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The Context Principle, Sentential Priority, and the Pursuit of Truth

Frege’s attempt to show that arithmetic is a part of logic requires definitions of the numbers. What criteria determine whether his definitions are acceptable? It seems to stand to reason that the definition of, say, the number one must pick out the object to which we have been referring, all along, when we use the numeral “1.” However, Frege does not assume that there are objects to which we have been referring all along when we use numerals. Definitions of the numbers must be, at least in part, stipulative. But how, then, can a science based on Frege’s definitions be our science of arithmetic? The key to answering this question is Frege’s sentential priority view. To be accurate to our arithmetic, Frege’s definition does not need to preserve reference; what they need to preserve is, rather, the truth of sentences expressing the “well known properties of the numbers.”