An analytic proposition is one whose truth depends on relations of ideas or concepts, and not on what it says about the world or the way the world is. This has been expressed in a number of different ways. German philosopher Gottfried Wilhelm Leibniz (1646 – 1716) distinguished between what he called truths of reason and truths of fact. Scottish-English philosopher David Hume (1711 – 1776) distinguished between what he called relations of ideas and matters of fact. In the twentieth century, American Philosopher C.I. Lewis (1883 – 1964), for one, held that analytic truth comes from linguistic convention. Analytic truth and analytic propositions were central notions to the Logical Empiricists (also known as Logical Positivists) and members of the Vienna Circle.
At the beginning of his Critique of Pure Reason, following the distinction that had been made earlier by Leibniz, German philosopher Immanuel Kant (1724 – 1804) wrote:
In all judgments in which the relation of a subject to the predicate is thought (if I only consider affirmative judgments, since the application to negative ones is easy) this relation is possible in two different ways. Either the predicate B belongs to the subject A as something that is (covertly) contained in this concept A; or B lies entirely outside the concept A, though to be sure it stands in connection with it. In the first case, I call the judgment analytic, in the second synthetic. (A:6-7)
As an example of an analytic judgment, Kant gave "All bodies are extended." He seems to mean that we cannot think of a body without also thinking of an extension in space. He contrasts this example with “All bodies are heavy,” where the predicate ("is heavy"), he says, “is something entirely different from that which I think in the mere concept of body in general.” (A:7)
Expanding on that, Kant made a fourfold distinction—analytic vs. synthetic propositions or statements, and a priori vs. a posteriori ones. Analytic statements are those in which, Kant claimed, the predicate is contained in the subject, whereas in synthetic ones it is not. An example frequently given for an analytic statement is "All bachelors are unmarried males." If the definition of 'bachelor' is known, then the predicate 'is an unmarried male' follows from that definition. A priori statements are ones whose truth can be known before any experience with the world, whereas the truth of a posteriori ones is discovered through experience of the world.
These two distinctions made for four possibilities—Analytic a priori, analytic a posteriori, synthetic a priori, and synthetic a posteriori.
Two of those were, until recently, accepted by more-or-less everyone as noncontroversial—analytic a priori and synthetic a posteriori. Everyone agreed that there are no analytic a posteriori statements because analytic implies a priori, i.e. analytic implies that the truth of the statement is not derived from experience of the world.
The controversial category was synthetic a priori statements. Kant held that this category is not empty; he claimed that there are some synthetic a priori statements, i.e. there are some statements in which the predicate is not contained in the subject—the predicate tells us some new information beyond what is in the subject or what an analysis of the subject would reveal—but the truth of the statement is known a priori, meaning that we do not require empirical experience of the world in order to ascertain their truth. Kant offered the statements of simple arithmetic, such as 7+5=12, and certain statements of philosophy, as examples of synthetic a priori propositions, and he then went on to investigate how (he thought) synthetic a priori propositions are possible.
In his seminal essay, “Two Dogmas of Empiricism,"(1951)—one of the most important and influential essays in twentieth-century analytic philosophy—American philosopher-logician Willard Van Orman Quine (1908 – 2000) attacked the notion of analyticity. The main point of his argument was that the different notions of analyticity are circular, and that no satisfactory account of analyticity has been given. (See Wikipedia, "Two Dogmas of Empiricism," for a summary of this essay and of its critics.) If this claim is accepted—it was rejected or at least challenged by Paul Grice and P.F. Strawson, Hilary Putnam, Scott Soames, and others—then the program of the Logical Empiricists tends to fall apart because the distinction between analytic and synthetic statements falls apart. That distinction was crucial to the Logical Empiricists because they held that all true statements are either based on logic or on positive experience of the world. But if the distinction between analytic and synthetic statements cannot be maintained, that tends to shatter the underpinnings of the program of the Logical Empiricists (Logical Positivists).
Kant argued that arithmetic is synthetic. He claimed that the predicate "=12" is not contained in the subject "7+5" of the statement "7+5=12."
The formalists in mathematics—especially David Hilbert and Gottlob Frege, and all who followed in their wake—rejected that claim about arithmetic and mathematics, holding mathematics and arithmetic to be either formal (Hilbert) or reducible to logic and set theory (Frege). The empiricists, especially Hume and the Logical Empiricists (Logical Positivists) and their followers (most of whom considered themselves to be intellectual descendants of Hume) rejected the claim that there are any possible statements of any form that are synthetic a priori. So for the empiricists and logical positivists, there are only two kinds of statements, analytic ones and synthetic ones; moreover, they claimed, all analytic statements are a priori and all synthetic statements are a posteriori, so analytic = a priori, and synthetic = a posteriori.
The notion that arithmetic can be a formal system was refuted, however, by the work of Czechoslovakian mathematician-logician Kurt Gödel (1906 – 1978). In what has come to be known as Gödel's Proof he showed that the axiomatic method, when applied to the arithmetic of cardinal numbers, cannot show both the consistency and the completeness of the axiomatized system. In other words, simple arithmetic cannot be reduced to or comprehended in an axiomatic system. Given any set of axioms for arithmetic, there are true statements of arithmetic that cannot be derived from those axioms. In addition, no proof of the formal consistency of such a set of axioms is possible.
Whether or not Gödel's Proof refutes the claim that arithmetic is analytic can be debated. But at the least Gödel's Proof shows that arithmetic cannot be reduced to or comprehended in a formal axiomatic system. Altnough it may not prove it, this does tend to lend support to Kant's claim that the statements of arithmetic are synthetic propositions (assuming that the analytic-synthetic distinction can be maintained; an assumption that is suspect after the work of Quine.)
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