Stephen Neale in his book “Facing Facts??? presents a proof that he claims: (a) ‘demonstrates conclusively that any supposedly non-truth-functional operation must satisfy an exacting logical condition in order to avoid collapsing into a truth-function, and that (b) any theory of facts, states of affairs, situations, or propositions must satisfy a corresponding condition if such entities are not to collapse into a unity.’[1] This proof, based on remarks by Kurt Gödel, is also claimed by Neale to be more compelling in strength than both the argument as it was originally presented by Gödel, and proofs originating with

Alonzo Church and developed by W.V. Quine. One reason for this added strength is that the proof ‘requires no supplementation with a precise semantics for definite descriptions.’[2] Another reason is that Neale’s proof relies on the weaker notion of ‘Godelian equivalence’ as opposed to the logical equivalence used in the Quinean style proofs. This essay will provide exposition on Neale’s presentation of these older slingshots, examining what he takes and discards from each. The following points will then be argued: i) that the proof, contrary to Neale’s claim, does indeed require a supplementation with a precise semantics for definite descriptions – namely, a Russellian supplementation; ii) that because of this requirement, Neale’s proof is no more compelling than both the original Godelian argument or Quinean proof; iii) and that if we are unwilling to see philosophical significance in the older versions of the slingshot, we should see no greater significance in Neale’s new version. I will conclude by arguing that Neale’s claim a) stands in need of revision, and his claim b) is reduced to a triviality.

Neale arrives at the discovery of his proof by the combination of insights provided by Gödel and Quine. He takes what he thinks is best concerning the two approaches and discards what he feels limits the effectiveness of each. Thus he arrives at his proof which he believes avoids the limitations of the other two. One could say he borrows the style of the proof from Quine, and the substance from Gödel. Starting with the Gödel argument, I will examine what Neale borrows and what he discards from both, and then consider Neale’s deductive proof in its final form.

The Gödel argument in its original form is based upon three assumptions[3]:

G1 Where descriptions are singular terms: a=?x(x=a • Fx) and Fa stand for the same fact.

G2 Any sentence that stands for a fact can be put into predicate-argument form.

G3 The Principle of Composition: “the signification of a composite expression, containing constituents which themselves have a signification, depends only on the signification of these constituents (not on the manner in which this signification is expressed)”.

The argument is such that where the sentences: i) Fa ii) a?? b iii) Gb are all true and represent a particular fact: ƒ_I , ƒ_II , and ƒ_III respectively, by the repeated application of G1 and G3 it can be shown that ƒ_I = ƒ_II = ƒ_III,. and hence that all facts reduce to a single fact. The point is also made that the argument would not go through under a Russellian interpretation of descriptions because the equivalence asserted between a=?x(x=a • Fx) and Fa does not hold.[4]

Assumptions G1 and G3 have a semantic nature. G1 assumes a referential semantics of definite descriptions. The principle of composition is likewise semantic in nature insofar as it states that the semantic value of a composite expression depends only on the signification of these constituents. That this argument contains these semantic assumptions is one of the things Neale finds to be dissatisfactory. Neale will claim that his proof, based on this argument, will avoid the problem of having semantic assumptions and as such will achieve a greater generality.

Neale dissatisfaction with this argument goes further however. He says: “it is unclear whether [G1]-[G3] and the assumption that definite descriptions are singular terms form a consistent set, and so unclear whether the argument is valid only in a boring sense.???[5] The reason he thinks that the set of assumptions may not form a consistent set is because it can be shown by them that the fact represented by Fa is the same as the fact that a=a. He says: “unless one already holds that all true sentences stand for the same fact, it is implausible in the extreme to hold that Fa and a=a stand for the same fact???[6] This scruple will be important once we come to examine whether Neale’s version of the proof is any improvement over the Gödelian original.

The key features of this argument that Neale will borrow are the sentences i) Fa ii) a?? b iii) Gb, which will serve as premises in his proof, and the notion of what Neale comes to call Gödelian Equivalence, namely, the equivalence asserted between a=?x(x=a • Fx) and Fa. What were assumed to be true sentences in Gödel’s argument will be premises in a syntactic proof. Gödelian equivalence will be justified on a supposedly semantically neutral rule of inference called i-conv. I will leave the precise explication of this for later – first I want to turn to the Quinean proof and examine what Neale borrows and discards.

Neale brings to our attention the Quinean strategy of constructing slingshot arguments:

(i) take an arbitrary n-place connective

, an arbitrary extensional sentence ?, and an arbitrary compound sentence ( . . . ? . . .); and then (ii) examine the deductive consequences of replacing the occurrence of ? in ( . . . ? . . .) by another sentence ?? obtained directly from ? using inference principles known to be valid in truth-functional contexts. The inference principles are applied not to ( . . . ? . . .) itself but to an occurrence of the extensional sentence ? within such a sentence, i.e. a sentence ? within the scope of [7]

The aim of the Quinean strategy was to show:

…that if one could saddle ? with one more plausible inference principle, perhaps one could derive an outright inconsistency; and if one could make the additional inference principle sufficiently innocuous, perhaps the inconsistency would generalize to all non-extensional connectives, not just the modal ones.[8]

The point here is that within the context of a fully syntactic, deductive proof, one could examine the constraints on any arbitrary connective. This would include any connective that operated in the sense of: ‘the fact that’, and thereby giving relevance to Neale’s larger thesis concerning the limitations on theories of facts. We can see where this is going. Neale is going to borrow this strategy and apply it to the content we saw in the Godel argument. But before we examine this, we are going to continue to consider the results of the Quinean attempt as presented by Neale.

The Quinean strategy will lay out a number of rules of inference. As rules of inference, they are supposed to be purely syntactic[9] and hence, Neale will argue, won’t suffer many of the objections he raised against the Godel argument. These rules can be stated as follows – it is sufficient for the moment to simply cut and paste Neale’s exposition here. Independent exposition and critique of these rules will follow naturally upon the examination of the proofs by Quine and Neale.

The Principle of Substitutivity for Material Equivalents (psme ) can be put thus: · For present purposes this should be read as saying that if two sentences ? and ? have the same truth-value and ?(?) is a true[10]

The Principle of Substitutivity for Singular Terms (psst ) can be put thus:

· This just says that if two singular terms ? and ? have the same extension (i.e. if ?=? is a true identity statement) and ?(?) is a true sentence containing at least one occurrence of ?, then ?(?) is also true, where ?(?) is the result of replacing at least one occurrence of ? in ?(?) by ?.

the Principle of Substitutivity for Logical Equivalents (psle ): · For present purposes this should be read as saying that if two sentences ? and ? are logically equivalent and ?(?) is a true sentence containing at least one occurrence of ?, then ?(?) is also true, where ?(?) is the result of replacing at least one occurrence of ? in ?(?) by ?.

On the basis of *14.15 and *14.16, we can add a fourth inference rule (actually, a triple of rules) to our collection, ?-substitution : · ?-subs is a valid rule of inference when the description ?x? occurs in an extensional context. As shorthand for this, let us say that extensional contexts are +?-subs (as opposed to -?-subs ).

Neale will use the convention to the effect that where any of these rules of inference are valid in extensional contexts, a + sign will be affixed to the abbreviation of the name of the rule of inference. Where they are not, the – sign is affixed. For sake of continuity with Neale’s presentation, I will also adopt this convention.

Given these rules, we can now give an exposition of the Quinean proof as presented by Neale[11]:

1	[1] ? ? ?	premiss
2	[2] ?	premiss
2	[3] (1 = ?x((x=1 • ?) V (x=0 • ?)))	2, +psle
1,2	[4] ?x((x=1 • ?) V (x=0 • ?))
	=?x((x=1 • ?) V (x=0 • ?))	1, def. of ‘?x‘
1,2	[5] (1 = ?x((x=1 • ?) V (x=0 • ?)))	3, 4, +?-subs
1,2	[6] ?	5, +psle .

This proof demonstrates that for any arbitrary connective , where the connective is +psle and +?-subs it must also be +psme, which means that the connective has been demonstrated to be extensional.

We are encouraged to relate the significance of this proof back to the discussion concerning facts by:

interpreting as e.g. “the fact that ? = the fact that ( )” or “the statement that ? corresponds to the fact that ( )” or “the fact that ? caused it to be the case that ( )”, “it is a moral (/legal/constitutional) requirement that ( )”, or any other connective that, prima facie, someone might be tempted to view as +psle and +?-subs [12]

Neale sees advantages in this approach over the Godelian. Quine’s proof avoids the assumption G3 because the notion of a deductive proof does not rely on the principle of composition – a semantic notion. The premises are not assumed to be true, as was the case with the Godelian argument[13]. Hence no appeal to the semantic content of the premises is being assumed. Also, the argument is fully general with respect to any connective. This aspect of the Quinean strategy will allow Neale to widen the scope of his claims regarding the significance of his proof (i.e. to all intensional connectives)

However, there are problems with this proof that cause Neale to see it as dissatisfactory. What these dissatisfactions are can be more clearly seen when we consider the following incomplete proof, out of which the above proof evolved:

1 [1] ? premiss 2 [2] ? premiss 3 [3] ? premiss 3 [4] (a = ?x(x=a • ?)) 3, +psle 1 [5] ?x(x = a • ?)=?x(x=a • ?) 1, 2, def. of ?x 1,2,3 [6] (a = ?x(x=a • ?)) 4, 5, +?-subs[14] 1,2,3 [7] ? 6, +psle .

The difference between this proof and the one above is that the second demonstrates “that ? leads to ? when ? and ? are true. What we want to know is whether ? leads to ? when ? and ? are materially equivalent, not just when they are both true.???[15] This is what the final Quinean proof manages to avoid.[16]

Of the incomplete proof Neale says:

If it is to be taken seriously, it must be supplemented with a precise semantics for definite descriptions. First, lines [4] and [6] both contain definite descriptions, and the notion of logical equivalence is invoked in getting from line [3] to line [4] and from line [6] to line [7]: on some treatments of descriptions the logical equivalences obtain, but on others they do not. Secondly, line [5] is meant to be justified by the semantics of descriptions assuming ? and ? are both true. (It should also be noted that the treatment of descriptions assumed by the argument will determine whether or not psst should replace ?-subs as the rule of inference invoked in getting from lines [4] and [5] to line [6].)

To unpack what Neale is saying here: his general point is that the incomplete proof still contains too much semantic content for his liking. He says that the Russellian account of definite descriptions underpins the logical equivalence used to get from line 3 to line 4. Hence the argument is not semantically neutral in this respect. Secondly, the statements ? and ? need to be true in order for the proof to go through[17], which is another semantic assumption. The complete proof avoids the second problem by using a variant of the Kronecker function[18] but it still relies on the Russellian semantics for the contained descriptions[19].

But given these considerations, we can intimate what Neale will attempt with his Godelian reconstruction – the presentation of which is to come. His aim will be to construct a proof that has all the advantages of the Quinean strategy (its generality), while avoiding the semantic content required by the actual proof Quine provides. The specific means of doing this will be by avoiding the use of the logical equivalence which, Neale claims, necessitates the Russellian interpretation. He will do this by employing those elements of the Godelian argument as stated above, and by introducing two new rules of inference:

iota-introduction and iota-elimination : · Here ?(x) is an extensional formula containing at least one occurrence of the variable x, ?(x/?) is the result of replacing every occurrence of x in ?(x) by the (closed) singular term ?

These two rules are together given the singular name: ?-conv where in extensional contexts both rules of inference are valid.

Neale now has all the elements he requires to construct his proof which he labels: , of what he calls the Descriptive Constraint. I won’t reproduce the full proof here; however, its most significant features can be described thus. Its premises are the sentences: i) Fa ii) a?? b iii) Gb iv) (Fa). The only rules of inference it employs are +?-subs and +?-conv. From these premises and these rules of inference, it deduces the sentence (Gb). In the other sections of the proof the second premise is a = b instead of a?? b and the first and the third premises are replaced with their negations so as to obtain the full result, namely: that where any arbitrary connective is both +?-subs and +?-conv it must also be +psme i.e. extensional.

Again we are told to draw philosophical consequences from this proof

by interpreting (. . .) as, e.g., “the fact that ? = the fact that (. . .)”, “the statement that ? corresponds to the fact that (. . .)”, “the fact that ? caused it to be the case that (. . .)”, or any other connective that, at least prima facie, someone might be tempted to view as +?-SUBS and +?-CONV. The friend of facts needs a theory according to which these connectives are either -?-SUBS or -??CONV.[20]

The key point made by Neale in the above is that the connective: ‘The fact that…’ is likely to be +?-SUBS and +?-CONV.[21] If it weren’t likely, then the whole exercise would have been fruitless when considering the relevance of the proof for theories of facts.

The other significant claim that Neale makes for this proof is that:

in all crucial respects _{1 2} is semantically neutral; most importantly, and unlike Quine’s connective proof, _{1 2} requires no supplementation with a precise semantics for definite descriptions. This is because (a) it nowhere appeals to alleged logical equivalences involving descriptions—on some theories of descriptions Fa and a=?x) x=a •Fx) are logically equivalent, but no appeal is made to this interesting fact in the course of setting out _{1 2} —and (b) the apparent identities involving descriptions—e.g. at lines [8] and [9] of ₁ are obtained by appeal to ?-subs not by any specific assumption about the semantics of descriptions.[22]

I will dispute both a) and b) in this presentation, and thereby the claim that the proof is indeed neutral with respect to the semantics of definite descriptions. Once this is established we can go on to look if this proof gains anything over the Godelian and Quinean slingshots – and see that it doesn’t.

To begin with the second, that: the apparent identities involving descriptions—e.g. at lines [8] and [9] of ₁ are obtained by appeal to ?-subs not by any specific assumption about the semantics of descriptions. To give an example of the use of ?-subs from the proof – lines 4,5 and 8.

1	[4]a = ?x(x=a • Fx)	1, ?-conv
2	[5]a = ?x(x=a • x?? b)	2, ?-conv
1,2	[8] ?x (x = a • Fx) = ?x(x=a • x?? b)	4, 5, ?-subs

Neale claims that this move presupposes no assumptions concerning the semantics of definite descriptions. But when we turn to his exposition concerning the rule of inference we can see that the rule itself is tied directly to a Russellian semantics. Its validity comes from the following proof provided by Neale[23]:

1	[1]c =?xRx	premiss
2	[2]Sc	premiss
1	[3] x( y(Ry ? y=x) • c=x)	1, def. of ?x
4	[4] ( y(Ry ? y=?) • c=?)	assumption
4	[5]c =?	4, • -elim
2,4	[6]S?	2, 5, psst .
4	[7] y(Ry ? y=?)	4, • -elim
2,4	[8] ( y(Ry ? y=?) • S?)	6, 7, • -intr
2,4	[9] x( y(Ry ? y=x) • Sx)	8, eg
1,2	[10] x( y(Ry ? y=x) • Sx)	3, 4, 9, ei
1,2	[11]S?xRx	10, def. of ?x.

As we can see on line three, the definition of ?x employed is the Russellian definition. Neale confirms in his own exposition that this is a rule is to be used exclusively for Russellian descriptions. Given this, it is hard to understand Neale’s claim that makes no assumptions concerning the semantics of definite descriptions. Perhaps, it could be argued that where descriptions are being treated as singular terms, ?-subs could be replaced by psst. But even if we grant this, Neale still fails to attain neutrality in his proof with respect to definite descriptions.

The other respect in which Neale thought he had avoided semantic assumptions was in the use of ?-conv. Recall that the Quinean proof, Neale claimed, failed to remain neutral in respect of its semantic assumptions because it made the following move:

3	[3] ?	premiss
3	[4] (a = ?x(x=a • ?))	3, +psle

And Neale said that this logical equivalence holds on some theories of definite descriptions (namely, the Russellian), but not on others. If we compare these two lines of the Quinean proof to two taken from Neale’s:

10	[10] (Fa)	premiss
10	[11] (a = ?x(x=a • Fx))	10, +?-conv

we see something strikingly similar. In fact, it is the exact same logical equivalence, except now it is justified by ?-conv, not psle. What are we to think?

Neale never provides a derivation of ?-conv in the same manner as he does for ?-subs, however in his exposition of the rule of inference he does have this to say:

?-intr and ?-elim are valid rules of inference in extensional contexts. As shorthand for this, let us say that extensional contexts are +?-intr and +?-elim . (Any adequate theory of descriptions, it seems to me, must be compatible with this fact, as Russell’s theory is. On Russell’s account, the fact that extensional contexts are +?-intr and +?-elim follows immediately from the fact that extensional contexts are +psle.)[24]

This provides us with a hint as to how Neale has arrived at this rule of inference. It follows from the fact that (Fa) and (a = ?x(x=a • ?)) are logically equivalent and can be substituted in extensional contexts. However, Neale has already admitted that this logical equivalence (the exact same one used in the Quine proof) does not hold under some interpretations of definite descriptions. ?-conv, since it is derived from psle must suffer the same limitations, namely it’s reliance on a Russellian semantics for definite descriptions.

Given that so much turns on this point it’s worth seeing exactly why this is so. I’ll start with the reason why the logical equivalence in the incomplete proof presupposes a Russellian semantics for definite descriptions and then I’ll look at ?-conv and show why it still suffers from the same limitation.

The definition of logical equivalence, in standard first order theories, is simply that if is a tautology then A and B are said to be logically equivalent. Neale is right to say that the logical equivalence of Fa and (a = ?x(x=a • Fx)) is dependent on a Russellian semantics for definite descriptions. For suppose that we take a referential semantics, then there is an interpretation where the bi-conditional: (Fa (a = ?x(x=a • Fx))) is false. This is because ?x(x=a • Fx) simply refers to the object a and so the right hand side of the bi-conditional is equivalent to a=a and is always true. But Fa need not always be true. Hence on a referential semantics, the bi-conditional is not a tautology and by definition, not logically equivalent. Where a Russellian semantics is assumed the bi-conditional is a tautology. This is because the right hand side is no longer an identity statement, but a generalisation of the form: x( y((y=a • Fx) ? y=x) • x=a). And this says, roughly, that there is at least one thing which is identical to a and it is also F. This is true where Fa is true and false wherever Fa is false. Hence the bi-conditional is always true, and the two logically equivalent.

Now to look at ?-conv – Neale believes it to be neutral with respect to the semantics for definite descriptions, but we can show this is not so. With any valid argument, a tautology can be created by making a conditional with the conjunction of the premises as the antecedent and the conclusion as the consequent. So since the argument a = ?x(x=a • Fx) therefore Fa is valid (by means of the elimination form of ?-conv), the conditional ((a = ?x(x=a • Fx)) Fa) should be a tautology. But under a referential semantics for definite descriptions, it’s not. The antecedent reduces to a=a, which is true under any interpretation, but Fa need not be. Hence the conditional is not a tautology and the rule of inference invalid on a referential semantics. The rule of inference is totally valid where a Russellian semantics is used for reasons similar to those given above.

Just to make sure all the loose ends are tied up here – Neale uses the elimination form of ?-conv as the very final step on line 16 of his proof.[25] Hence the proof, in order to be valid, relies on a Russellian semantics just as much as the Quinean proof did.

Once we see that Neale hasn’t avoided assumptions concerning the semantics of definite descriptions, we can begin to examine to what extent his reconstruction of the Godelian argument is really an improvement on both that argument and the Quinean proof.

Beginning with the incomplete Quinean proof, we could re-construct it in the following fashion (note that it is exactly the same except for its justifications):

1	[1] ?	premiss
2	[2] ?	premiss
3	[3] ?	premiss
3	[4] (a = ?x(x=a • ?))	3, +?-conv
1	[5] ?x(x = a • ?)=?x(x=a • ?)	1, 2, def. of ?x
1,2,3	[6] (a = ?x(x=a • ?))	4, 5, +?-subs
1,2,3	[7] ?	6, +?-conv

Why? Because the particular logical equivalence used in the incomplete Quinean proof is exactly the same as that which is licensed by ?-conv and because it relies on a Russellian interpretation as much or as little as Neale’s. Of course, it still suffers from the problem that line five is justified by the presumption of the truth of the premises. When we move to final form of the Quinean proof that involves use of the Kronecker function we find we can no longer justify lines four and seven in the same way, as a different logical equivalence is being used.

In order to progress, I shall borrow from the Nealean strategy by which ?-conv was created. Consider the following rule of inference.

? – KRON INTR: T[?(x/?)] ————————– T[? = ?x((x=1 • ?(x/?) ) v (x=0 • (?(x/?))))]

Like ?-conv, ?-kron intr is justified by appeal to a specific logical equivalence (the particular logical equivalence that was good enough for the final Quinean proof). As such, it is a valid rule of inference in extensional contexts. It follows from the fact that extensional contexts are +psle. Also like ?-conv, ?-kron intr relies on a Russellian interpretation of the descriptions for its validity (but unlike Neale I am stating this explicitly). The top and bottom can be reversed so as to get the elimination version. Both together can be simply called ?-kron[26].

The final Quinean proof can now be restated:

1	[1] ? ? ?	premiss
2	[2] ?	premiss
2	[3] (1 = ?x((x=1 • ?) V (x=0 • ?)))	2, +?-kron
1,2	[4] ?x((x=1 • ?) V (x=0 • ?))
	=?x((x=1 • ?) V (x=0 • ?))	1, def. of ‘?x‘
1,2	[5] (1 = ?x((x=1 • ?) V (x=0 • ?)))	3, 4, +?-subs
1,2	[6] ?	5, +?-kron

And it has now been proved that where an extensional operator is ?-kron and ?-subs, it must also be +psme. Might we ask – was this too easy? Would Neale think this result as interesting as his Gödelian proof? On the one hand, there is reason to think he might. In his essay on Quine he writes:

At this juncture it is worth reminding ourselves of the force of slingshot arguments on a standard Russellian analysis of descriptions. The Church-Quine slingshot succeeds in showing only that any S-connective that is +psle and +?-subs is also +psme. Gödel’s by contrast, shows something more worrying: any connective that is +?-conv and +?-subs is also +psme (this is more worrying on the obvious assumption that every “Godelian equivalence???, as given by ?-conv, is also a logical equivalence, but not vice versa).[27]

We see that it is because the Godelian equivalence is weaker than logical equivalence that has Neale so impressed. Kronecker equivalence, as I call it, is similarly weak. Hence any friend of facts has a new descriptive constraint alongside the one proposed by Neale.

Would the friend of facts be now doubly concerned? Unlikely. This is the ‘other hand’ implied above – the reason why Neale might not be too interested in ?-kron. He goes on to write following directly on from the above quotation:

This fact will be of interest if we find connectives or contexts that are –psle, +?-subs, and +?-conv, because defenders of such connectives will have no recourse to the most common rejoinder made explicitly of facts and situations…[28]

‘The most common rejoinder made explicitly of facts and situations’ of which he speaks concerns the idea of “material content???. The point is made clear enough by what he says in the following:

Fa and F?x(x = a), understood as (14), are in fact logically equivalent in standard systems, but one feels that this fact does not get to the heart of the matter, that Fa and F?x(x = a) are more tightly bound to one another than that, that there is a more interesting semantic (and perhaps syntactic) relation that holds between them, a relation that does not hold between Fa and, say, (16), to which it is also logically equivalent:

(16)	Fa •(Gb V Gb).

The obvious difference is that (16) contains a predicate and a singular term not present in Fa, expressions that contribute what we might call “material content”.[29]

As such, the common rejoinder is to the effect that where logically equivalent statements have different material content, it is strongly intuitive to think that they represent a different fact. Hence, most friends of facts would not consider a connective representing ‘the fact that’ to be +psle.

Neale makes the point when discussing Godelian equivalence that one would only see Fa and a = ?x(x=a • Fx) as representing distinct facts on a Russellian account of descriptions because if treated as singular terms, they seem to share the same material content. On a Russellian treatment however, Neale says: “Gödel’s description ?x(x=a • Fx) is not even a contender for being cast aside as a verbose rendering of a???[30] This is because, on a Russellian treatment, the statement: a = ?x(x=a • Fx) is not a genuine identity statement. It is rather a generalised statement in the matrix of which an identity statement occurs. To show the Russellian version: x( y((y=a • Fx) ? y=x) • x=a).

Returning to ?-kron – we can see why then that the fact theorist would not be particularly concerned (and Neale not interested). It is certainly arguable that the Kronecker equivalence does not share the same material content – but even if it does, the equivalence relies on a Russellian semantics for definite descriptions and, as Neale has gone to great pains to point out, on a Russellian interpretation, we are no longer compelled to see the sharing of material content. As such, for the fact theorist to be concerned about the (slightly) modified Quinean proof, they would first need to subscribe to a Russellian semantics for definite descriptions, just to get the proof going. But it is precisely because they subscribe to the Russellian interpretation that they would deny FIC is ?-kron for the reasons already articulated.

But what then of Neale’s proof? Well, as we have established, it is not neutral with respect to definite descriptions. It too, like the Quinean proof, assumes a Russellian semantics. As such it suffers from exactly the same problem. For the proof to be valid, we must adopt the Russellian semantics, but if we do adopt a Russellian semantics then the common rejoinder against slingshot theories is open to us. We can see that in the end, it has no advantage over the Quinean proof.

It’s a similar situation when we return to consider the Godelian argument and whether Neale’s proof has any advantage. We will see that it does not.

Recall that Neale saw two problems with the Godelian proof that i) it contained semantic assumptions and that ii) the assumptions seemed to contradict one another and therefore the argument was valid in only a trivial sense. For Neale’s proof to be any better it would have to avoid at least one of these problems. It avoids neither. It has already been established that Neale’s proof does not avoid the first problem insofar as it assumes the Russellian semantics for definite descriptions. With respect to the second problem some further exposition will be needed. Consider the following proof that Neale provides that makes first use of his new rule of inference ?-conv.

1 [1]Fa premiss 2 [2] (Fa) premiss 1 [3]a = ?x( x=a • Fx) 1, ?-conv 2 [4] (a = ?x(x=a • Fx)) 2, +?-conv 1,2 [5] (a = a) 3, 4, +?-subs .

Of this proof Neale says the following:

…notice that if were to be read as “the fact that Fa = the fact that ( )”, then if Fa is true and stands for a fact—the fact that Fa—we obtain the unpromising result that the fact that Fa is identical to the fact that a = a.[31]

Neale goes on to say that this is “surely a catastrophe for anyone hoping to get philosophical work out of facts.???[32]

But this is exactly the same result that caused Neale to question the importance of the Godelian argument in its original form. The problem was that where one assumed that descriptions were singular terms, the fact that Fa could be shown to be the same as a=a. Neale thought this an implausible result. He said that one would only think it plausible if one already assumed that all true sentences stood for the same fact – which was exactly what the original Godelian argument was trying to demonstrate (hence the triviality). As such, Neale concluded that the assumption of Godelian Equivalence did not sit well with the assumption that descriptions were singular terms.

So the question here is why should Neale be so impressed this time around? Where we interpret the arbitrary connective as ‘the fact that’ we get exactly the same result that Neale thought implausible in the Godelian case. If it was a case that such a result would lead one to question the compatibility of assumptions in the original Godelian argument, why would we not do the same here? We should be similarly forced to question the compatibility between the rule of inference ?-conv (which is performing the role of Godelian Equivalence) and the assumption of a Russellian semantics for definite descriptions. If the assumption of a Russellian semantics had not been made then there would be no problem. But again, as has been shown, Neale’s proof is not neutral in this respect.

More likely, the problem is with the concept of Godelian equivalence and the belief that a free rider can sit within the scope of the description operator. Since it seems to lead to such unintuitive results, the appropriateness of its use should be seriously examined.

In any case we can see that, as it was in the case of the comparison with the Quinean proof, the assumption of a Russellian semantics causes Neale’s proof to be no better than the Godelian argument that was its inspiration.

Given that it has been shown that Neale’s proof does not avoid the problems he attributes to the Godelian and Quinean versions of the slingshot argument, it is reasonable to ask whether anything has been achieved at all by presenting the insights of Godel in the Quinean format. With respect to the debate concerning facts, it seems pretty clear that nothing has. By combining the strategies of the two approaches, he ends up with a proof which suffers from both their flaws, as opposed to one that avoids them. There is no reason to conclude the argument any more significant than its predecessors for theories of facts.

With respect to Neale’s broader claims, quoted at the beginning of the essay we can say the following: the claim that the proof ‘demonstrates conclusively) that any supposedly non-truth-functional operation must satisfy an exacting logical condition in order to avoid collapsing into a truth-function’ could at least be salvaged if the condition that a Russellian semantics is being assumed. There may, for instance, be a modal theorist who accepts the Russellian picture and would then have to accept Neale’s logical condition. But of his second claim that any theory of facts, states of affairs, situations, or propositions must satisfy a corresponding condition if such entities are not to collapse into a unity, we can say that it is trivial. For the proof will only be valid on a Russellian semantics of definite descriptions, and as such the condition is automatically fulfilled.

In conclusion we can state very clearly now what Neale was aiming for, and what went wrong. Probably the most important part of his strategy was to avoid the use of logical equivalence in his proof. The reason for this, as we have seen, was because fact theorists had already provided a rejoinder against the use of logical equivalence in slingshot proofs and hence these proofs were not considered to be of any significance. And although he does indeed avoid using logical equivalence, he does not escape assumptions concerning the semantics of definite descriptions. Without this neutrality the significance of the proof is seriously undermined and prevents it from being any improvement upon the older slingshots.

Bibliography:

Facing Facts . Stephen Neale.

Oxford University Press.

Oxford Scholarship Online.

Oxford University Press. 2001 http://0-www.oxfordscholarship.com.opac.library.usyd.edu.au:80/oso/public/content/philosophy/0199247153/toc.html

Neale, Stephen “On a Milestone of Empiricism??? in Knowledge Language and Logic, ed. Alex Orenstein and Petr Kotanko,

Boston Studies in the Philosophy of Science, Vol. 210. Kluwer Publishers,

Dordrecht,

Correspondence between Adrian Heathcote and Stephen Neale

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[1] Facing Facts . Stephen Neale.

Oxford University Press.

Oxford Scholarship Online.

Oxford University Press. 2001 http://0-www.oxfordscholarship.com.opac.library.usyd.edu.au:80/oso/public/content/philosophy/0199247153/toc.html, p2

 

[2] Ibid., 184

 

[3] Full exposition of these assumptions can be found on p 130 of Facing Facts.

 

[4] This is because on a Russellian treatment of definite descriptions the two propositions do not share the same material content. This notion is discussed below.

 

[5] Ibid., p132

 

[6] Ibid., p132

 

[7] Ibid., p169

 

[8] Ibid., p169

 

[9] Adrian Heathcote has pointed out the confusion with respect to these rules of inference concerning the need for the premise in both PSST and PSLE to be true. The confusion seems to persist with respect to the incomplete Quinean proof as Neale says the premises need to be true in order for the proof to be valid. Rules of inference should be truth preserving such that if the premises are true then the conclusion must also be valid. But a proof can be valid whether or not the premises are true. This essay will give Neale the benefit of the doubt on this matter and read the rules such that he meant that they are truth preserving – but not requiring the truth of the premise to be valid. However, as Heathcote says, if Neale has made this mistake then it throws significant doubt on his claim that he has achieved a purely formal version of the Godelian slingshot.

 

[10] Ibid., p151

 

[11] This is a revised version of the proof sent to Adrian Heathcote.

 

[12] Ibid., p170

 

[13] See note 9 above

 

[14] In ‘On a milestone of empiricism??? Neale presents the justification of this line of the proof as the ambiguous – “substitutivity of identicals??? which would have been appropriate in Facing Facts also because one of his objections to the argument is that ones choice of a semantics of definite descriptions would determine whether i-subs or psst would be used at this point. So in the original presentation of the proof it makes a relevant point to initially leave it as ambiguous.

 

[15] Ibid., p172 This claim is one of Neale’s more baffling. As stated in note 9 above, this is not a valid proof at all, even with a Russellian semantics of definite descriptions because no valid argument relies on the truth of its premises to be valid.

 

[16] Note 9 is again relevant to this conclusion.

 

[17] I use the phrase ‘go through’ to be deliberately ambiguous – the incomplete proof could never be valid.

 

[18] In an earlier presentation found in facing facts, Neale had altered the function so as achieve greater generality, but it means that a semantic interpretation of the function must be given to the variables a and b as 1 and 0, which is presumably another thing concerning the proof that Neale would disapprove, though he makes no mention of it. Heathcote says further in his letter to Neale, even where the function is presented with a and b replaced with 1 and 0, there would still be need of an interpretation containing numbers. Formal proofs should be irrespective of their possible interpretations.

[19] This point is made by Neale on p 173 of Facing Facts. Line three of the complete Quinean proof is equivalent under the Russellian interpretation to: (12)

x( y(((y=a • ?) V (y=b • ?)) ? y=x) • x=a). And he says in a note: “Using Quine’s (1936) criterion, on a Russellian treatment of descriptions, ? and (12) have the same truth-value for all interpretations of their non-logical vocabulary. (Equivalently, following Tarski, the sentences are true in exactly the same models; equivalently, they are interdeducible.)??? Adrian Heathcote disputes this point and argues that lines 2 and 3 in the complete proof are materially equivalent and not logically equivalent.

 

[20] Ibid., p 185

 

[21] The actual justification of this claim comes when Neale discusses a number of different theories of facts in the last chapter of the book. It is doubtful that he establishes the likelihood of this claim. However, this is not a topic for this essay. If my argument is successful then the point is moot anyway.

 

[22] Ibid., p 184-185

 

[23] The proof can be found on p159 of Facing Facts

 

[24] Ibid., p180

 

[25] It’s significant that the proof uses the elimination form of ?-conv because the introduction form is neutral. (This can be seen easily from the fact that wherever Fa is true, a=a is true).

 

[26] I’m having a bit of fun here. Adrian Heathcote has pointed out that the equivalence involving the Kronecker function is not logical at all, but material – and what’s more, does not rely on a Russellian semantics. As such, ?-kron as a rule of inference couldn’t be used validly. But this is not too crucial to the point I am making, which is simply to use the creation of ?-kron to bring into sharp relief Neale’s strategy of introducing a new rule of inference and why it’s so disastrous to the success of his venture that this new rule does not avoid a Russellian semantics.

 

[27] Neale, Stephen “On a Milestone of Empiricism??? in Knowledge Language and Logic, ed. Alex Orenstein and Petr Kotanko,

Boston Studies in the Philosophy of Science, Vol. 210. Kluwer Publishers,

Dordrecht, p 282

 

[28] Ibid., p283

 

[29] Facing facts Op cit., p126

 

[30] Ibid., p128

 

[31] Ibid., p181

 

[32] Ibid., p 181