By Herbert B. Enderton

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**Additional resources for A Mathematical Introduction to Logic (2nd Edition)**

**Sample text**

The victim was an old acquaintance of Brown's. " Brown states "I didn't do it. I didn't even know the guy. " Clark says "I didn't do it. " Assume that the two innocent men are telling the truth, but that the guilty man might not be. Who did it? 13. An advertisement for a tennis magazine states, "If I'm not playing tennis, I'm watching tennis. " We can assume that the speaker cannot do more than one of these activities at a time. What is the speaker doing? ) 14. Let S be the set of all sentence symbols, and assume that v : S + {F,T} is a truth assignment.

In other words, the set of wffs isfreely generated from the set of sentence symbols by the five operations. PROOF. TO show that the restriction of EA is one-to-one, suppose that where a,B, y, and S are wffs. Delete the first symbol of each sequence, obtaining Then we must have a! The same argument applies to Ev, E, , and E, ; for E, a simpler argument suffices. A similar line of reasoning tells us that the operations have disjoint ranges. For example, if where a, #J, y, and S are wffs, then as in the above paragraph we have a = y.

Let a*be the result of interchanging A and V and replacing each sentence symbol by its negation. Show that a*is tautologically equivalent to ( 1 a). Use the induction principle. Remark: It follows that if a +=I /3 then a* +=I /3*. 10. Say that a set C1of wffs is equivalent to a set C2of wffs iff for any wff a , we have El a iff E2 a. Show that the following hold. (a) A finite set of wffs has an independent equivalent subset. (b) An infinite set need not have an independent equivalent subset. *(c) Let E = {ao,a1 , .