T TT TT TT TT TT TT T T TT FF TT FF FF FF T T FF TT FF TT FF FF T T FF FF FF FF FF FF FF TT TT TT TT TT TT FF TT FF TT TT FF TT FF FF TT TT TT FF TT FF FF FF TT TT FF TT
We have highlighted the rows where the premises are all true. Note that for these, the conclusion is true. Thus, in any kind of situation in which all the premises are true, the conclusion is true. This is equivalent, we have noted, to our definition of valid: necessarily, if all the premises are true, the conclusion is true. So this is a valid argument. The third column of the analyzed sentences (the column for(R^Q)) is there so that we can identify when the conclusion is true. The conclusion is a conditional, and we needed to know, for each kind of situation, if its antecedentP, and if its consequent(R^Q), are true. The third column tells us the situations in which the consequent is true. The stipulations on the left tell us in what kind of situation the antecedentPis true.
5.6 Problems
1. Translate the following sentences into our logical language. You will need to create your own key to do so.
a. Ulysses, who is crafty, is from Ithaca.
b. If Ulysses outsmarts both Circes and the Cyclops, then he can go home. c. Ulysses can go home only if he isn’t from Troy.
d. Ulysses is from Ithaca but not from Troy. e. Ulysses is not both crafty and from Ithaca.
a. Premise: ((P→Q) ^ ¬Q). Conclusion: ¬P.
b. Premises: ((P→Q) ^ (R→S)),(¬Q ^ ¬S). Conclusion: (¬P ^ ¬R). c. Premises: ((R ^ S) → T),(Q ^ ¬T). Conclusion: ¬(R ^ S). d. Premises: (P → (R → S)),(R ^ P). Conclusion: S.
e. Premises: (P → (R → S)),(¬S ^ P). Conclusion: ¬R.
3. Make truth tables for the following complex sentences. Identify which are tautologies. a. (((P→Q)^ ¬Q) → ¬P)
b. ¬(P ^ Q) c. ¬(¬P →¬Q) d. (P ^ ¬P) e. ¬(P ^ ¬P)
4. Make truth tables to show when the following sentences are true and when they are false. State which of these sentences are equivalent.
a. ¬(P^Q) b. (¬P^¬Q) c. ¬(P→Q) d. (P^¬Q) e. (¬P^Q) f. ¬(¬P→¬Q)
5. Write a valid argument in normal colloquial English with at least two premises, one of which is a conjunction or includes a conjunction. Your argument should just be a paragraph (not an ordered list of sentences or anything else that looks like formal logic). Translate the argument into propositional logic. Prove it is valid.
6. Write a valid argument in normal colloquial English with at least three premises, one of which is a conjunction or includes a conjunction and one of which is a conditional or includes a conditional. Translate the argument into propositional logic. Prove it is valid.
7. Make your own key to translate the following argument into our propositional logic. Translate only the parts in bold. Prove the argument is valid.
“I suspect Dr. Kronecker of the crime of stealing Cantor’s book,” Inspector Tarski said. His assistant, Mr. Carroll, waited patiently for his reasoning. “For,” Tarski said, “The thief left cigarette ashes on the table. The thief also did not wear shoes, but slipped silently into the room. Thus,If Dr. Kronecker smokes and is in his stocking feet, then he most likely stole Cantor’s book.” At this point, Tarski pointed at Kronecker’s feet. “Dr. Kronecker is in his stocking feet.” Tarski reached forward and pulled from Kronecker’s pocket a gold cigarette case. “And Kronecker smokes.” Mr. Carroll nodded sagely, “Your conclusion is obvious:
6.1 An argument from Hobbes
In his great work, Leviathan,the philosopher Thomas Hobbes (1588-1679) gives an important argument for government. Hobbes begins by claiming that without a common power, our condition is very poor indeed. He calls this state without government, “the state of nature”, and claims
Hereby it is manifest that during the time men live without a common power to keep them all in awe, they are in that condition which is called war; and such a war as is of every man against every man…. In such condition there is no place for industry, because the fruit thereof is uncertain: and consequently no culture of the earth; no navigation, nor use of the commodities that may be imported by sea; no commodious building; no instruments of moving and removing such things as require much force; no knowledge of the face of the earth; no account of time; no arts; no letters; no society; and which is worst of all, continual fear, and danger of violent death; and the life of man, solitary, poor, nasty, brutish, and short.[8]
Hobbes developed what is sometimes called “contract theory”. This is a view of government in which one views the state as the product of a rational contract. Although we inherit our government, the idea is that in some sense we would find it rational to choose the government, were we ever in the position to do so. So, in the passage above, Hobbes claims that in this state of nature, we have absolute freedom, but this leads to universal struggle between all people. There can be no property, for example, if there is no power to enforce property rights. You are free to take other people’s things, but they are also free to take yours. Only violence can discourage such theft. But, a common power, like a king, can enforce rules, such as property rights. To have this common power, we must give up some freedoms. You are (or should be, if it were ever up to you) willing to give up those freedoms because of the benefits that you get from this. For example, you are willing to give up the freedom to just seize people’s goods, because you like even more that other people cannot seize your goods.
We can reconstruct Hobbes’s defense of government, greatly simplified, as being something like this:
If we want to be safe, then we should have a state that can protect us.
If we should have a state that can protect us, then we should give up some freedoms.
Therefore, if we want to be safe, then we should give up some freedoms. Let us use the following translation key.
P: We want to be safe.
Q: We should have a state that can protect us. R: We should give up some freedoms.