“Proof Theory and Meaning: on second order logic,” pp 157–170 in *Logica 2007 Yearbook*, edited by Michal Pelis, Filosofia, 2008.

Second order quantification is puzzling. The second order quantifiers have natural and compelling inference rules, and they also have natural models. These do not match: the inference rules are sound for the models, but not complete, so either the proof rules are too weak or the models are too strong. Some, such as Quine, take this to be no real problem, since they take “second order logic” to be a misnomer. It is not logic but set theory in sheep’s clothing, so one would not expect to have a sound and complete axiomatisation of the theory.

I think that this judgement is incorrect, and in this paper I attempt to explain why. I show how on Nuel Belnap’s criterion for logicality, second order quantification can count as properly logic so-called, since the quantifiers are properly defined by their inference rules, and the addition of second order quantification to a basic language is conservative. With this notion of logicality in hand I then diagnose the incompleteness of the proof theory of second order logic in what seems to be a novel way.

Do you like this, or do you have a comment? Then please
*share* or *reply* on Twitter, or
email me.

← Proofnets for S5: sequents and circuits for modal logic | Writing Archive | Invention is the Mother of Necessity: modal logic, modal semantics and modal metaphysics →

I’m *Greg Restall*, and this is my personal website. I teach philosophy and logic as Professor of Philosophy at the University of Melbourne. ¶ From August 2021, I will be the Shelby Cullom Davis Professor of Philosophy at the University of St Andrews. ¶ Start at the home page of this site—a compendium of recent additions around here—and go from there to learn more about who I am and what I do. ¶ This is my personal site on the web. Nothing here is *in any way* endorsed by the University of Melbourne.

- School of Historical and Philosophical Studies, The University of Melbourne, Parkville 3010, Australia.
- greg@consequently.org
- keybase.io/consequently, to sign or encrypt a message to send to me privately.
- @consequently on Twitter.
- @consequently on Instagram.
- @consequently on GitHub.

To receive updates from this site, you can subscribe to the RSS feed of all updates to the site in an RSS feed reader, or follow me on Twitter at @consequently, where I’ll update you if anything is posted.