``` Test 1: Solutions, marking scheme, etc. Problem 4 (10 marks): An awful lot of marks for a VERY short proof: A /\ (¬A \/ B) == A /\ B <==> < ax. 10 > 2.5 marks A == ¬A \/ B == A \/ (¬A \/ B) == A /\ B ------ <==> < |- ax. (9)! |- T (2.1.15)! > 1 mark for ax. 9, 1 mark for saying it's a THEOREM, and roughly 0.7 plus 0.7 marks for mentioning 2.1.15 and saying it's a theorem A == A /\ B == ¬A \/ B == T \/ B ------ \_____________/ <==> < ax. (10) > 0.5 mark ( B == A \/ B == ¬A \/ B ) == T \/ B \___________/ -- |- 2.4.12! |- 2.4.7! (so the above line is also a theorem (but I guess not necessary to say this last thing) // 3.6 marks The last line is, of course, our much-used metatheorem (***) "If |- P and |- Q then |- (P == Q)." There are MANY other ways to do problem 4, but perhaps all of them are longer in the sense of having more <==>s in them. The above proof has only three <==>s. Many students had proofs with 8-10 <==>s in them. Those are too long, and I started deducting marks when proofs got that long, often. Other marking schemes were used for other proofs. There must be at least six proofs, any two of which are significantly different from each other. Most proofs use ax. 10 twice, 2.4.12 once, ax. 9, 2.1.15, and 2.4.7. Tourlakis' beloved "Redundant True" always makes a proof longer than it has to be. (It's really just 2.1.21.) Deductions were made whenever people used the above metatheorem (***) and did not explicitly say that they were dealing with THEOREMS. It's not enough just to write the number of the theorem followed by an exclamation point, for instance. People who wrote phrases at the start like "Let A, B be any wffs." got a 0.5 bonus. Ganong loves English, and people who explicitly say, at least sometimes, what is going on. Several people did what I explicitly warned against in class one week: They entirely omitted the <==> in the left margin. I already explained in class why that is a ridiculous thing to do. It's like writing a stack of numbers 13 + 71 119 - 35 7(12) without =s at the left margin. One is not SAYING anything. If you want to say that numbers are equal, you must SAY that (normally, using an = ). If you are claiming that certain wffs are syntactically equivalent, you must SAY that (normally using <==>). I made heavy deductions for this: -0.3 -0.5 -0.7 -0.7 -0.7 for the first five such omissions of <==>, roughly. People who put numbers (1), (2), etc. down the left margin, treating a bullet proof as if it were a Hilbert proof, lost 0.1 for each number on the far left. People who left angle brackets < > off annotations for the steps in a bullet proof lost 0.3 each time they did that. I never did that in class. Tourlakis neveer does it in his book. People who put <==>s or annotations way over at the right side of the page lost 0.1, 0.1, 0.2, 0.2, 0.2 for successive times they did this. I never did it in class and GT never does it in his text. People who wrote <--> for <==> lost 0.2, 0.3, 0.3, 0.3. Writing a |-- at the beginning of a bullet proof doesn't work; people lost 0.3 for this. Look at the non-English that results: |-- A <==> B is then pronounced "A is a theorem is syntactically equivalent to B." That's not English. People who put lots of extra periods in solutions lost 0.1 for two extra periods. Here, for problem 4, is a Second proof: A /\ (¬A \/ B) <==> < ax. (10) > 2.5 marks A == ¬A \/ B == A \/ (¬A \/ B) <==> < |- ax. (9)! |- T (2.1.15)! > 3.4 marks, roughly A == ¬A \/ B == T \/ B <==> < |- 2.4.7! |- 2.1.15! Then 2.1.21. > .7 mark, .7 mark, .7 mark, roughly A == ¬A \/ B <==> < 2.4.12 > 1.5 mark A == A \/ B == B ----------- <==> < ax. (10) > 0.5 mark A /\ B. // The first proof is better (more stream-lined, economical -- no need for "redundant T =="). It's also possible to include the right-hand side conjunction in every line of Proof number 2 above. Then the last step above is missing, but in the last line the student must observe that the wff there is precisely ax. (10), for 0.7-1 mark. And then the earlier mark totals are adjusted as well. Problem 5 (5 marks): I warned everyone that I would ask for an annotated Hilbert proof, testing among other things whether you could use and properly annotate steps using Leibniz. This is the problem testing that. I said in class that such a proof might have only one or two lines. My intention in this Hilbert proof was to ask for the absolutely easiest, shortest, simplest proof I could possibly think of. And then I get people writing nonsensical six-to-ten line proofs. One cannot force people to take easy marks. They will always find a way to avoid doing so. Some people gave proofs here that were 6, 7, 8, 9, 10 lines long. Nice way to kill yourself on a short-time-limit test like this one. I asked for a TOTALLY DETAILED annotated streamlined Hilb. pf. of the wff below, and said that the proof could be VERY short. Prove that |-- (A /\ T) \/ B == (A /\ ¬_|_) \/ B. Call this U to Call this V to save space below. save space below. Pf: (1) T == ¬_|_ < 2.4.5 > (2) U == V < (1) and Leibniz, with "C" being (A /\ p) \/ B and substitution for p, with p chosen not to occur in A or B > That's it. Both lines obvious. The only visual difference between the two sides of the biconditional to be proven is summarized in theorem scheme 2.4.5, so one just cites it in line 1 and uses primary inference rule Leibniz in line 2 with the obvious wff "C". I don't have the marking scheme in front of me, but probably 1 mark for mentioning 2.4.5 in line 1, and a bunch of marks allotted to the "does not occur" part of the annotation for line 2. Problem 6 (9 marks): Bullet-prove this theorem: A --> (B /\ C) == (A --> B) /\ (A --> C) VERY short proof, unless you decided to use ax. 11 to get rid of the arrows. Then it becomes much longer. This problem was intended to be VERY easy and fast. People were supposed to sit and look at the thing and say to themselves "Should I use ax. 11 or 2.4.11?" They were supposed to see that the former would lead to complications difficult to assess, but the latter leads to a short proof. Three easy steps: Proof via 2.4.11 (by far the shortest): A --> (B /\ C) <==> < 2.4.11 > 3 marks ¬A \/ (B /\ C) <==> < 2.4.23(i) > 3 marks (¬A \/ B) /\ (¬A \/ C) <==> < 2.4.11 twice > 2 marks for 2.4.11, 1 mark for "twice" (A --> B) /\ (A --> C). Proofs using ax. 11 one, two or three times are much longer and can take various forms; no one marking scheme handled all of those proofs. Problem 7 (8 marks): Two wffs were given; one was a theorem and one was not. You were asked in part (a) to identify the theorem if there was a theorem there, and to show that it is a theorem either by giving a bullet proof of it or by using facts from the course and naming them. (a) B was the theorem.. B was (p --> (p --> p)). One writes down the two-line truth table and notices that B is a tautology, hence, by Completeness, is a theorem. 1 mark for SAYING that B is a tautology, 1 mark for giving the truth table, 1 mark for concluding that B is a theorem, 1 mark for mentioning Completeness. (I think this was the basic marking scheme. I don't have it in front of me as I type this.) Very fast and easy. People did rather badly on this. A bullet proof of B is also rather short: 2.4.11 twice, then ax. 7, then ax. 9. (b) A is not a theorem. (1 mark) Given any state v with v(p) = f, v(A) = f. (1 mark for saying this.) So A is not a tautology. (1 mark) So A is not a theorem, by Soundness. (1 mark) (If it were a theorem, then by Soundness it would be a tautology. But it isn't. So it isn't.) I warned people explicitly about page 1 of the test. Anyone who took seriously the test preparation file posted several days before the test would certainly have ensured that he/she could rattle off perfectly, in no time at all, statements of Soundness and Completeness of Boolean Logic, and similarly rattle off certain definitions from the class notes or the course page. I will predict on the basis of years of experience that lots of people ignored my warnings. 15 marks out of 50 went for problems 1-3, very easy and fast for people who prepared. Lots of people messed up the definitions badly. Many also messed up Leibniz, in Problem 3. People did very well on "Name". Almost everyone got 3 marks out of 3 for writing her own name correctly. A couple of people lost a tenth of a mark for illegibility or not dotting an i or so. One makes "a"s that look like "u"s, which causes the graders unnecessary trouble looking for names on class lists, etc. If you scroll down about 70 lines in this file, you will find some statistics on how people did on this test. Don't look there if it might be psychologically bad for you to look. Here are some "statistics" for Test 1: 50 people wrote the test. The max. possible score was, as you know, 50. Here is some info about the "raw" (i.e., "unfiddled") scores: Average: 24.3 or 24.7 out of 50. (I don't have the grades file in front of me now.) Low scores: (Recall that 3 marks were given for writing one's name properly.) Six people got scores below 10 out of 50. High scores: Three people got scores between 40 and 43. Those high scores are a little low. Usually in a class the size of ours in a course like this one I would expect a couple of people to score over 45, and maybe four more over 40. This means that I will indeed "fiddle" the marks upward. ```