Chapter 3 The Boolean Connectives So far, we have discussed only atomic claims. To form complex claims, fol pro- vides us with connectives and quantifiers. In this chapter we take up the three simplest connectives: conjunction, disjunction, and negation, corresponding to simple uses of the English and, or, and it is not the case that. Because they Boolean connectives were first studied systematically by the English logician George Boole, they are called the Boolean operators or Boolean connectives. The Boolean connectives are also known as truth-functional connectives. truth-functional connectives There are additional truth-functional connectives which we will talk about later. These connectives are called “truth functional” because the truth value of a complex sentence built up using these connectives depends on nothing more than the truth values of the simpler sentences from which it is built. Because of this, we can explain the meaning of a truth-functional connective in a couple of ways. Perhaps the easiest is by constructing a truth table,a truth table table that shows how the truth value of a sentence formed with the connec- tive depends on the truth values of the sentence’s immediate parts. We will give such tables for each of the connectives we introduce. A more interesting Henkin-Hintikka game way, and one that can be particularly illuminating, is by means of a game, sometimes called the Henkin-Hintikka game, after the logicians Leon Henkin and Jaakko Hintikka. Imagine that two people, say Max and Claire, disagree about the truth value of a complex sentence. Max claims it is true, Claire claims it is false. The two repeatedly challenge one another to justify their claims in terms of simpler claims, until finally their disagreement is reduced to a simple atomic claim, one involving an atomic sentence. At that point they can simply examine the world to see whether the atomic claim is true—at least in the case of claims about the sorts of worlds we find in Tarski’s World. These successive challenges can be thought of as a game where one player will win, the other will lose. The legal moves at any stage depend on the form of the sentence. We will explain them below. The one who can ultimately justify his or her claims is the winner. When you play this game in Tarski’s World, the computer takes the side opposite you, even if it knows you are right. If you are mistaken in your initial assessment, the computer will be sure to win the game. If you are right, though, the computer plugs away, hoping you will blunder. If you slip up, the computer will win the game. We will use the game rules as a second way of explaining the meanings of the truth-functional connectives. 67
26
Embed
Chapter 3 The Boolean Connectives - Stanford · PDF fileChapter 3 The Boolean Connectives So far, we have discussed only atomic claims. To form complex claims, fol pro-vides us with
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Chapter 3
The Boolean Connectives
So far, we have discussed only atomic claims. To form complex claims, fol pro-
vides us with connectives and quantifiers. In this chapter we take up the three
simplest connectives: conjunction, disjunction, and negation, corresponding
to simple uses of the English and, or, and it is not the case that. Because they Boolean connectives
were first studied systematically by the English logician George Boole, they
are called the Boolean operators or Boolean connectives.
The Boolean connectives are also known as truth-functional connectives. truth-functional
connectivesThere are additional truth-functional connectives which we will talk about
later. These connectives are called “truth functional” because the truth value
of a complex sentence built up using these connectives depends on nothing
more than the truth values of the simpler sentences from which it is built.
Because of this, we can explain the meaning of a truth-functional connective
in a couple of ways. Perhaps the easiest is by constructing a truth table, a truth table
table that shows how the truth value of a sentence formed with the connec-
tive depends on the truth values of the sentence’s immediate parts. We will
give such tables for each of the connectives we introduce. A more interesting Henkin-Hintikka game
way, and one that can be particularly illuminating, is by means of a game,
sometimes called the Henkin-Hintikka game, after the logicians Leon Henkin
and Jaakko Hintikka.
Imagine that two people, say Max and Claire, disagree about the truth
value of a complex sentence. Max claims it is true, Claire claims it is false. The
two repeatedly challenge one another to justify their claims in terms of simpler
claims, until finally their disagreement is reduced to a simple atomic claim,
one involving an atomic sentence. At that point they can simply examine the
world to see whether the atomic claim is true—at least in the case of claims
about the sorts of worlds we find in Tarski’s World. These successive challenges
can be thought of as a game where one player will win, the other will lose. The
legal moves at any stage depend on the form of the sentence. We will explain
them below. The one who can ultimately justify his or her claims is the winner.
When you play this game in Tarski’s World, the computer takes the side
opposite you, even if it knows you are right. If you are mistaken in your initial
assessment, the computer will be sure to win the game. If you are right,
though, the computer plugs away, hoping you will blunder. If you slip up, the
computer will win the game. We will use the game rules as a second way of
explaining the meanings of the truth-functional connectives.
67
68 / The Boolean Connectives
Section 3.1
Negation symbol: ¬
The symbol ¬ is used to express negation in our language, the notion we
commonly express in English using terms like not, it is not the case that, non-
and un-. In first-order logic, we always apply this symbol to the front of a
sentence to be negated, while in English there is a much more subtle system
for expressing negative claims. For example, the English sentences John isn’t
home and It is not the case that John is home have the same first-order
translation:
¬Home(john)
This sentence is true if and only if Home(john) isn’t true, that is, just in case
John isn’t home.
In English, we generally avoid double negatives—negatives inside other
negatives. For example, the sentence It doesn’t make no difference is problem-
atic. If someone says it, they usually mean that it doesn’t make any difference.
In other words, the second negative just functions as an intensifier of some
sort. On the other hand, this sentence could be used to mean just what it
says, that it does not make no difference, it makes some difference.
Fol is much more systematic. You can put a negation symbol in front of
any sentence whatsoever, and it always negates it, no matter how many other
negation symbols the sentence already contains. For example, the sentence
¬¬Home(john)
negates the sentence¬Home(john)
and so is true if and only if John is home.
The negation symbol, then, can apply to complex sentences as well as to
atomic sentences. We will say that a sentence is a literal if it is either atomicliterals
or the negation of an atomic sentence. This notion of a literal will be useful
later on.
We will abbreviate negated identity claims, such as ¬(b = c), using =, asnonidentity symbol ( =)
in b = c. The symbol = is available on the keyboard palettes in both Tarski’s
World and Fitch.
Semantics and the game rule for negation
Given any sentence P of fol (atomic or complex), there is another sentence
¬P. This sentence is true if and only if P is false. This can be expressed in
terms of the following truth table.
Chapter 3
Negation symbol: ¬ / 69
P ¬Ptrue false
false true
truth table for ¬
The game rule for negation is very simple, since you never have to do game rule for ¬anything. Once you commit yourself to the truth of ¬P this is the same as
committing yourself to the falsity of P. Similarly, if you commit yourself to
the falsity of ¬P, this is tantamount to committing yourself to the truth of
P. So in either case Tarski’s World simply replaces your commitment about
the more complex sentence by the opposite commitment about the simpler
2. The sentence ¬P is true if and only if P is not true.
3. A sentence that is either atomic or the negation of an atomic sentence
is called a literal.
Section 3.1
70 / The Boolean Connectives
Exercises
3.1 If you skipped the You try it section, go back and do it now. There are no files to submit,
but you wouldn’t want to miss it.
3.2ö
(Assessing negated sentences) Open Boole’s World and Brouwer’s Sentences. In the sentence file
you will find a list of sentences built up from atomic sentences using only the negation symbol.
Read each sentence and decide whether you think it is true or false. Check your assessment. If
the sentence is false, make it true by adding or deleting a negation sign. When you have made
all the sentences in the file true, submit the modified file as Sentences 3.2
3.3ö
(Building a world) Start a new sentence file. Write the following sentences in your file and save
the file as Sentences 3.3.
1. ¬Tet(f)2. ¬SameCol(c, a)
3. ¬¬SameCol(c, b)
4. ¬Dodec(f)5. c = b
6. ¬(d = e)
7. ¬SameShape(f, c)
8. ¬¬SameShape(d, c)
9. ¬Cube(e)10. ¬Tet(c)
Now start a new world file and build a world where all these sentences are true. As you modify
the world to make the later sentences true, make sure that you have not accidentally falsified
any of the earlier sentences. When you are done, submit both your sentences and your world.
3.4.
Let P be a true sentence, and let Q be formed by putting some number of negation symbols
in front of P. Show that if you put an even number of negation symbols, then Q is true, but
that if you put an odd number, then Q is false. [Hint: A complete proof of this simple fact
would require what is known as “mathematical induction.” If you are familiar with proof by
induction, then go ahead and give a proof. If you are not, just explain as clearly as you can
why this is true.]
Now assume that P is atomic but of unknown truth value, and that Q is formed as before.
No matter how many negation symbols Q has, it will always have the same truth value as a
literal, namely either the literal P or the literal ¬P. Describe a simple procedure for determining
which.
Chapter 3
Conjunction symbol: ∧ / 71
Section 3.2
Conjunction symbol: ∧
The symbol ∧ is used to express conjunction in our language, the notion we
normally express in English using terms like and, moreover, and but. In first-
order logic, this connective is always placed between two sentences, whereas in
English we can also conjoin other parts of speech, such as nouns. For example,
the English sentences John and Mary are home and John is home and Mary
is home have the same first-order translation:
Home(john) ∧ Home(mary)
This sentence is read aloud as “Home John and home Mary.” It is true if and
only if John is home and Mary is home.
In English, we can also conjoin verb phrases, as in the sentence John slipped
and fell. But in fol we must translate this the same way we would translate
John slipped and John fell :
Slipped(john) ∧ Fell(john)
This sentence is true if and only if the atomic sentences Slipped(john) and
Fell(john) are both true.
A lot of times, a sentence of fol will contain ∧ when there is no visible
sign of conjunction in the English sentence at all. How, for example, do you
think we might express the English sentence d is a large cube in fol? If you
guessed
Large(d) ∧ Cube(d)
you were right. This sentence is true if and only if d is large and d is a cube—
that is, if d is a large cube.
Some uses of the English and are not accurately mirrored by the fol
conjunction symbol. For example, suppose we are talking about an evening
when Max and Claire were together. If we were to say Max went home and
Claire went to sleep, our assertion would carry with it a temporal implication,
namely that Max went home before Claire went to sleep. Similarly, if we were to
reverse the order and assert Claire went to sleep and Max went home it would
suggest a very different sort of situation. By contrast, no such implication,
implicit or explicit, is intended when we use the symbol ∧. The sentence
WentHome(max) ∧ FellAsleep(claire)
is true in exactly the same circumstances as
FellAsleep(claire) ∧WentHome(max)
Section 3.2
72 / The Boolean Connectives
Semantics and the game rule for ∧
Just as with negation, we can put complex sentences as well as simple ones
together with ∧. A sentence P ∧ Q is true if and only if both P and Q are true.
Thus P ∧ Q is false if either or both of P or Q is false. This can be summarized
by the following truth table.
P Q P ∧ Q
true true true
true false false
false true false
false false false
truth table for ∧
The Tarski’s World game is more interesting for conjunctions than nega-
tions. The way the game proceeds depends on whether you have committedgame rule for ∧to true or to false. If you commit to the truth of P ∧ Q then you have
implicitly committed yourself to the truth of each of P and Q. Thus, Tarski’s
World gets to choose either one of these simpler sentences and hold you to the
truth of it. (Which one will Tarski’s World choose? If one or both of them are
false, it will choose a false one so that it can win the game. If both are true,
it will choose at random, hoping that you will make a mistake later on.)
If you commit to the falsity of P ∧ Q, then you are claiming that at least
one of P or Q is false. In this case, Tarski’s World will ask you to choose one of
the two and thereby explicitly commit to its being false. The one you choose
had better be false, or you will eventually lose the game.
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 Scruffy are in the next room and the sound of a cat fight suddenly
breaks out. If we say Either Pris bit Scruffy or Scruffy bit Pris, we would not
be wrong if each had bit the other. So this would be translated as
Bit(pris, scruffy) ∨ Bit(scruffy, pris)
We will see later that the expression either sometimes plays a different logical
function.
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.
P Q P ∨ Q
true true true
true false true
false true true
false false false
truth table for ∨
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
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).
Exercises
3.8ö
If you skipped the You try it section, go back and do it now. You’ll be glad you did. Well,
maybe. Submit the file Sentences Game 2.
Chapter 3
Remarks about the game / 77
3.9ö
Open Wittgenstein’s World and the sentence file 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, find 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.
3.10ö
Open Ramsey’s World and start a new sentence file. Type the following four sentences into the
file:
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 modified world
as World 3.10. Submit both files.
Section 3.4
Remarks about the game
We summarize the game rules for the three connectives, ¬, ∧, and ∨, in
Table 3.1. The first 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 final 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 first 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
true you Choose one of
P ∨ Q P, Q that
false Tarski’s World is true.
true Tarski’s World Choose one of
P ∧ Q P, Q that
false you is false.
Replace ¬P¬P either — by P and
switch
commitment.
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 ∨ ¬Pwill 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 justified 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
Exercises
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 first 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 first 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 different 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 first 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 first conjunct is a disjunction. As a result, the truth
conditions for the two are quite different. This is analogous to the difference
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 symbolscope 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 different things. The first is a conjunction of literals, the first 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:
P ∧ Q ∧ R
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 usingleaving out parentheses
parentheses, since the result is not ambiguous, and similarly for disjunctions.
Second, it allows you to leave off 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.
Remember
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