proof theory and meaning: on second order logic

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 [7, page 66–68], 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. The idea that inference rules can truly define connectives is compelling. Once we commit to inference rules such as these: p q [∧I] p∧ q p∧ q [∧E1] p p∧ q [∧E2] q the sense of ‘∧’ is specified. We now understand logicians’ conjunction. Once we learn the inference rules, we learn how to use the connectives in deductive reasoning. One does not need to be a disciple of Wittgenstein to think that there is a connection between meaning and use, and in inference rules like these the connection with use is present to hand. The [∧I] rule tells us how to get a conjunction, and the [∧E] rules tell us what we can do with a conjunction when we have it. Inference rules speak to the matter of the use of the concept of conjunction without positing some ‘semantic value’ to correlate with the concept. ∗This paper is a draft, and comments from readers are very welcome. ¶ Thanks to Allen Hazen, Lloyd Humberstone, Penelope Maddy, Albert Visser, Alasdair Urquhart, Heinrich Wansing and audiences at the University of Melbourne Logic Group, Logica 2007, Logic Colloquium 2007 for discussion on these topics. ¶ This research is supported by the Australian Research Council, through grant dp0343388, and Arvo Pärt’s Te Deum.