In George Boolos (ed.), Method, Reason and Language: Essays in Honor of Hilary Putnam. Cambridge University Press. pp. 235-260 (1990)
AbstractDiscourse carries thin commitment to objects of a certain sort iff it says or implies that there are such objects. It carries a thick commitment to such objects iff an account of what determines truth-values for its sentences say or implies that there are such objects. This paper presents two model-theoretic semantics for mathematical discourse, one reflecting thick commitment to mathematical objects, the other reflecting only a thin commitment to them. According to the latter view, for example, the semantic role of number-words is not designation but rather the encoding of cardinality-quantifiers. I also present some reasons for preferring this view.
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