Analogies are like rational equalities

2008-08-17 19:19

A brief reddit exchange motivates me to argue that the analogies

(1) P is to Q as R is to S
(2) P is to R as Q is to S

are equivalent. Thanks to symmetry, I need only show that (1) entails (2). I claim that (1) means there is a sentence with two holes φ(x,y) such that φ(P,Q) and φ(R,S) are both true. For example, Davis Square is to Boston as Flushing is to New York, because I go to Davis Square for frozen dumplings when I am in Boston but I go to Flushing for frozen dumplings when I am in New York.

Because “and” above is commutative, (1) is equivalent to

(3) R is to S as P is to Q.

More importantly, we have

(4) P is to P as R is to R

for any P and R, because we can take φ(x,y) to be x=y. Now, suppose (1) holds. Take φ(x,y) to be

(5) P is to x as R is to y.

Then (4) and (1) together yield (2). For example, Davis Square is to Flushing as Boston is to New York, because Davis Square is to Davis Square as Flushing is to Flushing, and Davis Square is to Boston as Flushing is to New York. QED.

This proof is problematic: What is a “sentence with two holes”, and how can we splice things such as fish and bicycles into sentences, so as to draw analogies involving fish and bicycles (as opposed to the words “fish” and “bicycles”, which neither swim nor roll)? For that matter, what (sort of parametricity) stops us from taking φ(x,y) to be

(6) x either is or is not y,

thus concluding all sorts of intuitively baseless analogies? Perhaps these two difficult problems are to each other as quotation is to higher-order unification.