers where the power function  x is defined for natural numbers. The language L(X ) includes {, , +, −, x, :,  x }. N = The internal expansion where the power function is defined on all integers.

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