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Volume 10, No. 6 - September 2000  
Table of contents for this issue

New Model High School
Was Your Yearbook an Unintended Reality?

IN PHILOSOPHY AS IN POLITICS, les extrêmes se touchent. In their playful, oracular, Gallic way, Jacques Derrida and the deconstructionists have insisted that language ("the text") bears no determinate relation to the world. In its austere, rigorous, Teutonic way, modern logic seems to tell us something alarmingly similar.

The branch of logic that studies the relationship between language and the world is known as model theory. Suppose you start with a bunch of sentences and ask: What sort of world would make those sentences come true? In model theory, a world, or "model," consists of a set of objects along with various subsets corresponding to properties of those objects and the relations among the objects. To take a trivial example: The model might be the set of students at Riverdale High along with subsets corresponding to properties like "being a member of the chess club" or "being a member of the football team" and to relations like "x has a higher grade point average than y."

Now let's say you are considering the sentence "Every chess-club member has a higher grade point average than any football-team member." Riverdale High might be a model for this sentence -- that is, a world that makes the sentence come true. But does a model for this sentence have to consist of actual students? Not at all. Suppose Riverdale publishes a yearbook containing little photographs of all the students. Then the set of those photographs would furnish a perfectly good alternative model. Of course, the properties and relations would have to be reinterpreted. The relation "x has a higher grade point average than y" would now be interpreted as referring to a set of photographs, not students: The photo of Archie would "have a higher grade point average" than the photo of Jughead if and only if Archie had a higher grade point average than Jughead.

When we talk about the students at Riverdale High, we mean to refer to flesh-and-blood kids, not to little two-dimensional photographs. The model containing actual students is thus the intended model, even though the model comprising the photographs makes precisely the same sentences come true. This raises an interesting question: Can our use of language always distinguish intended from unintended models? In other words, can it guarantee that we are hooking onto the "right" reality?

Model theory was originally developed in the early decades of the twentieth century to investigate the foundations of mathematics. From the beginning, its findings seemed paradoxical. In 1919, the Norwegian logician Thoralf Skolem, building on work done by Leopold Löwenheim, proved that if a theory couched in formal logic has any model that makes it true, then it also has a nonstandard model whose objects are the natural numbers (0, 1, 2, and so on).

It was the Harvard philosopher W.V. Quine who first realized the metaphysical significance of all this. In a 1968 public lecture at Columbia, Quine asked whether Skolem's result might not vindicate the ancient Pythagorean dictum "All is number." If any formalizable theory, even our scientific theory of the world, can be interpreted as referring to the natural numbers, then perhaps numbers are world enough. Finally, though, Quine pulled back from neo-Pythagoreanism, declaring, in his ontologically relativistic way, that "there is no absolute sense in saying that all the objects of a theory are numbers, or that they are sets, or bodies, or something else."

This line of reasoning was taken further by the Harvard philosopher Hilary Putnam. In a series of papers beginning with a 1976 address to the American Philosophical Association, Putnam outlined what he called his "model-theoretic argument against realism." Even if there were such a thing as the True Theory of the World, he observed, Skolem's result showed that this theory would not be able to single out its own intended reference. So much for the realist doctrine that truth is based on a fixed correspondence between language and the world.

But what of the claim made by some philosophers that causal interactions determine how our words latch on to bits of reality? Surely, when I say "Alpha Centauri," I succeed in referring to a unique object because my use of the term is linked to the object by a complicated causal chain involving photons and telescopes and astronomy textbooks. Here Putnam pointed out that adding a "causal theory of reference" to our True Theory of the World amounts to just that -- adding more theory. The Skolem argument will still apply, meaning that the enlarged theory is guaranteed to have nonstandard interpretations where the term "cause" refers to God knows what. In one such model, our use of the term "Alpha Centauri" might be "causally linked," in an unintended interpretation of "causally," to the Eiffel Tower.

Philosophical apostates like Richard Rorty have seized on these model-theoretical arguments as evidence for their claim that the idea of language representing a mind-independent reality has collapsed. Putnam does not endorse this radical reading. (See his new book, The Threefold Cord: Mind, Body, and World [Columbia].) Still, it is clear that logic can be a formidable weapon against logocentrism. The next time you hear someone say, "Il n'y a pas d'hors texte," it will probably be an untenured professor of English at some Midwestern university. But it might be a model theorist talking French.



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