Section 5 presents some concluding remarks.Ī definition is generally understood as a statement or a set of statements that determine the meaning of an expression. Section 4.2 outlines an alternative, inferentialist account of the semantics of structural definitions. 4.1, we propose two model-theoretic explications that capture the informal accounts presented in the previous section. Section 4 contains a more systematic assessment of these two ways of thinking about structural definitions. 3.1) and as definitions of higher-order concepts (Sect.
3, we then turn to a more detailed survey of the two conceptions of structural definitions, namely as definitions of the meaning of primitive terms of a theory (Sect.
Section 2 presents a brief exposition of the logic of implicit and structural definitions. The central aim in the following will be to survey these two conceptions of structural definitions both in the history of mathematics and in selected philosophical debates. the concepts of a natural or real number system. More precisely, we propose that both in modern mathematics and in subsequent philosophical debates one can distinguish between two ways of understanding structural definitions, namely (1) as specifications of the meaning of the primitive terms of a theory and (2) as definitions of higher-order concepts of mathematical objects, e.g. In the context of the paper, we will label this sort of definition “structural definitions”.Ī central claim of this paper is that it is unclear how such definitions should be understood semantically. In particular, the focus here will be on a kind of definition used in formal axiomatics, which in general terms is usually understood as providing a definition of the primitives terms of a given axiomatic theory. We will address this issue by distinguishing between several types of definitions which are usually associated with the label “implicit definition”. The aim of this paper is to make more precise what implicit definitions are. Boghossian 1997).ĭespite the relevance of this notion, it is surprising that it is commonly used very informally in these philosophical debates. Finally, the notion features prominently in epistemology, in particular, in Boghossian’s work on epistemic analyticity (e.g. Footnote 2 Concerning the latter field, implicit definitions play a central role in neologicism and the discussion of abstraction principles as well as in mathematical structuralism.
As examples of the importance of this notion one can mention the debate on the meaning of theoretical terms in science, the contributions by Michael Friedman (and others) on the relativized a priori, as well as recent work in the philosophy of mathematics. Footnote 1 It is also ubiquitous in contemporary philosophical debates, ranging from the fields of philosophy of science and mathematics to philosophy of logic and language. The notion of implicit definition is a fundamental concept in modern mathematics, in particular in formal axiomatics.
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