proof theory and meaning: on the context of deducibility

I examine Belnap’s two criteria of existence and uniqueness for evaluating putative definitions of logical concepts in inference rules, by determining how they apply in four different examples: conjunction, the universal quantifier, the indefinite choice operator and the necessity in the modal logic s5. This illustrates the ways that definitions may be evaluated relative to a background theory of consequence, and the ways that different accounts of consequence provide us with different resources for making definitions. The idea is compelling to many of us: The concept of conjunction is given fully in its rules of inference. Everything we need to know about the behaviour of “and” in the logicians’ sense than is given by rules like these. p q [∧I] p∧ q p∧ q [∧E1] p p∧ q [∧E2] q The virtues of this view are simple: When we learn 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 be the referent (or extension or whatever) of the concept. ∗This paper is a draft, and comments from readers are very welcome. Please check the webpage http://consequently.org/writing/ptm-context for the latest version of the paper, to post comments and to read comments left by others. ¶ 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 Ethel’s self-titled album.