Logical inferentialists contend that the meanings of the logical constants are given by their inference rules. Not just any rules are acceptable, however: inferentialists should demand that inference rules must reflect reasoning in natural language. By this standard, I argue, the inferentialist treatment of quantification fails. In particular, the inference rules for the universal quantifier contain free variables, which find no answer in natural language. I consider the most plausible natural language correlate to free variables—the use of variables in the language of informal mathematics—and argue that it lends inferentialism no support.
"Inferentialism and Quantification." Notre Dame J. Formal Logic 58 (1) 107 - 113, 2017. https://doi.org/10.1215/00294527-3768059