# Translation practice

Key to symbols
~ = negation
& = conjunction
--> = conditional implication
E = existential quantifier
A = universal quantifier

1. There is no free lunch.
~Ex[F(x) & L(x)]
F(x) = x is free
L(x) = x is lunch

2. All sin is a form of lying.
Ax[S(x) --> L(x)]
S(x) = x is a sin
L(x) = x is a form of lying

3. If a thing is worth doing, it is worth doing badly.
Ax[W(x) --> B(x)]
W(x) = x is worth doing
B(x) = x is worth doing badly

Note that we don't have any way of representing adverbial modification---we have to introduce a new predicate ``worth doing badly'', even though there is an obvious semantic connection between this predicate and the predicate ``worth doing''.

4. All that glitters is not gold
a. ~Ax[G(x) --> A(x)] OR
b. Ax[G(x) --> ~A(x)]
G(x) = x glitters
A(x) = x is gold

This sentence is AMBIGUOUS: it can either mean that not everything that glitters is gold (the ``standard'' interpretation of the saying), or that none of the things that glitter are gold. These two interpretations correspond to the logical representations in (a) and (b), respectively, which differ in the relative scope of the universal quantifier and negation.

Note that the interpretation in (a), in which negation has ``wide scope'' with respect to the universal quantifier (its scope includes the universal quantifier), less directly mirrors the syntactic structure than the interpretation in (b), in which it has ``narrow scope'' with respect to the quantifier. This means that somewhere between the mapping from syntax to semantics, some operations must come into play that ``shift'' the relative ordering of constituents. Figuring out what these operations are and how they are constrained is one of the primary goals of work at the syntax-semantics interface.

5. All that is gold does not glitter.
a. ~Ax[A(x) --> G(x)] OR
b. Ax[A(x) --> ~G(x)]
G(x) = x glitters
A(x) = x is gold

Again, the English sentence is ambiguous, though the ``wide scope negation'' interpretation (a) seems a bit harder to get in this case. To convince yourself that such readings are in fact possible (and common), just keep your eyes open when you're reading the newspaper---these things show up all the time. Here's an example from the Tribune that I came across during the fall: ``In Washington, the rule is that all senators don't have to advise and consent, even though the Constitution says they do.'' It was quite clear from the context of the article that this sentence was not claiming that no senators have to advise and consent; rather, it was claiming that not all senators have to advise and consent.

6. Sometimes a cigar is just a cigar.

This example is extremely tricky. The message that it conveys is that while cigars may often be more than just cigars (phallic symbols or whatever), this isn't always true: sometimes they're nothing other that what they appear to be. As far as the cigars go, it's clear that we have universal quantification here: this sentence is making a claim about all cigars (or cigar-like things). Unfortunately, the only way that we can capture all this extra meaning in the system we have developed so far is to do so by brute force, and in a way that completely ignores the contribution of sometimes, as shown in the following logical representation:

Ax[C(x) --> ~O(x)]
C(x) = x is a cigar
O(x) = x is something other than a cigar

This formula says that ``everything that is a cigar is nothing other than a cigar'', which is an almost accurate paraphrase of the sentence we started with.

An alternative strategy would be to complicate our model just a bit so that we can deal with expressions like ``sometimes'' (obviously, this is something we need to do anyway, since we know what ``sometimes'' means). In other words, we have to add a new class of objects into our domain: instants of time. Once we do this, we can treat instants of time the same that we treat other objects in the domain. In particular, we can introduce relations between objects and instants of time and we can quantify over them. This gives us the option of characterizing the meaning of this sentence as below.

EtAy[C(y,t) --> ~O(y,t)]
C(x,t) = x is a cigar at time t

In this logical representation, ``C'' and ``O'' denote relations between objects and times such that e.g. C(a,t) is true if and only if a is a cigar at time t. The claim that's being made here, then, is something like this: ``There are times when every cigar is nothing other than a cigar'' (which in turn gives rise to the very accurate implicature that there are also times when every cigar is something other than a cigar...).

The important point is that the formal system we have devised allows us to do things like introduce quantification over instants of time, as long as we make explicit what the interpretations of the new expressions we've introduce are, with respect to our extended model (in which the domain of discourse includes both objects and instants of time).

FOOTNOTE: Some of you may wonder if we're cheating by introducing the predicate O (``something other than a cigar'') into the truth conditions --- after all, it's not part of the English sentence. If you are wondering this, then you're right: this is cheating. As an exercise, figure out whether you could get essentially the same results by using a more direct translation (i.e., one that literally translates the English into predicate logic), and taking pragmatic information into account. HINT: What kinds of truth conditions would you get if you translated (6) literally?

7. He who can, does. He who cannot, teaches.
AxAy[C(x,y) --> D(x,y)] & AxAy[~C(x,y) --> T(x,y)]
C(x,y) = x can do y
D(x,y) = x does y
T(x,y) = x teaches y

There are a couple of interesting things about this example. First, note that the pronouns are actually interpreted as variables bound by some kind of universal quantification. (To be more precise, this is probably a kind of ``generic'' quantification, which would differ from universal quantification in allowing exceptions. But for now, universal quantification is the closest approximation we can get.) Second, the predicates in these examples are ``implictly'' two place predicates, where the second argument is also bound by a universal quantifier. So the best ``logical paraphrase'' of this example is something like ``for every pair of things x and y, if x can do y, then x does y; and for every pair of things x and y, if x can't do y, then x teaches y.'' This characterization of the meaning of the saying is captured by the logical representation above. What the logical reprsentation loses, however, is the connection between the predicates ``can do'' and ``does'', which are obviously very close in meaning.

8. The only completely consistent people are the dead.
Ax[(C(x) & P(x)) --> D(x)]
C(x) = x is completely consistent
P(x) = x is a person

For an additional exercise, try translaint Only completely consistent people are dead. Does this sentence have the same truth conditions as (8)?

9. You have to be over 30 to enjoy Proust.
Ax[Ey[P(y) & E(x,y)] --> A(x,30)]
E(x,y) = x enjoys y
A(x,y) = x is at least as old as y
P(x) = x is one of Proust's writings
30 = 30 years of age [a ``degree of age'' that represents 30 years]

This meaning of this sentence could have been symbolized in a simpler way, by introducing a predicate like ``T(x)'' meaning ``x is 30 years old'', and by allowing the constant ``p'' to stand for the writings of Proust, but the more complex symbolization I used here makes a couple of points, one which illustrates a problem with the system and one that represents a positive aspect.

First, if we do introduce existential quantification over Proust's writings, as I have done here, we have to say that this quantifier can only have narrow scope with respect to the universal: this sentence doesn't really have an interpretation in which it's talking about some specific writing (e.g., Remembrance of Things Past). This is a fact about the English sentence, but it's not expected if we represent its meaning with existential quantification---in principle, we would expect a wide scope existential interpretation as well. This is a problem with our system as its set up.

Second, characterizing ``over 30'' in terms of the two place predicate ``x is at least as old as y'' shows that we can introduce new types of objects into our domain if we want: ``x is as old as y'' is not a relation between two objects, but rather a relation between an object and some kind of abstract ``amount'' (which I've represented with the individual constant ``30''). Expanding the system in this way is not a problem---in fact, it's something that we clearly want to be able to do, since natural language seems to make reference to things like ``amounts'', as well as actual objects in the world. All that is necessary, in order to construct a coherent semantics, is to make sure that we specify exactly what types of objects our model contains and that we give the appropriate interpretations for expressions of logic that refer to (or describe relations between) those objects.

10. No one but a blockhead writes except for money.
~Ex[~B(x) & (W(x) & ~M(x))]
B(x) = x is a blockhead
W(x) = x writes
M(x) = x writes for money

This example is interesting because it illustrates the semantic connection between expressions like ``but'' (when attached to a quantifier) and ``except'' and negation: in effect, this sentence says that ``no one who is not a blockhead both writes and doesn't write for money''. Another example that illustrates this point is (i):

i. Every boy except/but Frank did the rhumba.

The best representation of the truth conditions of this sentence is (ii), where =(x,y) means ``x has the same interpretation as y'', B(x) means ``x is a boy'', R(x) means ``x did the rhumba'', and f corresponds to Frank:

ii. Ax[(B(x) & ~=(x,f)) --> R(x)]

Here, ``except'' or ``but'' gets interpreted as a restriction on what the value of the variable x can't be: specifically, it can't correspond to Frank. Examples like this show that we can make such restrictions explicit by using ``except'', ``but'', or other expressions like ``save'' (everyone save Frank; this is a bit archaic, of course); what's more interesting is that in normal discourse, such restrictions are often implicit. Consider a sentence like (iii):

iii. Everyone is taller than Frank.

If the logical representation of this sentence is (iv), then it would end up contradictory, since it could never be the case that everything in the domain is taller than Frank: Frank isn't taller than Frank! (This point was brought up by Carolyn Gottfurcht before class the other day; we'll talk about it more next week.)

iv. Ax[T(x,f)]
T(x,y) = x is taller than y
f = Frank

What we really want the logical representation of this sentence to be is something like (v), which correponds to ``everyone except Frank is taller than Frank''.

v. Ax[~=(x,f) --> T(x,f)]

The important thing here is that this information (the ``except Frank'' part), while not part of the utterance, is part of the truth conditions of the sentence. What we need to add to our system, then, is some means of bringing contextual information into truth conditional representations of meaning.