Slingshot argument

From Wikipedia, the free encyclopedia

In logic, a slingshot argument is an argument that tries to show that all true sentences stand for the same thing, a formal object that can for convenience and without loss of generality be referred to as the truth value True. Under the assumption of bivalence for logical values, the slingshot argument has the additional consequence that all false sentences stand for the same thing, in a corresponding manner referred to as the truth value False. In various formulations of the slingshot argument, the terms denote, designate, or refer to may be used instead of the term stand for. In light of these variations, the thrust, range, and impact of the slingshot may vary widely with the theory of reference that is held to be in force in a particular field of application.

This type of argument was dubbed the "slingshot" by philosophers Jon Barwise and John Perry (1981) due to its simplicity and minimal presuppositions. Various versions of the slingshot have been given by Charles Peirce, Gottlob Frege, Alonzo Church, W. V. Quine, and Donald Davidson, but Stephen Neale (1995) claims that its most compelling version was put forth by Kurt Gödel (1944).

Some types of slingshot arguments are used to counter the position that logically distinct sentences denote logically distinct objects. The nature of the opposition will of course vary with the definition, or lack thereof, of the concept logically distinct. The corresponding logical objects are in various accounts referred to as facts, propositions, realities, situations, states of affairs, truth conditions, truthmakers, and so on.

1 Preliminary considerations
2 The argument
3 Responses to the argument
4 See also
5 References
6 Bibliography

[edit] Preliminary considerations

The diversity of conclusions reached by a diversity of thinkers on this score is partly due to the diversity of their jumping off points, logically speaking, in other words, the variety of assumptions, not always fully expressed and critiqued at the outset, from which they set out. For example, consider one of C.S. Peirce's more emphatic statements of the principle:

Finally, and in particular, we get a Seme of that highest of all Universes which is regarded as the Object of every true Proposition, and which, if we name it [at] all, we call by the somewhat misleading title of "The Truth". (C.S. Peirce "Apology for Pragmaticism", The Monist, 16, 492–546 (1906), Collected Papers, CP 4.539.)

It is possible, but certainly not automatic, that Peirce means the same thing by his statements as Frege means by his statements of what we call the same idea. And it is far less immediate that either of them has in mind what goes under that banner today. These questions require careful examination.

What Peirce means by his statement naturally depends on what he means by such terms as Object, Proposition, Seme, and Universe. For Peirce in particular the clarification of these terms is set in the frame of reference that is formed by his theories of signs, information, and inquiry. But without going too far into those questions, his example makes it clear that a corresponding statement applies to any other thinker who makes use of the sort of argument that is here placed under the catch-all category of a slingshot.

Taking Peirce as an object example also serves to highlight one of the critical dimensions of variation in slingshots, namely, the extent to which objects of reference are one or many. Peirce's notion of denotation follows the classical notion that allows for general denotation and plural reference. In short, signs in general and sentences in particular are allowed to have many objects, collectively referred to as their denotation or their extension. Furthermore, signs and sentences may have many different types of objects, appropriately qualified as to the particular mode of reference in question. Considerations of this sort considerably blunt the force of many oppositions between one-truth slings and many-truth arrows, and so they will be discussed below under the heading of blunt object mediations.

[edit] The argument

One version of the argument (Perry 1996) proceeds as follows.

Assumptions:

Substitution. If two terms designate the same thing, then substituting one for another in a sentence does not change the designation of that sentence.
Redistribution. Rearranging the parts of a sentence does not change the designation of that sentence, provided the truth conditions of the sentence do not change.
Every sentence is equivalent to a sentence of the form F(a). In other words, every sentence has the same designation as some sentence that attributes a property to something. (For example, "All men are mortal" is equivalent to "The number 1 has the property of being such that all men are mortal".)
For any two objects there is a relation that holds uniquely between them. For example, if the objects in question are denoted by "a" and "b", the relation in question might be R(x, y), which is stipulated to hold just in case x = a and y = b.)

Let S and T be arbitrary true sentences, designating Des(S) and Des(T), respectively. (No assumptions are made about what kinds of things Des(S) and Des(T) are.) It is now shown by a series of designation-preserving transformations that Des(S) = Des(T). Here, " $ι x$ " can be read as "the x such that".

1.	$S$
2.	$φ(a)$	assumption 3
3.	$a = \iota x (\phi (x) \land x=a)$	redistribution
4.	$a = \iota x (\pi (x,b) \land x=a)$	substitution, assumption 4
5.	$π(a, b)$	redistribution
6.	$b = \iota x (\pi (a,x) \land x=b )$	redistribution
7.	$b = \iota x (\psi (x) \land x=b )$	substitution, assumption 3
8.	$ψ(b)$	redistribution
9.	$T$	assumption 3

Note that (1)-(9) is not a derivation of T from S. Rather, it is a series of (allegedly) designation-preservating transformation steps.

[edit] Responses to the argument

As Gödel (1944) observed, the slingshot argument does not go through if Bertrand Russell's famous account of definite descriptions is assumed. Russell claimed that the proper logical interpretation of a sentence of the form "The F is G" is:

Exactly one thing is F, and that thing is also G.

Or, in the language of first-order logic:

$\exists x (\forall y (F(y) \leftrightarrow y = x) \land G(x))$

When the sentences above containing $ι$ -expressions are expanded out to their proper form, the steps involving substitution are seen to be illegitimate. Consider, for example, the move from (3) to (4). On Russell's account, (3) and (4) are shorthand for:

3'.	$\exists x (\forall y ((\phi(y) \land y=a) \leftrightarrow y = x) \land a = x)$
4'.	$\exists x (\forall y ((\pi (y,b) \land y=a) \leftrightarrow y = x) \land a = x)$

Clearly the substitution principle and assumption 4 do not license the move from (3') to (4'). Thus, one way to look at the slingshot is as simply another argument in favor of Russell's theory of definite descriptions.

If one is not willing to accept Russell's theory, then it seems wise to challenge either substitition or redistribution, which seem to be the other weakest points in the argument. Perry (1996), for example, rejects both of these principles, proposing to replace them with certain weaker, qualified versions that do not allow the slingshot argument to go through.

[edit] See also

[edit] References

Barwise, Jon, and Perry, John (1981), "Semantic innocence and uncompromising situations", Midwest Studies in the Philosophy of Language, VI.

Gödel, Kurt (1944), "Russell's mathematical logic", in Schillp (ed.), The Philosophy of Bertrand Russell, Evanston and Chicago: Northwestern University Press, pp. 125-53.