Slingshots and boomerangs - response to article by Graham Oppy in this issue, p. 121, on Godel's Slingshot

Mind, Jan, 1997 by Stephen Neale, Josh Dever

A "slingshot" proof suggested by Kurt Godel (1944) has been recast by Stephen Neale (1995) as a deductive argument showing that no non-truth-functional sentence connective can permit the combined use, within its scope, of two truth-functionally valid inference principles involving definite descriptions. According to Neale, this result provides indirect support for Russell's Theory of Descriptions and has broader philosophical repercussions because descriptions occur in non-truth-functional constructions used to motivate talk about (e.g.) necessity, time, probability, causation, obligation, facts, states of affairs, and propositions.

We develop Neale's claims and rebut Graham Oppy's (1997) criticism of Neale. In particular, we (i) work out the details of several formal, philosophical, and historical points raised by Neale, Oppy, and Quine, (ii) explore the consequences of Godel's Slingshot for specific theories of facts, (iii) demonstrate the integrity of Godel's Slingshot and the claims Neale bases on it, and the falsity of all of Oppy's main claims.

1. The generalized, description-neutral, operator version of Godel's slingshot

We begin by restating Neale's negative conclusions. The original slingshot arguments of Church, Davidson, and Quine do not establish the collapses their authors sought--the collapse of facts into one Great Fact, the collapse of modal distinctions, the collapse of non-truth-functional sentence connectives into the class of truth-functional connectives. (i) If the complex descriptions (or class abstracts) being substituted at key points in slingshots are treated in accordance with Russell's (independently motivated) Theory of Descriptions, then (a) statements containing descriptions are not identity statements, (b) descriptions are not singular terms, and (c) the Principle of Substitutivity for Singular Terms (PSST) has no application. (ii) If the descriptions are treated as singular terms, implausible semantic properties must be attributed to them in order to obtain the logical equivalences necessary to get the slingshots loaded. In short, the simultaneous reliance on logical equivalences involving descriptions and a treatment of descriptions as singular terms renders these slingshots (as originally stated) incapable of producing collapses.

The situation is similar, Neale argued, if one attempts to obtain a collapse with a slingshot suggested by Godel (1944) that uses not logical equivalence but a formally tighter notion--Godelian equivalence--holding between sentences like "Fa" and "a = ([Iota]x) (x = a.Fx)". (i) As above. (ii) If the relevant descriptions are treated as singular terms, implausible semantic properties must be attributed to them in order to obtain the relevant Godelian equivalences. In short, the simultaneous reliance on Godelian equivalences involving descriptions and a treatment of descriptions as singular terms renders Godel's slingshot (as originally stated) incapable of producing a collapse.

Despite these negative conclusions, Neale argued for the philosophical significance of a generalized, description-neutral, operator version of Godel's slingshot. Neale claimed--and we agree--that this proof delivers a definite constraint on the logic of non-truth-functional S-connectives (but not the eliminative constraint sought by Quine and Davidson). He also claimed--and we will soon prove--that this constraint has philosophical consequences (e.g., it can be used to narrow down the range of plausible theories of facts).

We want to clarify one inferential issue surrounding the operator version of Godel's slingshot. Neale's procedure (following Quine 1953b, 1960) was as follows: (i) take an arbitrary n-place sentence connective ??, an arbitrary truth-functional sentence [Phi], and a compound sentence ??(...[Phi]...); (ii) examine the logic of ?? by looking at what happens when the occurrence of the sentence [Phi] in ??(...[Phi]...) is replaced by a sentence [Phi]' obtained directly from [Phi] using inference principles that are valid in truth-functional contexts. Note: the inference principles are applied not to ??(...[Phi]...) itself but to the occurrence of [Phi] in such a sentence.

The purpose of proceeding in this abstract way is to avoid thinking about any particular sentence connective, to avoid being sidetracked by preconceived ideas about the logic of this or that connective, which may have something to do with, say, one's views about facts, states of affairs, causes, or necessity.

The relevant inference principles are the Principle of Substitution of Material Equivalents (PSME), the Principle of Substitution of Singular Terms (PSST), and three principles involving definite descriptions ([Iota]-SUBS, [Iota]-INTR, and [Iota]-ELIM). These principles can be stated without mentioning scope, extensionality, semantic innocence, or direct reference. Properties corresponding to particular inference rules can be ascribed to particular S-connectives. For example, truth-functional S-connectives are +PSME: