564 Lecture 11 Sept. 30, 1999

Gondy's if-then statement:

"No one answered any question that everyone attempted"

ÂxÂy((Question(x)&Person(y)&Attempted(x,y)) --> ~¤z(Person(z) & Answered(z,x)))

ÂxÂy((Question(x)&Person(y)) --> (Attempted(x,y) --> ~¤ z(Person(z) & Answered(z,x))))

(P&Q) --> R ¿<=>? P-->(Q-->R)

2 The shortcomings of predicate calculus

We've already seen the problem that we can't map the predicate logical formulas onto the syntactic structure for English trees in any straightforward way. This means that our stated goal of compositionality -- having the syntax and the semantics work hand-in-hand, with a transparent correspondence between the two -- is not met by the predicate calculus.

Here's some more problems. These have to do with the fact that first-order predicate logic is so-called because the only constants and variables it contains are constants and variables representing individual entities.

1. English(indeed, any natural language) can refer to things besides individuals

a. Susan is healthy.

b. Swimming is healthy.

c. John has all the properties of Santa Claus.

d. Red has all the properties of green.

"healthy" is a predicate that picks out a set of individuals, so "Healthy(s)" is a perfectly good predicate-logical formula. However, what if we want to apply something like "healthy" to an activity? 1b indicates that the activity of swimming is healthy, but we don't have terms which can represent activities, only ones for individuals. The word "property" is quantified over in 1c, but we don't have any terms that can stand for properties. And finally, in 1d, we're talking about properties of properties, another step away from individuals. We need a logical language that will let us talk about all of these things – essentially, we need more types.

Now, deSwart's point is clear, although to a certain extent there are ways around most of her specific examples within the bounds of predicate logic. Consider (1c). Here's a possible version of 1c in predicate logic:

2. c. Âx((Property(x)& Have(sc, x)) --> Have(j, x))

Because individuals can be abstract entities, it seems to me that quantifying over properties in first-order logic isn't so much a problem. However, this means that something like "generosity" has to be an individual, not a predicate -- so something like "Generous(j)" wouldn't be a good representation of "John is generous", but rather "John has generosity" (Have(j,g)). Essentially, all things that we've been representing as predicates would have to be properties. What about something like "Santa Claus is married to Mrs. Claus." If "being married to Mrs. Claus" is a property of Santa, we would have to represent "being married to Mrs. Claus" as an individual, x, and say that Santa has that property (Have(sc,mmrsclause)). This quickly gets silly, and uninformative, besides.

For the point 1b, it might just be that the surface syntactic representation isn't exactly what it looks like. For instance, what if the syntax were really something like

3. b. [[PRO_i Swimming] [is [healthy for PRO_i]]

where PRO stands for an invisible pronoun, like "one". In that case, we could say that what this sentence really means is that if one swims, then one is healthy, or something like:

4. Âx(Swim(x)-->Healthy(x))

and we wouldn't have done any violence to first-order predicate calculus. However, we're still subject to the same compositionality problems -- how can we get this formula out of this syntax? And get a formula that looks identical out of the different syntax of "Cats are felines" ?

More seriously, what about situations like the following:

5. a. There was a red book on the table.

b. There was a small elephant at the zoo.

Now, the relationship between "red" and "book" is different from the relationship between "small" and "elephant". So, the predicate-notations in 6 are truthful, while those in 7 are not:

6. a. {x | x is a red book}is a subset of {x | x is a book}

b. {x | x is a red book}is a subset of {x | x is red}

7. {x | x is a small elephant}is a subset of {x | x is an elephant}

##{x | x is a small elephant}is a subset of {x | x is small}

The meaning of "small" depends on the scale of the set it's predicated of (the set of elephants, here), and is not independently defined. These are called "relative" adjectives, as opposed to "absolute" adjectives like "red". (They'd pose a real problem for a theory like that outlined above, where properties are individuals; it might be alright to talk about something "having" redness, but it'd be nonsensical to talk about something "having" smallness).

Similarly, "Milly swam slowly" is relative to either Milly's ability to swim, or the average speed of swimming humans in general.

8. "Milly swam as fast as she could, while the dolphin obligingly

swam slowly so she could keep up."

Finally, in the modification problem section, we have words like "very" or "extremely" (or any adverb that itself can modify an adjective or an adverb): we need to be able to modify modifiers, indicating the degree to which the modifier applies (even when the modifier itself is applying relativistically).

The solution to these problems involves introducing other types of constants into the theory: constants that can refer to sets, and to sets of sets, and so on. The introduction of these elements into the theory is the introduction of new types of entities, hence going beyond the level of first-order predicate logic is entering the realm of type theory.

This move will solve problems with words like "small" and "very", as well as with quantifiers that we haven't been able to deal with yet, like "three" (in "Three cats") or "most" in "Most students"). However, there's another class of items which we haven't been able to deal with which are exemplified below:

9. a. The former athlete marveled at how far he had fallen.

b. The alleged killer was seen in San Jose.

Consider the subset relation in these cases:

10. a. #{x | x is a former athlete}is a subset of {x | x is an athlete}

b. #{x | x is an alleged killer}is a subset of {x | x is a killer}

c. #{x | x is a former athlete}is a subset of {x | x is former}

d. #{x | x is an alleged killer}is a subset of {x | x is alleged}

As in the cases with "small" above, the subset problem arises when you try to talk about the set denoted by the adjective, but not only that, you run into problems when you try to talk about the subset relation with respect to the head noun as well! In order to resolve these problems, semanticists turn to the notion of intensionality – you have to be able to talk about the concept of being an athlete or a killer separately from the denotation of the predicates, the set of killers or the set of athletes. Similarly, the notion of intensionality allows us to resolve the problem illustrated in 11:

11. c. Sue believes she met Greta Gustafsson/Greta Garbo.

If you're an old movie buff, you'll know that Greta Gustafsson and Greta Garbo were the same person. However, Sue could believe that she met one without believing that she met the other. Unfortunately, the theory we've developed so far doesn't really allow for the difference between sense and reference to come into play, which it clearly needs to do here, since so far we've talked about proper names as if they were simply reference to individuals.

And finally, we've got the problem alluded to above: predicate calculus only allows for two quantifiers, when natural language has tons more, and they can be compositionally combined:

12. Exactly two students laughed.

Most students laughed.

Many student laughed.

A few students laughed (*if any).

Few students laughed (if any).

Consider deSwart's attempt at representing "most" as a first-order predicate logical quantifier:

13. MxPx is true iff for most individuals x in the universe of discourse U it is true that x has the property P.

This doesn't work. Consider what would happen if we tried to translate 14a as 14b or c.

14. a. Most students laughed.

b. Mx(Student(x) & laugh(x))

c. Mx(Student(x) --> laugh(x))

The problem is that we can't talk about "most students" as meaning "most individuals are students", which is what first-order calculus would necessitate. First order calculus can only talk about individuals, not the set of individuals denoted by a particular predicate, which is obviously what is needed. Again, we've got the problem of relativity here: "most" needs to be defined with respect to the set of students (just like "small" does). Checking every individual for some assignment in the universe of discourse isn't going to cut it; what we need to do is say, take the set of students and check each individual in that set. But the way our assignment functions etc. work right now, that can't happen. So, ultimately, we need to move beyond a first-order theory. This will also help solve our compositionality problem, too.