Problems from Chapter 5
5.1.2 is a good sort of test/exam question to ask students; "find all the errors" (without telling you how many there are) is also good.
A particular line of a derivation is to be regarded as an ERROR if and only if, ASSUMING that all lines ABOVE it are OK (which of course they may not be, but assume they are anyway when answering such a question), a rule has been wrongly used to get THAT line.
5.1.3 is a cute question. At first it looked undoable to me, but after a minute or so I began to see the light.
Note that there is an error in 5.2.1: The wff P should be the first premise in a list of four premises.
Also: If the program Symlog HELPS you, that is, of course, good. A problem like this is good from MY point of view because it is an example of a derivation that can be done entirely FORWARD. Typically, derivations/proofs/arguments (for that matter, problem-solvings generally) require "backwards-forwards" maneuvering: One makes progress for a while and gets stuck, looks to the end to remind oneself of the goal, tries to work backward from it, gets a new idea, goes back to the place one was stuck, goes forward a bit, etc.
5.2.2, on the other hand, goes naturally BACKWARD. :-) So do all the fellows in 5.2.3.
5.3.1 seems oddly worded to me, but maybe that is because I never work with Symlog . I think what is really being asked here is, "Which rule of SD is likely to be written to the left of the conclusion, to justify writing it on the last line of the deriv.?"
As always, explain answers to True-False questions. To show the truth of a general statement, you need a general argument. To show its falsity, you need only one explicit counterexample. And as always, part of the problem may be figuring out whether the statement is a general statement or is the negation of a general statement.
Just look at the answer in the back of the book for 5.5.11, just for laughs. (I noted the length of the derivation and the forest of subderivations; you need not read the damned thing.)
Problems from Chapter 3
I don't like the wording of 3.1.4: There IS no truth-functional "relationship" among atoms. I would say, rather, "Each of the following sentences is an English translation of a certain SL sentence, i.e., is true under a given arbitrary t.v.a. iff that SL sentence is true under that t.v.a. Write down the SL sentence."
In 3.1.5, of course, give reasons.
Have a three-second look also at 3.3.1 just to see the notation used there. In a problem like 3.3.3, just do as many as you can stomach (maybe none, maybe one or two...).