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Induction and Comparison

Frege proved an important result, concerning the relation of arithmetic to second-order logic, that bears on several issues in linguistics. Frege’s Theorem illustrates the logic of relations like PRECEDES(x, y) and TALLER(x, y), while raising doubts about the idea that we understand sentences like ‘Carl is taller than Al’ in terms of abstracta like heights and numbers. Abstract paraphrase can be useful—as when we say that Carl’s height exceeds Al’s—without reflecting semantic structure. Related points apply to causal relations, and even grammatical relations like DOMINATES(x, y). Perhaps surprisingly, Frege provides the resources needed to recursively characterize labelled expressions without characterizing them as sets. His theorem may also bear on questions about the meaning and acquisition of number words. Introduction Suppose that Carl is just barely taller than Bob, who is just barely taller than Al. It follows that Carl is taller than Al, but not that Carl is just barely taller than Al. Yet if Carl is much heavier than Bob, who is heavier than Al, then Carl is much heavier than Al. And if Al is Bob’s father, while Bob is Carl’s father, then Al is Carl’s grandfather. Though if Al is Bob’s brother, while Bob is Carl’s, then Al and Carl are brothers. Speakers of English recognize such inferential relations. This raises questions about (i) how the relevant sentences are understood, and (ii) the kinds of inference that are germane. Perhaps ‘Carl is taller than Al’ means roughly that Carl’s height exceeds Al’s height, and speakers make relevant inferences about heights and exceeding. But 1 I’ll urge a different perspective, based on Frege’s (1879, 1884, 1892, 1893) reasons for inventing the modern logic—presupposed by most semanticists—that lets us deal with relations like SUCCESSOROF(x, y) and GREATERTHAN(x, y). This requires a