Remarks about the game

.

LetPbe 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 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 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 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)

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 of PandQ. 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

. . . .

¬Cube(a)∧ ¬Cube(b)∧ ¬Cube(c)

I 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. I 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

J 4. Play until Tarski’s World says that you have lost. Then click on Back 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.

J 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.

J 6. Save your sentence file asSentences Game 1when you are done.

. . . .

1. IfPandQare sentences of fol, then so isP∧Q.

2. The sentenceP∧Qis true if and only if bothPandQare true.

Exercises