. 16
( 107 .)


Parentheses must be used whenever ambiguity would result from their
omission. In practice, this means that conjunctions and disjunctions must
be “wrapped” in parentheses whenever combined by means of some other

You try it
1. Let™s try our hand at evaluating some sentences built up from atomic
sentences using all three connectives §, ∨, ¬. Open Boole™s Sentences and
Wittgenstein™s World. If you changed the size or shape of f while doing
Exercises 3.6 and 3.9, make sure that you change it back to a large tetra-

Chapter 3
Ambiguity and parentheses / 81

2. Evaluate each sentence in the ¬le and check your assessment. If your as-
sessment is wrong, play the game to see why. Don™t go from one sentence
to the next until you understand why it has the truth value it does.

3. Do you see the importance of parentheses? After you understand all the
sentences, go back and see which of the false sentences you can make true
just by adding, deleting, or moving parentheses, but without making any
other changes. Save your ¬le as Sentences Ambiguity 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Congratulations


To really master a new language, you have to use it, not just read about it. The exercises and problems
that follow are intended to let you do just that.

3.12 If you skipped the You try it section, go back and do it now. Submit the ¬le Sentences
‚ Ambiguity 1.

3.13 (Building a world) Open Schr¨der™s Sentences. Build a single world where all the sentences
‚ in this ¬le are true. As you work through the sentences, you will ¬nd yourself successively
modifying the world. Whenever you make a change in the world, be careful that you don™t
make one of your earlier sentences false. When you are ¬nished, verify that all the sentences
are really true.

3.14 3.15
(Parentheses) Show that the sentence (More parentheses) Show that
‚ ‚
¬(Small(a) ∨ Small(b)) Cube(a) § (Cube(b) ∨ Cube(c))

is not a consequence of the sentence is not a consequence of the sentence

¬Small(a) ∨ Small(b) (Cube(a) § Cube(b)) ∨ Cube(c)

You will do this by submitting a coun- You will do this by submitting a coun-
terexample world in which the second terexample world in which the second
sentence is true but the ¬rst sentence is sentence is true but the ¬rst sentence is
false. false.
3.16 (DeMorgan Equivalences) Open the ¬le DeMorgan™s Sentences. Construct a world where all the
‚ odd numbered sentences are true. Notice that no matter how you do this, the even numbered
sentences also come out true. Submit this as World 3.16.1. Next build a world where all the
odd numbered sentences are false. Notice that no matter how you do it, the even numbered
sentences also come out false. Submit this as World 3.16.2.

Section 3.5
82 / The Boolean Connectives

3.17 In Exercise 3.16, you noticed an important fact about the relation between the even and odd
 numbered sentences in DeMorgan™s Sentences. Try to explain why each even numbered sentence
always has the same truth value as the odd numbered sentence that precedes it.

Section 3.6
Equivalent ways of saying things
Every language has many ways of saying the same thing. This is particularly
true of English, which has absorbed a remarkable number of words from other
languages in the course of its history. But in any language, speakers always
have a choice of many synonymous ways of getting across their point. The
world would be a boring place if there were just one way to make a given
Fol is no exception, even though it is far less rich in its expressive capaci-
ties than English. In the blocks language, for example, none of our predicates
is synonymous with another predicate, though it is obvious that we could
do without many of them without cutting down on the claims expressible in
the language. For instance, we could get by without the predicate RightOf by
expressing everything we need to say in terms of the predicate LeftOf, sys-
tematically reversing the order of the names to get equivalent claims. This is
not to say that RightOf means the same thing as LeftOf”it obviously does
not”but just that the blocks language o¬ers us a simple way to construct
equivalent claims using these predicates. In the exercises at the end of this
section, we explore a number of equivalences made possible by the predicates
of the blocks language.
Some versions of fol are more parsimonious with their basic predicates
than the blocks language, and so may not provide equivalent ways of express-
ing atomic claims. But even these languages cannot avoid multiple ways of
expressing more complex claims. For example, P § Q and Q § P express the
same claim in any ¬rst-order language. More interesting, because of the su-
per¬cial di¬erences in form, are the equivalences illustrated in Exercise 3.16,
known as DeMorgan™s laws. The ¬rst of DeMorgan™s laws tells us that the
DeMorgan™s laws
negation of a conjunction, ¬(P § Q), is logically equivalent to the disjunction
of the negations of the original conjuncts: ¬P ∨ ¬Q. The other tells us that
the negation of a disjunction, ¬(P ∨ Q), is equivalent to the conjunction of
the negations of the original disjuncts: ¬P § ¬Q. These laws are simple con-
sequences of the meanings of the Boolean connectives. Writing S1 ” S2 to
indicate that S1 and S2 are logically equivalent, we can express DeMorgan™s

Chapter 3
Equivalent ways of saying things / 83

laws in the following way:

¬(P § Q) ” (¬P ∨ ¬Q)
¬(P ∨ Q) ” (¬P § ¬Q)

There are many other equivalences that arise from the meanings of the
Boolean connectives. Perhaps the simplest is known as the principle of double double negation
negation. Double negation says that a sentence of the form ¬¬P is equivalent
to the sentence P. We will systematically discuss these and other equiva-
lences in the next chapter. In the meantime, we simply note these important
equivalences before going on. Recognizing that there is more than one way of
expressing a claim is essential before we tackle complicated claims involving
the Boolean connectives.


(Double negation and DeMorgan™s Laws) For any sentences P and Q:

1. Double negation: ¬¬P ” P

2. DeMorgan: ¬(P § Q) ” (¬P ∨ ¬Q)

3. DeMorgan: ¬(P ∨ Q) ” (¬P § ¬Q)


3.18 (Equivalences in the blocks language) In the blocks language used in Tarski™s World there are
‚ a number of equivalent ways of expressing some of the predicates. Open Bernays™ Sentences.
You will ¬nd a list of atomic sentences, where every other sentence is left blank. In each blank,
write a sentence that is equivalent to the sentence above it, but does not use the predicate
used in that sentence. (In doing this, you may presuppose any general facts about Tarski™s
World, for example that blocks come in only three shapes.) If your answers are correct, the odd
numbered sentences will have the same truth values as the even numbered sentences in every
world. Check that they do in Ackermann™s World, Bolzano™s World, Boole™s World, and Leibniz™s
World. Submit the modi¬ed sentence ¬le as Sentences 3.18.

3.19 (Equivalences in English) There are also equivalent ways of expressing predicates in English.
 For each of the following sentences of fol, ¬nd an atomic sentence in English that expresses
the same thing. For example, the sentence Man(max) § ¬Married(max) could be expressed in

Section 3.6
84 / The Boolean Connectives

English by means of the atomic sentence Max is a bachelor.
1. FatherOf(chris, alex) ∨ MotherOf(chris, alex)
2. BrotherOf(chris, alex) ∨ SisterOf(chris, alex)
3. Human(chris) § Adult(chris) § ¬Woman(chris)
4. Number(4) § ¬Odd(4)
5. Person(chris) § ¬Odd(chris)
6. mother(mother(alex)) = mary ∨ mother(father(alex)) = mary [Notice that mother and
father are function symbols. If you did not cover Section 1.5, you may skip this sen-

Section 3.7

An important skill that you will want to master is that of translating from
English to fol, and vice versa. But before you can do that, you need to know
how to express yourself in both languages. The problems below are designed
to help you learn these related skills.
How do we know if a translation is correct? Intuitively, a correct translation
correct translation
is a sentence with the same meaning as the one being translated. But what
is the meaning? Fol ¬nesses this question, settling for “truth conditions.”
What we require of a correct translation in fol is that it be true in the same
circumstances as the original sentence. If two sentences are true in exactly
the same circumstances, we say that they have the same truth conditions. For
truth conditions
sentences of Tarski™s World, this boils down to being true in the very same
Note that it is not su¬cient that the two sentences have the same truth
value in some particular world. If that were so, then any true sentence of
English could be translated by any true sentence of fol. So, for example,
if Claire and Max are both at home, we could translate Max is at home by
means of Home(claire). No, having the same actual truth value is not enough.
They have to have the same truth values in all circumstances.


In order for an fol sentence to be a good translation of an English sen-
tence, it is su¬cient that the two sentences have the same truth values
in all possible circumstances, that is, that they have the same truth con-

Chapter 3
Translation / 85

In general, this is all we require of translations into and out of fol. Thus,
given an English sentence S and a good fol translation of it, say S, any other
sentence S that is equivalent to S will also count as an acceptable translation
of it, since S and S have the same truth conditions. But there is a matter of
style. Some good translations are better than others. You want sentences that
are easy to understand. But you also want to keep the fol connectives close
to the English, if possible.
For example, a good translation of It is not true that Claire and Max are
both at home would be given by

¬(Home(claire) § Home(max))

This is equivalent to the following sentence (by the ¬rst DeMorgan law), so
we count it too as an acceptable translation:

¬Home(claire) ∨ ¬Home(max)

But there is a clear stylistic sense in which the ¬rst is a better translation, since
it conforms more closely to the form of the original. There are no hard and
fast rules for determining which among several logically equivalent sentences
is the best translation of a given sentence.
Many stylistic features of English have nothing to do with the truth con-
ditions of a sentence, and simply can™t be captured in an fol translation. For
example, consider the English sentence Pris is hungry but Carl is not. This
sentence tells us two things, that Pris is hungry and that Carl is not hungry.
So it would be translated into fol as

Hungry(pris) § ¬Hungry(carl)

When it comes to truth conditions, but expresses the same truth function
as and. Yet it is clear that but carries an additional suggestion that and does but, however, yet,
not, namely, that the listener may ¬nd the sentence following the but a bit sur-
prising, given the expectations raised by the sentence preceding it. The words
but, however, yet, nonetheless, and so forth, all express ordinary conjunction,
and so are translated into fol using §. The fact that they also communicate
a sense of unexpectedness is just lost in the translation. Fol, as much as we
love it, sometimes sacri¬ces style for clarity.
In Exercise 21, sentences 1, 8, and 10, you will discover an important
function that the English phrases either. . . or and both. . . and sometimes play. either. . . or, both. . . and
Either helps disambiguate the following or by indicating how far to the left
its scope extends; similarly both indicates how far to the left the following
and extends. For example, Either Max is home and Claire is home or Carl


. 16
( 107 .)