In philosophical logic, a slingshot argument is an argument that purports to show that all true sentences stand for the same thing (e.g. the truth value True). The argument (if sound) also shows that all false sentences stand for (designate, refer to) the same thing (e.g. the truth value False). Philosophical logic is the study of the more specifically philosophical aspects of logic. ... An argument is an attempt to demonstrate the truth of an assertion called a conclusion, based on the truth of a set of assertions called premises. ... When someone sincerely agrees with an assertion, they might claim that it is the truth. ... Sentence, derived from Latin sententia (perception, in the subjective sense of how one feels reality is), has three common meanings: Sentence (linguistics) Sentence (mathematical logic) Open sentence (a term that mathematics teachers attempted to introduce, but not used by mathematicians) Sentence (law) Sentence (music) This is a disambiguation page — a... In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ... (This article discusses the soundess notion of informal logic. ... When someone sincerely agrees with an assertion, they are claiming that it is the truth. ... In general, a reference is something that refers or points to something else, or acts as a connection or a link between two things. ...

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 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). Simple slingshot A slingshot, also called a shanghai or a catapult (not to be confused with either the catapult siege engine or shepherds sling) is a small hand-powered projectile weapon. ... A philosopher is a person who thinks deeply regarding people, society, the world, and/or the universe. ... Kenneth Jon Barwise (June 29, 1942 - March 5, 2000) was a US mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used. ... See also John Perry. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ... Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician and logician who was responsible for some of the foundations of theoretical computer science. ... W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ... There are two Donald Davidsons: Donald Davidson (poet) Donald Davidson (philosopher) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Kurt Gödel Kurt Gödel [kurt gøːdl], (April 28, 1906–January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ...

Slingshot arguments are used against those who hold that, e.g. 'Snow is white' and 'Grass is green' denote (distinct) things such as facts, propositions, states of affairs, truth conditions, truthmakers etc. Look up Fact in Wiktionary, the free dictionary A Fact is any of the following: Something actual as opposed to invented. ... Proposition is a term used in logic to describe the content of assertions. ... State of affairs has some technical usages in philosophy, as well as being a phrase in everyday speech in English. ... In semantics, truth conditions are what obtain precisely when a sentence is true. ... A truthmaker for a true proposition is that concrete entity in virtue of which the proposition is true. ...

The argument

One version of the argument proceeds as follows. (This presentation follows closely that of Perry 1996.)

Assumptions:

1. (Substitution) If two terms designate the same thing, then substituting one for another in a sentence does not change the designation of that sentence.
2. (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.
3. Every sentence has an equivalent of the form F(a), i.e. every sentence is equivalent to (i.e. 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".)
4. 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) ('ιx' can be read as 'the x such that'):

1.	S
2.	φ(a)	assumption 3
3.		redistribution
4.		substitution, assumption 4
5.	π(a,b)	redistribution
6.		redistribution
7.		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.

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: The Right Honourable Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was an influential British logician, philosopher, and mathematician, working mostly in the 20th century. ... A definite description is a denoting phrase in the form of the X where X is a noun-phrase or a singular common noun that picks out a specific individual or object. ...

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

Or, in the language of first-order logic: First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ...

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'.
4'.

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.

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.
• Neale, Stephen. 1995. The philosophical significance of Gödel's Slingshot. Mind, Vol. 104, No. 416, pp. 761-825.
• Perry, John. 1996. Evading the slingshot. In Andy Clark et al. (eds.), Philosophy and Cognitive Science. The Netherlands. [1]