Monday, January 24, 2005

The Traditional Analysis of Reduplicative Propositions

Since I've referred several times to the traditional analysis of reduplicative propositions, I thought I would say something about that, for anyone who hasn't come across it before. (Geach famously says in Providence and Evil that the logic of these statements is still an unsolved problem; I don't recall what he thought the problem to solve was. But it is certainly true that there isn't much done on reduplicative propositions these days.) All this will be a bit rough.

A reduplicative proposition, in the broad sense, is an exponible proposition--in technical terms, it's something that can't be plugged directly into a categorical syllogism without translation into more than one categorical proposition; in non-technical terms, to examine it, you have to unfold its meaning more fully. Reduplicative propositions are typically marked by reduplicative particles, e.g.:

Man inasmuch as he is rational is risible.
Man as rational is risible.
Man insofar as he is rational is risible.
Man qua rational is risible.

But they need not be. For instance, the following proposition can be considered a reduplicative proposition:

The board at time t is blank.

We can see this by recognizing that, at least on some interpretations of this proposition, it could just as easily be formulated as:

The board, as it exists at time t, is blank.

This is certainly reduplicative.

Now, there are many sorts of reduplicative propositions. The general point of a reduplication is either (1) to apply to the subject a formal concept under which the predicate applies to it or (2) to give the cause of the predicate's applying to the subject. Propositions of type (1) are called specificative reduplicative propositions, or specifically reduplicative propositions, or simply specificative propositions. Propositions type (2) are called reduplicatively reduplicative propositions, or just plain reduplicative propositions. Roughly, the difference is that the reduplication in specificative propositions fiddles with the subject; in reduplicative propositions, it fiddles with how the predicate applies to the subject. I'll only look at the latter here.

Exponible propositions generally break down into elements that are called exponent propositions; they explain or expound what is going on in the exponible proposition. The traditional analysis isolates four exponents for every reduplicative proposition:

1) the fundamental proposition;
2) a proposition affirming the reduplicate term of the subject;
3) a proposition affirming the predicate of the reduplicate term;
4) a causal proposition (in a very broad sense of 'causal') giving the ground between subject and predicate.

As an example, take the reduplicative proposition, "Every human being as rational is risible." This has four exponents:

1) Every human being is risible.
2) Every human being is rational.
3) Every rational (thing) is risible.
4) Being rational is a cause of being risible.

The key exponent here is (4); in explaining the reduplicative proposition, we can just say, "Every human being's being rational is a cause of his/her being risible; therefore every human being is risible inasmuch as he/she is rational."

One reason we need to focus on (4) is that reduplication can do funny things with the predicates of the propositions. Take, for instance, the following set of reduplicative propositions:

1) Every man as rational is risible.
2) Every man as animal is not risible.

The second exponents, if we were to forget that these are reduplicative propositions, would read:

1) Every man is risible.
2) Every man is not risible.

These look like contradictions. They are not, however; this is because the fourth exponent shows that the second exponents of (1) and (2) need not be contradictions. To treat them as contradictions would be to commit the fallacy of equivocation: the fourth exponent, remember, is fiddling with the meaning of the predicates by telling us why and how the predicates apply to the subjects; and they do not apply in the same way, so there is a difference between the second exponents of (1) and (2). The actual contradictory of

1) Every man as rational is risible;

is

1') Every man as rational is not risible.

The actual contradictory of

2) Every man as animal is not risible;

is

2') Every man as animal is risible.

Thus there is no formal contradiction in saying, "Every man is (as rational) risible and (as animal) not risible."

I've been considering reduplicatively reduplicative propositions. Specificative propositions are somewhat different, since they don't involve the causal exponent; they tend to be somewhat simpler. The distinction between the two is sometimes important, because sometimes there can be considerable differences in meaning. A famous historical example is found in Christology. In orthodox Christology,

(A) Christ as man came to be

is, if taken reduplicatively, true, because it implies that the person (who is Christ) began to be man; if taken specificatively, false, because it implies that the person (who is Christ the man) began to be.

Reduplicative propositions, despite our tendency to ignore them, play an important role in many philosophical fields. Aristotle, for instance, argued that being qua being is the subject of metaphysics (when a term reduplicates itself, this is called a reflexive reduplication); Leibniz's principles of identity presuppose reduplicative analysis; Anscombe has noted that her use of the phrase 'under the description' in discussing intention is reduplicative. They are often used in resolving paradoxes. And so forth.

Roberto Poli has an interesting online discussion of reduplicative analysis, Qua-Theories (PDF; for HTML, here's Google's cache).

UPDATE: corrected a few typos.

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