ii. Jack, Jill really likes.

(i) and (ii) clearly differ in syntactic structure: whereas (i) has the SVO order typically associated with transitive sentences in English, (ii) has the OSV order associated with so-called ``topicalization'' or ``focus movement'' constructions. In terms of their function, (i) makes a simple assertion, while (ii) carries extra meaning which can be roughly paraphrased in the following way: ``it's Jack that Jill really likes.''

Even though (i) and (ii) differ in terms of function
and in terms of the contexts in which they could be
felicitously used, they do in fact have the same truth
conditions: both are true just in case Jill and Jack
stand in the ``really likes'' relation, where Jill is
the ``liker'' and Jack is the ``likee''. Since the goal
of the simple predicate logic system we've been working
with so far is just to represent truth conditional meaning,
both sentences get translated into predicate logic in
the same way (where *Like* = `really likes', *j* =
`Jill', and *k* = `Jack'):

iii. Like(j,k)

a. grandfather of

This relation is not symmetric, transitive, or reflexive,
nor does it have a true converse. This last point is
tricky: although we at first glance want to say that
the *grandchild of* relation is the converse of
*grandfather of*, this isn't exactly true. Although
it is true that if Pat is the grandfather of Mabel, then
Mabel is the grandchild of Mort, it's not necessarly
the case that if Mabel is the grandchild of Pat, Pat is
the grandfather of Mabel: Pat *could* in principle
be Mabel's grandmother.

b. have the same color as:

This relation is clearly symmetric and transitive:
my car has the same color as my bike if and only if my bike has
the same color as my car; and if my bike has the same color
as my wallet, then my car also has the same color as my
wallet. This relation is also reflexive: for anything
that we can apply it to, it is going to be true that
that thing has the same color as itself. The condition
``anything we can apply the relation to'' is crucial here:
in order to ``have the same color'' as something else, an
object must be monochrome (this is the difference between
*have the same color as* and *have the same colors
as*. So although we might not want to say that a
multicolored object has the same color as itself, this
doesn't matter: a multicolored object doesn't have the
same color as anything, so it's not the type of object that
we can apply the relation to in the first place.

c. sit next to

This relation *appears* to be symmetric (but see the
comments below), but it's not transitive or reflexive,
and it doesn't have a converse. It certainly
seems to be the case that if Jack is sitting next to
Jill, then Jill is sitting next to Jack (and so on for any
pair of objects in the relation), indicating symmetry.
However, it doesn't follow that
if Jill is sitting next to Pat, then Jack is sitting next
to Pat as well, however. While this would be the case if
they were sitting in a circle, it doesn't have to hold,
so transitivity isn't satisfied. Since no one can sit
next to him/herself, the relation is not reflexive (in fact,
it's *irreflexive*.

Although I characterized *sit next to* as symmetric,
and although it is probably typically used in a symmetric
way, it's actually false that this relation is, in a strict
sense, symmetric. The problem is that we often take *sit
next to* to mean something more general: *be next to*.
But while the latter is clearly symmetric, the former actually
isn't: the fact that Fred is sitting next to Frank doesn't
entail that Frank is sitting next to Fred, because Frank could
be standing up!

d. west of

This relation is transitive, but not reflexive or symmetric
(though see below).
If Chicago is west of New York, and LA is west of Chicago, then
it follows that LA is west of New York (and the same goes for
any three things in the relation). Nothing is west of itself,
however, and for any two things, if the first is west of the
other, then the second isn't west of the first. The second
is *east of* the first, however, thus *east of* is
the converse of *west of*.

The claim about nonsymmtry (technically ``antisymmetry'')
holds only if we're considering points in a plane, however.
If our model is such that points are on a sphere, then *west
of* __is__ actually symmetric: if you keep going west
from Chicago (following a certain route), you will end up in
New York again. So in this type of model, *west of*
would in fact be symmetric (and reflexive, as well; see
the comments on *differ from* below).

e. differ from

This relation is symmetric, but not transitive or reflexive. If the Purple Line L differs from the Red Line, then it follows that the Red Line differs from the Purple Line. The relation isn't transitive, though; if it were, then the fact that the Purple Line differs from the Red Line, and the Red Line differs from the Purple Line, would require the Purple Line to differ from itself, which isn't the case. This brings up an important point: if a relation is both symmetric and transitive, then it also has to be reflexive. This is whyf. earlier than

*Earlier than* is transitive, but not reflexive or symmetric.
If my dentist appointment is earlier than my dinner date, and
my dinner date is earlier than the movie I want to see, it follows
that my dentist appointment is earlier than the movie I want to see.
Of course, nothing can be earlier than itself (so no reflexivity),
and if some thing is earlier than some other thing, the reverse
clearly can't hold as well. Time, as they say, is transitive....

g. be married to

Symmetric, but not transitive or reflexive. Symmetry should be clear, as should the absence of reflexivity. Consider transitivity, however. It's possible that John could be married to Mary, and then could disappear and be presumed dead. Mary might then marry Steve. Even if John later on reappears from whereever he has been hiding, it clearly doesn't follow that John is married to Steve.

h. guest-host

If we take these to be relations, then they are the converses of one another: if x is the guest of y, then y is the host of x (and vice-versa). When we take them to be one-place predicates, however (x is a guest/host), they are not converses, because only n-place relations (n > 1) can have converses.

i. teacher-student

The analysis is the same as for *guest* and *host*.

a. ancestor of/mother of

The crucial difference between the *ancestor of* relation
and the *mother of* relation is that the former is transitive
but the latter is not. For any three objects x, y, and z,
if x is the ancestor of y, and y is the ancestor of z, then x is
also the ancestor of z. However, if x is the mother of y, and
y is the mother of z, then x is definitely not the mother of z
(though x is the *grandmother of*z.

An additional relation between the two relations is that the
second entails the first (but not vice-versa), if we take
*mother of* to mean *biological mother of*: for any pair of
objects, if x is the mother of y, then x is also the ancestor
y. Of course, if we apply *mother of* to cases of adoption
(as we typically do in normal usage), this entailment does not
go through.

b. be older than/be one year older than

Again, the crucial difference is transitivity:There is also an entailment relation here, regardless of
how we interpret *(one year) older than*: if x is
one year older than y, then x is older than y (but not the
other way around).

c. sister of/sibling of

Both relations are transitive, and neither is reflexive, but only the latter is symmetric: if x is a sibling of y, then y is a sibling of x (and vice-versa). It does necessarily follow that if x is the sister of y then y is the sister of x, however: x could by y's brother.

These relations also stand in an entailment relation, but
it is the reverse of the previous two: *sister of* entails
*sibling of*, but *sibling of* does not entail
*sister of* (for the same reason that *sister of* is
not symmetric).

d. be taller than/be as tall as

This pair is tricky. It's clear that both are transitive:
for any objects x, y, and z, if x is taller than y, and y is
taller than z, then x is taller than z; and if x is as tall as
y, and y is as tall as z, then x is as tall as z. It's also
clear that *taller than* is neither reflexive nor symmetric.
Finally, it seems pretty clear that *as tall as* is reflexive:
everything is as tall as itself.

Questions arise when we ask the question ``is *as tall as*
symmetric?'', however. The complexity stems from the truth conditional
interpretation this relation. If we take a sentence like ``Kim is as
tall as Lee'' to mean ``Kim is __exactly__ as tall as Lee'',
then the relation is clearly symmetric: if x is exactly as tall as
y, then y is exactly as tall as x. It's not so clear that this
is the right analysis of the truth conditions of *as tall as*,
however. If it were, then the sentence in (1) should be
contradictory.

1. Kim is as tall as Lee; in fact, Kim is even taller than Lee.

This doesn't seem to be the case, however. What seems to be the
case is that the truth conditions for *as tall as* should be
something more like ``__at least__ as tall as'', and that the
``exactly'' interpretation arises as an implicature. Evidence in
support of this comes from examples like (2), which would be
redundant if *as tall as* meant (truth conditionally) *exactly
as tall as*, and from the fact that (3), unlike (1), __is__
quite clearly contradictory.

2. Kim is exactly as tall as Lee.

3. Kim is exactly as tall as Lee; in fact, Kim is even taller than
Lee.

So, if *as tall as* means *at least as tall is*, it
must not be symmetric, because it could be true, for example, that
Kim is as tall as Lee, but false that Lee is as tall as Kim. This
would be the case in a context (a model) in which Kim is taller
than Lee.

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Chris Kennedy Last modified: Tue Feb 2 18:33:23 CST