Copyright © 1999 Massachusetts Institute of Technology
The thesis of extensionality says that every meaningful declarative sentence is equivalent to some extensional sentence. Understanding this thesis requires understanding the terms in italics.
Two sentences are equivalent if and only if they have the same truth-value in all possible circumstances. The following are equivalent because no matter how far Jody actually ran, the sentences are either both true or both false:
Jody just jogged
3.1 miles.
Jody just jogged 5 kilometers.
When the replacement of a component of a sentence with another component always results in a sentence that has the same truth-value as the original (true if the original is true and false if the original if false) this is a replacement that preserves truth-value.
Terms of different kinds have extensions of different kinds. The extension of a name or description is the individual or individuals to which the name or description applies. The extension of a one-place predicate such as "is a synapse" applies to the class of all the individuals, in this case, all the synapses, to which the predicate applies. The extension of an n-place predicate is a set of ordered n-tuples to which the predicate applies. The extension of a declarative sentence, which one can regard as a zero-place predicate, is its truth value.
Two terms are coextensive only when they have the same extension, two sentences that have the same truth value, two names that refer to exactly the same thing (or things), and so on.
A sentence is extensional only when each and every replacement of a component of a sentence with a coextensive term preserves truth value.
If "Stan's car" and "the oldest Volvo in North Carolina" are coextensive, then replacing the first with the second in (a) "Stan's car is in the driveway" preserves truth-value. So far as this replacement shows, then, (a) is extensional. A similar replacement in (b) "Hillis thinks that Stan's car is a new Jaguar" does not preserve truth value. Hillis does not think that the oldest Volvo in North Carolina is a new Jaguar. Statements about PROPOSITIONAL ATTITUDES such as thinking, believing, fearing, and hoping are typically not extensional.
If "The Mercury Track Team" and "The Vanguard Video Club" are coextensive because each expression names a group with exactly the same members, then replacing the first with the second in (c) "Nobody in the Mercury Track Team smokes cigars" preserves truth-value. A similar replacement in (d) "The Captain of the Mercury Track Team is Lou Silver" will not preserve truth-value if (d) is true and the Vanguard Video Club does not have a captain. Examples of this sort show at least that clubs are not sets.
"Vicki will discover the greatest prime number" and "Vicki will win the New Jersey Lottery" are coextensive because they are both false. Replacing the first with the second in (e) "It isn't so that Vicki will discover the greatest prime number" preserves truth-value. A similar replacement in (f), "It is absolutely impossible that Vicki will discover the greatest prime number," does not preserve truth value. However unlikely, it is still possible that Vicki will hit the jackpot. The existence of a largest prime number, in contrast, is not merely unlikely; it is impossible. Modal statements about what is possible or necessary, treated by MODAL LOGIC, are often not extensional. The connectives of standard propositional LOGIC such as not, and, or, if, and if and only if are truth-functional. Replacement in a truth-functional sentence of any component with another with the same truth-value preserves the truth-value of the original.
The thesis of extensionality, a version of REDUCTIONISM, says that every meaningful, declarative sentence is equivalent to some extensional sentence. This does not require that every sentence be extensional but rather that every nonextensional sentence about psychological attitudes, modality, laws, counterfactual conditionals, and so forth, have an extensional equivalent.
An historically important statement of the thesis of extensionality appears in Wittgenstein (1922): "A proposition is a truth-function of elementary propositions (Proposition 5)."
Wittgenstein's later philosophy abandons both elementary propositions and the thesis of extensionality. Russell expresses sympathy for the thesis in several places including "Truth-Functions and Others," Appendix C of Whitehead and Russell (1927).
Rudolf Carnap formulates the thesis of extensionality (hereafter abbreviated TOE) as a relation between extensional and nonextensional languages (see especially Carnap 1937, sect. 67). The truth of TOE promises a greater intelligibility of the world. Extensional languages have "radically simpler structures and hence simpler constitutive rules" than nonextensional languages (Carnap 1958: 42). If "the universal language of science" (Carnap 1937: 245) is extensional, therefore, we can discuss exhaustively every scientific phenomenon in a language that has a radically simple structure.
Carnap defends TOE by finding an extensional sentence about language that he thinks is equivalent to a given nonextensional sentence. The modal, nonextensional sentence "Necessary, if you steal this book, then you steal this book" is equivalent to the extensional sentence "'If you steal this book, then you steal this book' is true in a certain formal metalanguage L" (cf. Carnap 1958: 42). The psychological, nonextensional sentence "John believes that raccoons have knocked over the garbage cans" is equivalent, perhaps, to the extensional sentence "John is disposed to an affirmative response to some sentence in some language which expresses the proposition that raccoons have knocked over the garbage cans" (cf. Carnap 1947: 55).
Although he devotes much time and effort to defend TOE, Carnap does not claim to establish it. He regards it as a likely conjecture. The project, however, now appears to be more difficult than Carnap predicted. A successful translation of a nonextensional sentence must satisfy two requirements: (1) the new sentence must really be equivalent to the original, and (2) the new sentence must really be extensional. In the example above, if someone can have such a belief about raccoons without being disposed to respond to any sentences, (1) is violated. If one cannot understand affirmative response except as a nonextensional notion, then (2) is violated.
In terms of INTENTIONALITY, Chisholm (1955-56) formulates a thesis resembling that of Brentano (1874) about the distinctiveness of the psychological: (A) TOE is true for all nonpsychological sentences, and (B) TOE is false for all psychological sentences. In the 1950s, Chisholm defended (B) by attacking translations of sentences about believing that Carnap and others proposed. (See Hahn 1998 for a Chisholm bibliography.)
Clause (A), however, cannot be taken for granted. The project of finding extensional equivalents of nonextensional modal sentences also faces difficulties. Quine (1953), like Carnap, attempts syntactic translations that are about language. (See chapter 10, section 3, "The Problems of Intensionality," in Kneale and Kneale 1962.) Montague (1963) derives significant negative results that are beyond the scope of this article.
Brentano, F. (1874). Psychologie vom empirischen Standpunkt. Vienna.
Carnap, R. (1937). The Logical Syntax of Language. London: Routledge and Kegan Paul. First published in German in 1934.
Carnap, R. (1947). Meaning and Necessity. Chicago: University of Chicago Press.
Carnap, R. (1958). Introduction to Symbolic Logic and Its Applications. New York: Dover. First published in German in 1954.
Chisholm, R. M. (195556). Sentences about believing. Proceedings of the Aristotelian Society 56:125-148.
Hahn, L. E., Ed. (1998). The Philosophy of Roderick M. Chisholm: The Library of Living Philosophers. Peru, IL: Open Court.
Kneale, W., and M. Kneale. (1962). The Development of Logic. Oxford: Oxford University Press.
Montague, R. (1963). Syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability. Reprinted in R. H. Thomason, Ed., 1974, Formal Philosophy; Selected Papers of Richard Montague. New Haven: Yale University Press.
Quine, W. (1953). Three grades of modal involvement. Reprinted in Quine (1966), The Ways of Paradox. New York: Random House.
Whitehead, A. N., and B. Russell. (1927). Principia Mathematica. 2nd ed. Cambridge: Cambridge University Press.
Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. First published in German in 1921. Recent English translation by D. F. Pears and B. F. McGuinness (1961), London: Routledge and Kegan Paul.
Copyright © 1999 Massachusetts Institute of Technology