PHIL12A Section answers, 14 February 2011 Julian Jonker 1 How much do you know? 1. You should understand why a truth table is constructed the way it is: why are the truth values listed in the order they are? In principle, it doesn’t matter how they are listed, but there is a natural way to list them systematically, which we use. (a) My favourite slot machine gives me money if I line up four cherries in a row. When I pull the lever, each of the four displayed items can come up as either a cherry or a banana. How many different combinations can the machine display? List them. This is exactly like listing all possible truth values for four atomic sentences, except with say, cherry for True and banana for False. There are 2 × 2 × 2 × 2 = 24 = 16 possible combinations. Think of it in the following way: Suppose you were betting on just one item. It could come up as a cherry or banana, so there are two possible ‘combinations’. Now suppose you were betting on two items. The first item can be a cherry, but there are two possible ways for it to be a cherry if we include the possible values of the second item: cherry cherry cherry banana But this is true also if it comes up as a banana. banana cherry banana banana So there are four possible combinations. More particularly, we have 2 × 2 = 22 = 4 possibilities. If we add a third item, then the first item can come up in four different ways: if it comes up as a cherry, then there are two different ways the second item can come up (cherry or banana), and for each way the second item comes up there are two different ways the third item can come up. So all in all there are 2 × 2 × 2 = 23 = 8 possibilities. The same holds for a truth table. If you have n atomic sentences, you will have 2n rows in your truth table. In other words, the number of rows grows exponentially, which is why you will never be asked to do a truth table for more than a handful of atomic sentences! 1 (b) Suppose the machine displays three different items at a time – my goal is to line up three cherries. But each item could be either a cherry, a banana, a palm tree or a robot. How many combinations are possible? It would take a long time to list them all, but convince yourself that there is a way to list them systematically so that you are sure you don’t leave any out. What would the 17th item on your list be? There are 4 × 4 × 4 = 43 = 64 possible combinations. Think of the list as a sort of tree. For the first item, draw four branches for the different possibilities: cherry, banana, palm tree, robot. At the end of the cherry branch, draw another four branches for the possible values of the second item (cherry, banana, palm tree, robot), and then do this three more times at the end of each of the other branches. You should now have sixteen end points. At the end of each end point, draw four branches for the possible values of the third item. All in all you should have 64 end points now, which you could number if you wish. If you drew the branches in the order I mentioned them here (you didn’t have to, but you should have stuck with some kind of order), your 17th possibility would have been: banana cherry cherry 2. Draw up truth tables for the following sentences, and say whether they are tautologies or not. (a) (A _:B) _ (:A _ B) A B (A _: B) _ (: A _ B) T T TTFT T FTTT T F TTTF T FTTF F T FFFT T TFTT F F FTTF T TFTF (b) (Ex 4.4) :(B ^ :C ^ :B) B C :( B ^ : C ^ : B) T T T TFFTFFT T F T TTTFFFT F T T FFFTFTF F F T FFTFFTF 2 (c) (:A _ (:B _ C)) _ (:(A ^ B) _ C) A B C (: A _ (: B _ C)) _ (: (A ^ B) _ C) T T T FTTFTTT T FTTTTT T T F FTFFTFF F FTTTFF T F T FTTTFTT T TTFFTT T F F FTTTFTF T TTFFTF F T T TFTFTTT T TFFTTT F T F TFTFTFF T TFFTTF F F T TFTTFTT T TFFFTT F F F TFTTFTF T TFFFTF (d) (Ex 4.6) :[:A _:(B ^ C) _ (A ^ B)] A B C :[ : A _: (B ^ C) _ (A ^ B)] T T T F FTFFTTTTTTT T T F F FTTTTFFTTTT T F T F FTTTFFTTTFF T F F F FTTTFFFTTFF F T T F TFTFTTTTFFT F T F F TFTTTFFTFFT F F T F TFTTFFTTFFF F F F F TFTTFFFTFFF 2 Something slightly harder, if there’s time. A sentence S is tautological if and only if every row of its truth table assigns True to S. S is logically necessary if and only if it is true in every logically possible circumstance. S is TW-necessary if and only it is true in every world that could possibly be constructed in Tarski’s World. 1. Is the following sentence a tautology? ‘It is not the case that both all men are mortal and Socrates is a man, otherwise Socrates is mortal.’ Is it logically necessary? The sentence is logically necessary, since there is no possible world in which it is false. There may be some possible worlds in which Socrates is mortal, and in that case the sentence is true. In the other worlds, at least one of the following must be true: not all men are mortal (since Socrates isn’t), or Socrates isn’t a man and isn’t mortal like them. So in these cases the sentence is true. Now consider whether the sentence is a tautology or not. The atomic sentences are: ‘all men are mortal’, ‘Socrates is a man’, and ‘Socrates is mortal’, and they are combined by the connectives :, ^ and _. So we 3 could abbreviate the sentence in the following way: :(P ^ Q) _ R, where P stands for ‘all men are mortal’, Q stands for ‘Socrates is a man’, and R stands for ‘Socrates is mortal’. This sentence is not a tautology, as drawing up a truth table will show. The moral of the story is that our truth table method is not fine-grained enough to capture the meanings of predicates and names. There is logical structure embedded within the atomic sentences that cannot be captured using truth tables. (In fact, many logic textbooks teach logic in two stages: propositional logic, dealing only with complex sentences made up of atomic sentences connected by truth-functional connectives; and predicate logic, which allows you to get into the structure of atomic sentences by working with predicates, names and quantifiers. Truth tables are taught in propositional logic, but drop out of the picture with predicate logic.) 2. Name at least two sentences that are logically necessary but not tautological. For example: (a) All bachelors are unmarried men. (b) There are infinitely many prime numbers. 3. Name at least two sentences that are TW-necessary but not logically necessary. For example: (a) Cube(a) _ Tet(a) _ Dodec(a) (b) (Large(a) ^ Large(b)) ^ :Adjoins(a,b) 4. For each of the following, determine whether the sentence is a tautology, a logical necessity, or a TW necessity. (a) a=b _ b=c _ c=c This is not a necessity of any sort, though it is logically possible. (b) BackOf(a,b) _:BackOf(a,b) This is a tautology, and therefore also a logical necessity and a TW necessity. (c) :(Cube(b) ^ Cube(e)) _ Cube(b) A truth table shows that this is a tautology, and therefore also a logical necessity and a TW necessity. (d) :(Cube(b) ^ Cube(e)) _ SameShape(b,e) This is a TW necessity, since in Tarski’s World there are one of two possibilities: either b and e are the same kind of block, for example they are both tetrahedrons or they are both cubes, in which case the sentence is true; or they are different shapes. But if they are different shapes, then they are certainly not both cubes, which means that the sentence is true. The sentence is not a tautology, as a truth table will show; however it is a logical necessity, as far as I can tell. 4 (e) SameRow(a,b) _:(FrontOf(a,b) _ BackOf(a,b)) This is a TW necessity. Either a and b are in the same row; or it is not the case that either a is in front of b or b is in front of a. I believe this is a logical necessity; it is not a tautology. (f) SameCol(a,b) ^ SameRow(a,b) ^ a=b This is not a necessity of any kind, since it is false if a and b are not in the same place. The following would be a TW necessity: SameCol(a,b) ^ SameRow(a,b) _:(a=b), but it would not be a logical necessity. (Why?) 5. (Based on Ex 4.10) A sentence S is a logically possibility if it is true in some logically possible circumstances. S is a TW-possibility if there is at least one world that can be constructed in Tarski’s World and in which S is true.