An Inferentialist Account of (Implicit) Definition Dan Kapan

Abstract: PhDLX 2018: PhDs in Logic X April 13, 2018

This paper explores the intersection of two lines of thought. Each line of thought is potentially independently explicable, but the conclusion is more powerful by considering where they come together. The first line of thought concerns a philosophical account of definition. Definitions, the thought goes, are of particular philosophical significance not only in themselves but for the fact that all inquiry begins by introducing the terms in which that inquiry will be couched. Sometimes such definitions are given more-or-less explicitly, so Kant begins the Critique of Pure Reason by defining analytic and synthetic judgments as:

“In all judgments in which the relation of a subject to the predicate is thought (I take into consideration aﬃrmative judgments only, the subsequent applica- tion to negative judgments being easily made), this relation is possible in two diﬀerent ways. Either the predicate B belongs to the subject A; or B lies outside the concept A, although it does indeed stand in connection with it. In the one case I entitle the judgment analytic, in the other synthetic” (B10– 11; underlining and bold are my own).

Of course no one takes Kant to have actually given us a definite meaning yet with these words (even if all these concepts are clear). We only find out what he means by analytic and synthetic by reading further and understanding what this distinction allow him to conclude about reason, understanding, and which sorts of judgments he takes to be analytic and synthetic. Even in mathematical works, we often come to understand the meaning of a term by seeing how they are used, reasoned about, and what they let us prove. In other words, what the word means is more accurately specified by how we come to use it to reason. One contrast I am here invoking is a distinction between “explicit” definitions—in which a word or linguistic item generally is defined by saying explicitly what it means, i.e. specifying something like an equivalent expression that can take its place (given the proviso that this might involve some rephrasing)—and “implicit” definitions—in which a word or linguistic item generally acquires its meaning by saying something about how it is used or how it figures in true assertion. My contention is that the latter is the more ubiquitous and important notion, and that the principal way such items acquire meaning through implicit definition is via their use in good inference. I therefore want to defend an inferentialist account of implicit definition and my hope is to explain how this can help us understand cases of genuine implicit definition. By “genuine” implicit definition, I mean implicit definition that need not be reducible to an explicit definition conferring a meaning. One problem with this goal, however, is that there are well-known results in the logic of definitions that render it impossible. I have in mind, of course, Beth’s Theorem, which proves that something can be implicitly defined if and only if it can be explicitly defined. This leads me to the second line of thought of the paper. In response to certain paradoxes in logic—often the result of definition—philosophers and logicians have sought to limit the scope of what counts as good definition. For example the

1 Abstract: PhDLX 2018: PhDs in Logic X April 13, 2018

sentence which asserts of itself that absurdity follows—i.e. the Curry Sentence c =df. c →⊥— leads to a paradox. Granting that A → A is always valid, then we may reason that c → c and thus (since c =df. c →⊥) that c → (c →⊥). This of course reduces to c →⊥. This is equivalent to c, so we’ve just deduced that c. But c together with c →⊥ gives us ⊥ via modus ponens, but surely this is bad. How can merely saying what a sentence means gets us a substantive truth? Worse the “truth” here is absurdity, so it in fact gets us all the truths (and falsehoods). Surely a sentence like the Curry sentence is, if not meaningless, poorly defined. I agree that paradoxes like the above give us reason to draw new distinctions, but I don’t think such distinctions should amount to ruling out a large class of potentially good definitions. Paradoxes gives us reasons to draw new distinctions, but ideally not to simply deny the phenomena. Part of what Beth’s theorem gives us (more accurate, one half of it, which was already attributable to a result by Padoa) is that—given some presuppositions about the underlying logic—properly formed explicit definitions are guaranteed to be safe. The implication is that anything outside of those shouldn’t be thought of as just risky, but in fact as meaningless or confused. I examine the presuppositions of the underlying logic in order to draw new distinctions within what the standard account calls definition. In particular, I want to examine the space between properly formed explicit definition and good implicit definition to see what structure we might find there.1 These two lines of thoughts intersect in the thought that there might be good implicit definitions that don’t yet amount to properly-formed explicit definition. The second line of thought provides the resources for making some distinctions between these two sorts of definition, while the first gives us a philosophically informed understanding of meaning to make such distinctions plausible. I proceed as follows in the paper: I begin by rehearsing the standard account of definition and explaining how adequacy of definition is envisaged.2 Throughout the rest of the paper I do not attempt to challenge these criteria. I accept the orthodoxy, but try to draw some internal distinctions. Following this I rehearse some problems standardly associated with accounts of implicit definition.3 I also explain how the standard account understands implicit definition. Next, I turn to my own account of definition. I begin by explaining what an inferentialist theory of meaning is, and what it means to understand meaning in terms of role in inference.4 Next, I explain how I conceive of implicit definition. In contrast to the standard account of implicit definition, where it is the taking to be true of a sentence or theory containing the defined term, on my account it is the taking to be good of an implication or inferential relation in which such new vocabulary appears. Following this I introduce several other notions of definition and the connections between them. Under classical logic they all collapse into one another (as is well known from Beth’s Theorem), but I hope my analysis there makes plausible a layer between them.

1Work by Nuel Belnap and Anil Gupta shares a similar ambition here: i.e. to elucidate a diﬀerent notion of “good deﬁnition” (Gupta and Belnap, 1993). 2My main sources for examining the “standard account” may be found in: (Belnap, 1993; Gupta, 2015; Suppes, 1957). 3See e.g. (Hale and Wright, 2000, 2001; Horwich, 1997, 1998). 4I here rely heavily on e.g. (Brandom, 2009; Peregrin, 2014).

2 Abstract: PhDLX 2018: PhDs in Logic X April 13, 2018

References

Belnap, N. (1993). On rigorous deﬁnitions. Philosophical studies, 72(2):115–146.

Beth, E. (1953). On padoa’s method in the theory of deﬁnition. (15):330–339.

Brandom, R. (2009). Articulating reasons. Harvard University Press.

Gupta, A. (2015). Deﬁnitions. The Stanford Encyclopedia of Philosophy (Summer 2015).

Gupta, A. and Belnap, N. D. (1993). The revision theory of truth. MIT Press.

Hale, B. and Wright, C. (2000). Implicit deﬁnition and the a priori. In Boghossian, P. and Peacocke, C., editors, New essays on the a priori, pages 286–319. Oxford: OUP.

Hale, B. and Wright, C. (2001). The Reason’s Proper Study: Essays Toward a Neo-Fregean Philosophy of Mathematics. Clarendon Press.

Horwich, P. (1997). Implicit deﬁnition, analytic truth, and apriori knowledge. Nous, 31(4):423–440.

Horwich, P. (1998). Meaning. Oxford University Press.

Kant, I. (1999). Critique of pure reason. Cambridge University Press.

Padoa, A. (1967). Logical introduction to any deductive theory. In van Heijenoort, J., editor, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, pages 118–123. Harvard University Press.

Peregrin, J. (2014). Inferentialism: Why Rules Matter. Palgrave Macmillan.

Suppes, P. (1957). Introduction to logic. The University series in undergraduate mathematics. D. Van Nostrand Company, Inc.