Friday, October 5, 2007, 12-2 p.m.

Some notes on Symmetry and Symmetrical Predicates

This talk is inspired by the paper Gleitman et al (1996), “Similar, and similar concepts”, which solved the mysteries of apparent non-symmetrical behavior of symmetrical predicates like similar Tversky had suggested that ‘similar’ is not symmetrical, since subjects generally rate (1a) as holding to a higher degree than (1b).

(1) a.    North Korea is similar to Red China.   b. Red China is similar to North Korea.

Gleitman et al argue that ‘similar’ is symmetrical, and the difference in judgments reflects the independent contribution of figure-ground differences encoded in the syntax. They further argue in support of a robust linguistic distinction between symmetrical and asymmetrical predicates, illustrated by the fact that (2a) and (2b), with symmetrical meet, are close in meaning, but (3a) and (3b), with the “asymmetrical” drown, are not.

(2)  a.   John and Bill meet.        b.   John and Bill meet each other.

(3)  a.   John and Bill drown.           b.   John and Bill drown each other.

      The arguments in the paper are convincing; at the same time, their uses of the terms “symmetrical” and “asymmetrical” do not always fit the mathematical definitions in (4).

(4)  a. A relation R is symmetrical iff for all x, y:       if R(x,y), then R(y,x).

      b. A relation R is asymmetrical iff for all x, y:    if R(x,y), then ¬ R(y,x).

      c. A relation R is non-symmetrical iff if is not symmetrical.

This talk takes up the challenge of modifying and extending the mathematical definitions to accommodate extended uses of the terms in linguistic and cognitive contexts. For example, Gleitman et al refer to the ‘decidedly asymmetrical act of drowning someone’, although the possibility of (3b) shows that ‘drown’ is not an asymmetrical relation in the mathematical sense. I will explore the role of the event argument in references to ‘asymmetrical acts’, in the classification of otherwise ‘non-symmetrical’ action verbs like drown as ‘asymmetrical’, and in the perceived difference in meaning between the “near-synonyms” (2a) and (2b).

Another example of the issues to be discussed: The mathematical definition has no place for the notion of “sometimes symmetrical”, but in the context of linguistic work one wants to understand, not proscribe, natural locutions like “Love isn’t always symmetrical, but sometimes it is.” A natural way to do this, and to see symmetry as a graded notion: relations may be 100% symmetrical, 0% symmetrical, or anywhere between. With these and other modifications we can provide formal underpinnings for many of Gleitman et al’s proposals, or in some cases for friendly amendments to their proposals.

Reference:

Gleitman, Lila R., Henry Gleitman, Carol Miller, and Ruth Ostrin 1996. Similar, and similar concepts. Cognition 58:321-376.