Defining Quantifiers

November 16, 2024

Abstract: Suppose we have a language involving non-denoting singular terms. (The language of everyday mathematics provides one example. Terms like $\frac{n}{m}$ and $\underset{x\to \mathrm{\infty }}{lim}f(x)$ do not denote, for appropriate choices of $m$ and of $f$.) It is not too difficult to define inference rules for an appropriately free logic that incorporates non-denoting terms. If $t$ does not denote, then $\varphi (t)$ need not entail $\mathrm{\exists }x\varphi (x)$, and neither is $\varphi (t)$ entailed by $\mathrm{\forall }x\varphi (x)$. We now have a very good understanding of well-behaved cut-free sequent calculi for a variety of different free logics, appropriate for languages with non-denoting singular terms.

There is a tradition, in proof-theoretic semantics, of thinking of well-behaved inference rules for some concept as defining that concept. If the rules are well-enough behaved (whatever that notion of good behaviour might be—whether a notion of harmony, or conservative extension and unique definability, or something else—then we are tempted to take the concept that can be introduced by such rules to be defined by them, and hence to be apt for introduction to our vocabulary.

In this talk, I will show how, in a simple negative free logic, we are nonetheless able to—in some sense—define “outer” quantifiers $\mathrm{\Pi }$ and $\mathrm{\Sigma }$ where $\varphi (t)$ entails $\mathrm{\Sigma }x\varphi (x)$, and is entailed by $\mathrm{\Pi }x\varphi (x)$, since the rules for such quantifiers meet all the strictures for defining concepts in proof-theoretic semantics mentioned above. I will discuss the significance of this result for proof-theoretic semantics and our understanding of how inference rules might be used in definitions.

• This talk was a presentation at Topics in Free Logic, a workshop at the Munich Center for Mathematical Philosophy.

• Here are the slides for the talk.