degree relations: a puzzle

i've noticed a pattern among degree relations that calls for explanation. when i originally wrote this up for a term paper some five years ago, i came up with an unsatisfactory solution that was of little explanatory value. now, in the spirit of bertrand russell's words about it being a wholesome plan to stock one's mind with puzzles, i hereby record and publicly stock my puzzle. actually, there appears to be a theory of vector space semantics, applied to prepositions (of all things) that might present a solution to the puzzle. but i don't understand the formalism, so i'll leave it as a puzzle.

first question: what is a degree relation? well, it's the semantics corresponding to things like comparatives, which take two degrees and compare them. in english, these include the comparatives -er, more and less, as well as function words like so, such, too, and enough, as in john is too tall to operate an airplane, or mary is so tall that she bumps her head on lampposts. what degree relations do semantically is compare two degrees. for example, in the sentence above too compares the degree of john's height with the degree of height appropriate or permissible for operating airplanes, finding that the former exceeds the latter, or, more appropriate to the common view that degrees are extents on a scale and not merely points, that the maximum of john's height is greater than the maximum appropriate or permissible for operating airplanes. mathematically, degree relations can be characterized as functions from pairs of degrees to a truth value.

naturally, to be compared the degrees must be along the same scale. however, since chris kennedy's santa cruz dissertation, we speak of positive and negative degrees. john's height can be equivalently expressed with the adjectives short or tall.

john is taller than mary
mary is shorter than john
these sentences are logically equivalent, and both do the same thing: compare the degree of john's height with the degree of mary's height. they just do it from the perspectives of different sides of the height scale. tall starts from the low end of the scale and extends upward, making it a positive degree. short starts from the high end and extends downward, making it a negative degree.

with this distinction in mind, we can characterize the degree relations in terms of whether the two degrees they compare are of the same polarity or not.

consider the comparative:

john is taller than mary
john is shorter than mary

here two degrees of the same polarity are being compared. they can both be positive or both be negative. it doesn't matter, as long as they're the same. we can see this by the fact that we are either comparing the maximum degree, in the case of tall, or the minimum degree, in the case of short.

the same is true of comparatives more and less. and it's true of too. consider:

mary is too tall to play basketball
this compares two positive degrees. it says the maximum of mary's height exceeds the maximum of the height appropriate to basketball-playing. if we substituted short, then we would compare the minimums.

other degree relations require opposite polarities.

john is tall enough to play basketball
john is so tall that he can play basketball
both of these sentences compare the maximum of john's height with the minimum height appropriate for basketball. if we used short instead, both polarities would reverse and we would compare the minimum of john's height and the maximum height appropriate to basketball.

now that we have defined degree relations and classified them by whether they compare degrees of identical or opposite polarity, we arrive at the observation that forms the puzzle: same-polarity degree relations can be specified by an extent phrase consisting of a number and a measure, like three feet. opposite-polarity degree relations cannot.

john is three feet taller/less tall than mary
john is three feet too tall to play basketball
*john is three feet tall enough to play basketball
*john is three feet so tall that he can play basketball

these facts appear intuitively to be nonaccidentally related. there is no obvious semantic problem: if a minimum and a maximum can be determined on the same scale, we are able to find the difference between them. in fact, we can syntactically circumlocute, thus:

john is tall enough by three feet to play basketball
?john is so tall that he can play basketball, by three feet
okay, the latter is kind of awkward, but the point seems valid. what appears to be the problem is placing the measure phrase in the syntactically privileged specifier position in front of the degree phrase. and yet our intuition is that it is the semantics of degree relations that allows or disallows the measure phrase specifier.

good puzzle, no? chris kennedy liked it. i may one day read zwarts and winter's vector space semantics well enough to check for its applicability to the present puzzle. or perhaps someone else will come up with a solution.

or maybe someone has - i really don't keep up with the literature.

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