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Humanities Libertexts

2E: “If…then….” and “It is not the case that….” (Exercises)

  1. The answer to our card game was: you need only turn over cards 3 and 4.  This might seem confusing to many people at first.  But remember the meaning of the conditional:  it can only be false if the first part is true and the second part is false.  The sentence we want to test is “For each of these four cards, if the card has a Q on the letter side of the card, then it has a square on the shape side of the card”.  Let Q stand for “the card has a Q on the letter side of the card.”  Let S stand for “the card has a square on the shape side of the card.”  Then we could make a truth table to express the meaning of the claim being tested:
Q        S (Q→S)
T T T
T         F F
F         T T
F         F T

Look back at the cards. The first card has an R on the letter side.  So, sentence Q is false.  But then we are in a situation like the last two rows of the truth table, and the conditional cannot be false.  We do not need to check that card.  The second card has a square on it.  That means S is true for that card.  But then we are in a situation represented by either the first or third row of the truth table.  Again, the claim that (Q→S) cannot be false in either case with respect to that card, so there is no point in checking that card.  The third card shows a Q.  It corresponds to a situation that is like either the first or second row of the truth table.  We cannot tell then whether (Q→S) is true or false of that card, without turning the card over.  Similarly, the last card shows a situation where S is false, so we are in a kind of situation represented by either the second or last row of the truth table.  We must turn the card over to determine if (Q→S) is true or false of that card.

Try this puzzle again.  Consider the following claim about those same four cards:  If there is a star on the shape side of the card, then there is an R on the letter side of the card.  What is the minimum number of cards that you must turn over to check this claim?  What cards are they?

  1. Consider the following four cards in figure 2.2.  Each card has a letter on one side, and a shape on the other side.

cards p triangle hex diamond

 

 

Figure 2.2

 

For each of the following claims, in order to determine if the claim is true of all four cards, describe (1) The minimum number of cards you must turn over to check the claim, and (2) what those cards are.

  1. There is not a Q on the letter side of the card.
  2. There is not an octagon on the shape side of the card.
  3. If there is a triangle on the shape side of the card, then there is a P on the letter side of the card.
  4. There is an R on the letter side of the card only if there is a diamond on the shape side of the card.
  5. There is a hexagon on the shape side of the card, on the condition that there is a P on the letter side of the card.
  6. There is a diamond on the shape side of the card only if there is a P on the letter side of the card.

3. Which of the following have correct syntax?  Which have incorrect syntax?

  1. PQ
  2. ¬(PQ)
  3. (¬PQ)
  4. (P¬→Q)
  5. (P→¬Q)

4. Use the following translation key to translate the following sentences into a propositional logic.

Translation Key
Logic English
P Abe is able.
Q Abe is honest.
  1. If Abe is honest, Abe is able.
  2. Abe is not able.
  3. Abe is not able only if Abe is not honest.
  4. Abe is able, provided that Abe is not honest.
  5. If Abe is not able then Abe is not honest.

5. Make up your own translation key to translate the following sentences into a propositional logic. Then, use your key to translate the sentences into the propositional logic. Your translation key should contain only atomic sentences.   These should be all and only the atomic sentences needed to translate the following sentences of English.  Don’t let it bother you that some of the sentences must be false.

  1. Josie is a cat.
  2. Josie is a mammal.
  3. Josie is not a mammal.
  4. If Josie is not a cat, then Josie is not a mammal.
  5. Josie is a fish.
  6. Provided that Josie is a mammal, Josie is not a fish.
  7. Josie is a cat only if Josie is a mammal.
  8. Josie is a fish only if Josie is not a mammal.
  9. It’s not the case that Josie is not a mammal.
  10. Josie is not a cat, if Josie is a fish.

6. This problem will make use of the principle that our syntax is recursive.  Translating these sentences is more challenging.  Make up your own translation key to translate the following sentences into a propositional logic.  Your translation key should contain only atomic sentences; these should be all and only the atomic sentences needed to translate the following sentences of English.

  1. It is not the case that Tom won’t pass the exam.
  2. If Tom studies, Tom will pass the exam.
  3. It is not the case that if Tom studies, then Tom will pass the exam.
  4. If Tom does not study, then Tom will not pass the exam.
  5. If Tom studies, Tom will pass the exam—provided that he wakes in time.
  6. If Tom passes the exam, then if Steve studies, Steve will pass the exam.
  7. It is not the case that if Tom passes the exam, then if Steve studies, Steve will pass the exam.
  8. If Tom does not pass the exam, then if Steve studies, Steve will pass the exam.
  9. If Tom does not pass the exam, then it is not the case that if Steve studies, Steve will pass the exam.
  10. If Tom does not pass the exam, then if Steve does not study, Steve won’t pass the exam.

7. Make up your own translation key in order to translate the following sentences into English.  Write out the English equivalents in English sentences that seem (as much as is possible) natural.

  1. (RS)
  2. ¬¬R
  3. (SR)
  4. (¬S→¬R)
  5. ¬(RS)

[3] One thing is a little funny about this second example with unknown number a.  We will not be able to find a number that is evenly divisible by 4 and not evenly divisible by 2, so the world will never be like the second row of this truth table describes. Two things need to be said about this.  First, this oddity arises because of mathematical facts, not facts of our propositional logic—that is, we need to know what “divisible” means, what “4” and “2” mean, and so on, in order to understand the sentence.  So, when we see that the second row is not possible, we are basing that on our knowledge of mathematics, not on our knowledge of propositional logic.  Second, some conditionals can be false.  In defining the conditional, we need to consider all possible conditionals; so, we must define the conditional for any case where the antecedent is true and the consequent is false, even if that cannot happen for this specific example.

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