Quantity and Quantification
Daniel A. Bonevac · Noûs · 1985
Substitutional interpretations of the quantifiers have succeeded in dealing with finite and even countably infinite domains, but meet their nemeses in nondenumerable universes. Any theory requiring such a universe-set theory, for example-has thus remained immune to substitutional interpretation. The author argues that a slight revision of substitutional semantics enables it to handle domains of any cardinality. He alters the standard valuation recursion by incorporating extensions of substitutional models. He then uses variants of the Koening and Richard paradoxes to show that, though each parametric extension of the language may have only countably many parameters, there are uncountable many distinct parametric extensions. The author concludes by pointing out several other advantages of his new semantics.