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If we wanted to express the exclusive sense of or in the above example, we
could do it as follows:

[Home(john) ∨ Home(mary)] § ¬[Home(john) § Home(mary)]

As you can see, this sentence says that John or Mary is home, but it is not
the case that they are both home.
Many students are tempted to say that the English expression either . . . or
expresses exclusive disjunction. While this is sometimes the case (and indeed
the simple or is often used exclusively), it isn™t always. For example, suppose
Pris and Scru¬y are in the next room and the sound of a cat ¬ght suddenly
breaks out. If we say Either Pris bit Scru¬y or Scru¬y bit Pris, we would not
be wrong if each had bit the other. So this would be translated as

Bit(pris, scru¬y) ∨ Bit(scru¬y, pris)

We will see later that the expression either sometimes plays a di¬erent logical
Another important English expression that we can capture without intro-
ducing additional symbols is neither. . . nor. Thus Neither John nor Mary is
at home would be expressed as:

¬(Home(john) ∨ Home(mary))

This says that it™s not the case that at least one of them is at home, i.e., that
neither of them is home.

Semantics and the game rule for ∨
Given any two sentences P and Q of fol, atomic or not, we can combine them
using ∨ to form a new sentence P ∨ Q. The sentence P ∨ Q is true if at least
one of P or Q is true. Otherwise it is false. Here is the truth table.

true true true
truth table for ∨
true false true
false true true
false false false

The game rules for ∨ are the “duals” of those for §. If you commit yourself game rule for ∨
to the truth of P ∨ Q, then Tarski™s World will make you live up to this by
committing yourself to the truth of one or the other. If you commit yourself to
the falsity of P ∨ Q, then you are implicitly committing yourself to the falsity

Section 3.3
76 / The Boolean Connectives

of each, so Tarski™s World will choose one and hold you to the commitment
that it is false. (Tarski™s World will, of course, try to win by picking a true
one, if it can.)

You try it
1. Open the ¬le Ackermann™s World. Start a new sentence ¬le and enter the

Cube(c) ∨ ¬(Cube(a) ∨ Cube(b))

Make sure you get the parentheses right!
2. Play the game committed (mistakenly) to this sentence being true. Since
the sentence is a disjunction, and you are committed to true, you will
be asked to pick a disjunct that you think is true. Since the ¬rst one is
obviously false, pick the second.
3. You now ¬nd yourself committed to the falsity of a (true) disjunction.
Hence you are committed to the falsity of each disjunct. Tarski™s World
will then point out that you are committed to the falsity of Cube(b). But
this is clearly wrong, since b is a cube. Continue until Tarski™s World says
you have lost.
4. Play the game again, this time committed to the falsity of the sentence.
You should be able to win the game this time. If you don™t, back up and
try again.
5. Save your sentence ¬le as Sentences Game 2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Congratulations


1. If P and Q are sentences of fol, then so is P ∨ Q.

2. The sentence P ∨ Q is true if and only if P is true or Q is true (or both
are true).


3.8 If you skipped the You try it section, go back and do it now. You™ll be glad you did. Well,
‚ maybe. Submit the ¬le Sentences Game 2.

Chapter 3
Remarks about the game / 77

3.9 Open Wittgenstein™s World and the sentence ¬le Sentences 3.6 that you created for Exercise 3.6.
‚ Edit the sentences by replacing § by ∨ throughout, saving the edited list as Sentences 3.9.
Once you have changed these sentences, decide which you think are true. Again, record your
evaluations to help you remember them. Then go through and use Tarski™s World to evaluate
your assessment. Whenever you are wrong, play the game to see where you went wrong. If you
are never wrong, then play the game anyway a couple times, knowing that you should win. As
in Exercise 3.6, ¬nd the maximum number of sentences you can make true by changing the
size or shape (or both) of block f . Submit both your sentences and world.

3.10 Open Ramsey™s World and start a new sentence ¬le. Type the following four sentences into the
‚ ¬le:
1. Between(a, b, c) ∨ Between(b, a, c)
2. FrontOf(a, b) ∨ FrontOf(c, b)
3. ¬SameRow(b, c) ∨ LeftOf(b, a)
4. RightOf(b, a) ∨ Tet(a)

Assess each of these sentences in Ramsey™s World and check your assessment. Then make a single
change to the world that makes all four of the sentences come out false. Save the modi¬ed world
as World 3.10. Submit both ¬les.

Section 3.4
Remarks about the game

We summarize the game rules for the three connectives, ¬, §, and ∨, in
Table 3.1. The ¬rst column indicates the form of the sentence in question,
and the second indicates your current commitment, true or false. Which
player moves depends on this commitment, as shown in the third column.
The goal of that player™s move is indicated in the ¬nal column. Notice that commitment and rules
although the player to move depends on the commitment, the goal of that
move does not depend on the commitment. You can see why this is so by
thinking about the ¬rst row of the table, the one for P ∨ Q. When you are
committed to true, it is clear that your goal should be to choose a true
disjunct. But when you are committed to false, Tarski™s World is committed
to true, and so also has the same goal of choosing a true disjunct.
There is one somewhat subtle point that should be made about our way of
describing the game. We have said, for example, that when you are committed
to the truth of the disjunction P ∨ Q, you are committed to the truth of one
of the disjuncts. This of course is true, but does not mean you necessarily
know which of P or Q is true. For example, if you have a sentence of the form

Section 3.4
78 / The Boolean Connectives

Table 3.1: Game rules for §, ∨, and ¬

Form Your commitment Player to move Goal

you Choose one of
P∨Q P, Q that
Tarski™s World is true.

Tarski™s World Choose one of
P§Q P, Q that
you is false.

Replace ¬P
¬P either ” by P and

P ∨ ¬P, then you know that it is true, no matter how the world is. After all,
if P is not true, then ¬P will be true, and vice versa; in either event P ∨ ¬P
will be true. But if P is quite complex, or if you have imperfect information
about the world, you may not know which of P or ¬P is true. Suppose P
is a sentence like There is a whale swimming below the Golden Gate Bridge
right now. In such a case you would be willing to commit to the truth of the
disjunction (since either there is or there isn™t) without knowing just how to
play the game and win. You know that there is a winning strategy for the
game, but just don™t know what it is.
Since there is a moral imperative to live up to one™s commitments, the
use of the term “commitment” in describing the game is a bit misleading.
You are perfectly justi¬ed in asserting the truth of P ∨ ¬P, even if you do
not happen to know your winning strategy for playing the game. Indeed, it
would be foolish to claim that the sentence is not true. But if you do claim
that P ∨ ¬P is true, and then play the game, you will be asked to say which
of P or ¬P you think is true. With Tarski™s World, unlike in real life, you can
always get complete information about the world by going to the 2D view,
and so always live up to such commitments.

Chapter 3
Ambiguity and parentheses / 79


Here is a problem that illustrates the remarks we made about sometimes being able to tell that a sentence
is true, without knowing how to win the game.

3.11 Make sure Tarski™s World is set to display the world in 3D. Then open Kleene™s World and
 Kleene™s Sentences. Some objects are hidden behind other objects, thus making it impossible
to assess the truth of some of the sentences. Each of the six names a, b, c, d, e, and f are in use,
naming some object. Now even though you cannot see all the objects, some of the sentences in
the list can be evaluated with just the information at hand. Assess the truth of each claim, if
you can, without recourse to the 2-D view. Then play the game. If your initial commitment is
right, but you lose the game, back up and play over again. Then go through and add comments
to each sentence explaining whether you can assess its truth in the world as shown, and why.
Finally, display the 2-D view and check your work. We have annotated the ¬rst sentence for you
to give you the idea. (The semicolon “;” tells Tarski™s World that what follows is a comment.)
When you are done, print out your annotated sentences to turn in to your instructor.

Section 3.5
Ambiguity and parentheses
When we ¬rst described fol, we stressed the lack of ambiguity of this language
as opposed to ordinary languages. For example, English allows us to say things
like Max is home or Claire is home and Carl is happy. This sentence can be
understood in two quite di¬erent ways. One reading claims that either Claire
is home and Carl is happy, or Max is home. On this reading, the sentence
would be true if Max was home, even if Carl was unhappy. The other reading
claims both that Max or Claire is home and that Carl is happy.
Fol avoids this sort of ambiguity by requiring the use of parentheses, much
the way they are used in algebra. So, for example, fol would not have one
sentence corresponding to the ambiguous English sentence, but two:
Home(max) ∨ (Home(claire) § Happy(carl))
(Home(max) ∨ Home(claire)) § Happy(carl)
The parentheses in the ¬rst indicate that it is a disjunction, whose second
disjunct is itself a conjunction. In the second, they indicate that the sentence
is a conjunction whose ¬rst conjunct is a disjunction. As a result, the truth
conditions for the two are quite di¬erent. This is analogous to the di¬erence
in algebra between the expressions 2 + (x — 3) and (2 + x) — 3. This analogy
between logic and algebra is one we will come back to later.

Section 3.5
80 / The Boolean Connectives

Parentheses are also used to indicate the “scope” of a negation symbol
scope of negation
when it appears in a complex sentence. So, for example, the two sentences
¬Home(claire) § Home(max)
¬(Home(claire) § Home(max))
mean quite di¬erent things. The ¬rst is a conjunction of literals, the ¬rst of
which says Claire is not home, the second of which says that Max is home. By
contrast, the second sentence is a negation of a sentence which itself is a con-
junction: it says that they are not both home. You have already encountered
this use of parentheses in earlier exercises.
Many logic books require that you always put parentheses around any pair
of sentences joined by a binary connective (such as § or ∨). These books do
not allow sentences of the form:
but instead require one of the following:
((P § Q) § R)
(P § (Q § R))
The version of fol that we use in this book is not so fussy, in a couple of ways.
First of all, it allows you to conjoin any number of sentences without using
leaving out parentheses
parentheses, since the result is not ambiguous, and similarly for disjunctions.
Second, it allows you to leave o¬ the outermost parentheses, since they serve
no useful purpose. You can also add extra parentheses (or brackets or braces)
if you want to for the sake of readability. For the most part, all we will require
is that your expression be unambiguous.



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