Given any sentence Poffol (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.
Negation symbol:¬/ 69
P ¬P
true 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 ¬Pthis 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 sentence.
You try it . . . .
1. OpenWittgenstein’s World. Start a new sentence file and write the following
sentence.
¬¬¬¬¬Between(e,d,f)
2. Use theVerifybutton to check the truth value of the sentence.
3. Now play the game, choosing whichever commitment you please. What
happens to the number of negation symbols as the game proceeds? What happens to your commitment?
4. Now play the game again with the opposite commitment. If you won the
first time, you should lose this time, and vice versa. Don’t feel bad about losing.
5. There is no need to save the sentence file when you are done.
. . . .
CongratulationsRemember 1. IfPis a sentence offol, then so is¬P.
2. The sentence¬Pis true if and only ifPis not true.
3. A sentence that is either atomic or the negation of an atomic sentence is called aliteral.
Section 3.1
70 /The Boolean Connectives
Exercises
3.1
If you skipped theYou try it section, go back and do it now. There are no files to submit, but you wouldn’t want to miss it.3.2
➶
(Assessing negated sentences) OpenBoole’s WorldandBrouwer’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 asSentences 3.2
3.3
➶
(Building a world) Start a new sentence file. Write the following sentences in your file and save the file asSentences 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
✎
LetPbe a true sentence, and let Qbe formed by putting some number of negation symbols in front ofP. Show that if you put an even number of negation symbols, then Qis 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 thatPis atomic but of unknown truth value, and thatQis formed as before.
No matter how many negation symbolsQ has, it will always have the same truth value as a literal, namely either the literalPor the literal¬P. Describe a simple procedure for determining which.
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 likeand,moreover, andbut. 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 sentencesJohn and Mary are homeandJohn is home and Mary is homehave 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 sentenceJohn slipped and fell. But infolwe 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 cubein fol? If you guessed
Large(d)∧Cube(d)
you were right. This sentence is true if and only ifdis large anddis a cube—
that is, ifdis 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 homebeforeClaire went to sleep. Similarly, if we were to reverse the order and assertClaire went to sleep and Max went homeit 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 sentenceP∧Qis true if and only if bothPandQare true.
ThusP∧Qis false if either or both ofPorQis 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 committed game 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 ofPandQ. 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 ofP∧Q, then you are claiming that at least one ofPorQis false. In this case, Tarski’s World will askyouto 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.
You try it . . . .
1. OpenClaire’s World. Start a new sentence file and enter the sentence
¬Cube(a)∧ ¬Cube(b)∧ ¬Cube(c)
2. Notice that this sentence is false in this world, since c is a cube. Play the game committed (mistakenly) to the truth of the sentence. You will see that Tarski’s World immediately zeros in on the false conjunct. Your commitment to the truth of the sentence guarantees that you will lose the game, but along the way, the reason the sentence is false becomes apparent.
3. Now begin playing the game committed to the falsity of the sentence.
When Tarski’s World asks you to choose a conjunct you think is false, pick the first sentence. This is not the false conjunct, but select it anyway and see what happens after you chooseOK.
Conjunction symbol: ∧/ 73
4. Play until Tarski’s World says that you have lost. Then click onBack a
couple of times, until you are back to where you are asked to choose a false conjunct. This time pick the false conjunct and resume the play of the game from that point. This time you will win.
5. Notice that you can lose the game even when your original assessment
is correct, if you make a bad choice along the way. But Tarski’s World always allows you to back up and make different choices. If your original assessment is correct, there will always be a way to win the game. If it is impossible for you to win the game, then your original assessment was wrong.
6. Save your sentence file asSentences Game 1when you are done.
. . . .
CongratulationsRemember
1. IfPandQare sentences of fol, then so isP∧Q.
2. The sentenceP∧Qis true if and only if bothPandQare true.
Exercises
3.5
➶
If you skipped the You try it section, go back and do it now. Make sure you follow all the instructions. Submit the fileSentences Game 1.
3.6
➶
Start a new sentence file and openWittgenstein’s World. Write the following sentences in the sentence file.
1. Tet(f)∧Small(f) 2. Tet(f)∧Large(f) 3. Tet(f)∧ ¬Small(f) 4. Tet(f)∧ ¬Large(f) 5. ¬Tet(f)∧ ¬Small(f) 6. ¬Tet(f)∧ ¬Large(f) 7. ¬(Tet(f)∧Small(f)) 8. ¬(Tet(f)∧Large(f))
Section 3.2
74 /The Boolean Connectives
9. ¬(¬Tet(f)∧ ¬Small(f)) 10. ¬(¬Tet(f)∧ ¬Large(f))
Once you have written these sentences, decide which you think are true. Record your eval-uations, to help you remember. Then go through and use Tarski’s World to evaluate your assessments. Whenever you are wrong, play the game to see where you went wrong.
If you are never wrong, playing the game will not be very instructive. Play the game a couple times anyway, just for fun. In particular, try playing the game committed to the falsity of sentence 9. Since this sentence is true inWittgenstein’s World, Tarski’s World should be able to beat you. Make sure you understand everything that happens as the game proceeds.
Next, change the size or shape of block f, predict how this will affect the truth values of your ten sentences, and see if your prediction is right. What is the maximum number of these sentences that you can get to be true in a single world? Build a world in which the maximum number of sentences are true. Submit both your sentence file and your world file, naming them as usual.
3.7
➶
(Building a world) OpenMax’s Sentences. Build a world where all these sentences are true.
You should start with a world with six blocks and make changes to it, trying to make all the sentences true. Be sure that as you make a later sentence true you do not inadvertently falsify an earlier sentence.
Section 3.3
Disjunction symbol: ∨
The symbol ∨is used to express disjunction in our language, the notion we express in English usingor. In first-order logic, this connective, like the con-junction sign, is always placed between two sentences, whereas in English we can also disjoin nouns, verbs, and other parts of speech. For example, the English sentencesJohn or Mary is homeandJohn is home or Mary is home both have the same first-order translation:
Home(john)∨Home(mary) This folsentence is read “Home John or home Mary.”
Although the Englishoris sometimes used in an “exclusive” sense, to say exclusive vs. inclusive
disjunction thatexactlyone (i.e., one but no more than one) of the two disjoined sentences is true, the first-order logic∨is always given an “inclusive” interpretation: it means that at least one and possibly both of the two disjoined sentences is true. Thus, our sample sentence is true if John is home but Mary is not, if Mary is home but John is not, or if both John and Mary are home.
Disjunction symbol:∨/ 75
If we wanted to express the exclusive sense oforin 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 expressioneither . . . or expresses exclusive disjunction. While this is sometimes the case (and indeed the simpleoris 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 sayEither 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 expressioneithersometimes 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 homewould 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 sentencesPandQoffol, atomic or not, we can combine them using ∨to form a new sentenceP∨Q. The sentenceP∨Qis true if at least one ofPorQis 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 ofP∨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 file Ackermann’s World. Start a new sentence file and enter the sentence
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 first one is obviously false, pick the second.
3. You now find 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 ofCube(b). But this is clearly wrong, sinceb 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 file asSentences Game 2
. . . .
CongratulationsRemember
1. IfPandQare sentences offol, then so isP∨Q.
2. The sentenceP∨Qis true if and only ifPis true orQis true (or both are true).
Exercises
3.8
➶
If you skipped theYou try it section, go back and do it now. You’ll be glad you did. Well, maybe. Submit the fileSentences Game 2.