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Forgive me if it is not the right place to ask this question in SE sites. I am mostly active on SO but I thought it might be a better fit here.

I enrolled in a class this semester and there was a question asked that I could not answer. Here it is:

In terms of semantics, "crazy" does not mean the same as "sane".

The phrase "half crazy" means the same as "half sane".

The phrase "almost half crazy" does not mean the same as "almost half sane".

If we read their semantics off of trees compositionally from the way they are combined, what kind of tree structures can we assume for them?

Thank you.

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Trees are not going to solve the problem. This is not a syntactic issue; it's semantic.
Note, no syntax up my sleeve in the following.

Sane ... Crazy represents a semantic cline, with sane and crazy at opposite poles.

(They can both be used with absolutely, which can only modify a polar adjective:

  • He's absolutely sane/crazy.
  • It's absolutely freezing/broiling outside vs *It's absolutely hot/warm/tepid/cool/cold outside.)

The key word is almost, which is a vector quantity;
i.e, it points not only to a quantity on a cline, but a direction as well.

A simple example is the difference between Few people attended and A few people attended. The same absolute numbers may well be reported by both, but few is negative, while a few is positive.
This orientation refers to expectations or hopes on the part of the speaker.

Few people means 'fewer than hoped/expected';
a few people means 'more than hoped/expected'.


Almost means 'close to, but lower than, some clinal value'.
Half sane and half crazy both refer to the midpoint on the cline,
i.e, they do point to the same position on the cline.
But they refer to opposite directions on the cline.

Half sane means 'a sanity value of .5'; i.e, sane is UP, crazy is DOWN on the cline.
Half crazy means 'a craziness value of .5'; i.e, crazy is UP, sane is DOWN.

Since almost is always on the DOWN side of a cline position,
almost half crazy does not mean the same as almost half sane.
Almost half crazy announces a craziness value < .5, and thus a sanity value > .5.
This is saner than almost half sane, which announces a sanity value < .5.

Good luck doing this with syntax trees.

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  • Why would one want to "assume a type of tree structure" for them (and btw, what is meant by "them")? Why not a spreadsheet structure or an energy field structure? Structures are invented by theoreticians for their own purposes, and may or may not be useful for describing any phenomenon. Tree-like structures are often useful for hierarchical or genetic models, but they're hardly universal. – jlawler Jan 24 '14 at 21:49
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The question your teacher gave you has already been discussed by Barbara Partee in her introductory paper ‘Lexical Semantics and Compositionality’ (see reference at the end) - only she used almost half empty and almost half full. If we assume that the constituent structure of both expressions is as in (1), and also make the assumption in (2), the question then arises how come the two expressions differ in meaning.

(1) [almost [half [crazy/sane]]]

(2) The meaning of a complex expression is derived by combining the meaning
     of each of its elements with that element with which it forms a constituent.

Adopting (2) means that according to (1), we derive the meaning of almost half crazy by first combining the meanings of half and crazy and then combining the meaning of almost with that of half crazy. Since half crazy has the same meaning as half sane, it’s not clear how almost gives one meaning when combined with the first and another meaning when combined with the second.

One way to solve the problem is to deny that the expressions almost half crazy/sane have the structure in (1) and assign them instead the constituent structure in (3), which is also given by the tree in (4).

(3) [[almost half] [crazy/sane]]

(4)

          
           see tree on our website

According to (3) and (4), in deriving the meanings of the expressions almost half crazy/sane, the meaning of almost half combines each time with another meaning, and therefore it is not surprising that the two final expressions differ in meaning.

A second way to solve the problem is to accept the constituent structure in (1), but to deny that half crazy and half sane have the same meaning. Maybe the meanings of half crazy and half sane always combine with some approximation/precision value, and when such value is not expressed it is taken to be the absolute precision value (the value expressed by exactly). We can now assume that the meanings of half crazy and half sane are such that they give the same meaning when combining with the meaning of exactly but not when combining with the meanings of other approximation/precision expressions (like almost).

A third way to solve the problem is to accept the constituent structure in (1), but to deny the principle expressed by (2). We can now assume that in deriving the meanings of almost half crazy/sane, the meaning of almost is combined with the meaning of half, even though it doesn’t form a constituent with this element.

Reference: Invitation to Cognitive Science, 2nd edition. Daniel Osherson, general editor; in Part I: Language, Lila Gleitman and Mark Liberman, eds. MIT Press, Cambridge 1995, pp. 311-360.

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