Truth Tables for Propositions

Transcription

1 Truth Tables for Propositions 1. Truth Tables for 2-Letter Compound Statements: We have learned about truth tables for simple statements. For instance, the truth table for A B is the following: Conditional A B A B T T T T F F F T T F F T So, if I had told you that, If you come over and help me move my couch on Saturday, then I will buy you pizza, the ONLY way you might call me a liar is if you DID come over and help me, and I DID NOT buy you pizza. Now we will learn how to make truth tables for more complicated statements. Let s start with a compound statement which contains ONLY TWO distinct simple statements. Example #1: For instance, imagine that a bunch of people are invited to a party. You speculate about your friends: If Alice goes, then Brad will go; but, if Brad goes, then Alice WON T go. Let A = Alice goes to the party and B = Brad goes to the party, so that we get: (A B) (B A) The un-filled-in truth table for this proposition looks like this: A B (A B) (B A) T T? T F? F T? F F? Our job now is to figure out what goes in place of the question marks. We figure them out one line at a time. Let s start with the first line, where A and B are both true. In that case, we get the following: 1

2 Line 1: A=T, B=T (A B) (B A) becomes: (T T) (T T) (We replace the letters with T for True and F for False ) (T T) (T F) (We negate the last T ) T F (The first conditional is true, and the second one is false) F (A conjunction with one false conjunct is false) So, the right-hand column should have an F on the first line, like this: A B (A B) (B A) T T F T F? F T? F F? Let s do the other 3 lines: Line 2: A=T, B=F (A B) (B A) (T F) (F T) (T F) (F F) F T F Line 3: A=F, B=T (A B) (B A) (F T) (T F) (F T) (T T) T T T Line 4: A=F, B=F (A B) (B A) (F F) (F F) (F F) (F T) T T T 2

3 So, now that we have our truth values for lines 1-4, we can fill in the entire truth table: A B (A B) (B A) T T F T F F F T T F F T The bold values on the far right represents the truth table for the proposition. As we can see, your claim ONLY comes out true if Brad is the only one of the two who shows up, or if NEITHER Alice nor Brad show up. Example #2: We have seen this claim: Timmy is either five or six, but he s not both five AND six. Substituting F for Timmy is five and S for Timmy is six, we get the following symbolization: Again, these are the four possible combinations of truth and falsehood for F and S: F S T T? T F? F T? F F? But, typically, to save space we do not determine the truth table like we did in example #1. Usually, we put the T s and F s directly underneath the formula, all together, like this: T T T T T F T F F T F T F F F F First, let s determine the truth values of the disjunction and conjunction (the stuff inside of the parentheses), like this: T T T T T T T T F T F F F T T F F T F F F F F F 3

4 Above, the original truth values for the statements F and S are in black, and the truth values for the disjunction and the conjunction are in green. Note that the disjunction is only false on the fourth line, since disjunctions are only ever false when BOTH of their disjuncts are false. Also, the conjunction is only true on the first line, since conjunctions are only true when BOTH of their conjuncts are true. Next, we perform the negation on the stuff in parentheses on the right, like this: T T T F T T T T T F T T F F F T T T F F T F F F T F F F The solution for the truth values of the negation is in blue. For negation, we just write the opposite truth value of whatever is being negated (so the blue truth values are just the opposite of the green truth values listed underneath the conjunction). Last, let s solve for the main operator, the conjunction, : T T T F F T T T T T F T T T F F F T T T T F F T F F F F T F F F The bold, red letters are the truth function for the whole proposition, which was: Timmy is either five or six, but he s not both five AND six. The final truth table looks like this: F S T T F T F T F T T F F F 4

5 Notice that only the second and third lines are true. So, looking at the original truth values for the statements F and S (in black), we see that the whole statement is false ONLY WHEN it is true that Timmy is five, but false that he is six, OR when it is false that Timmy is five, but true when he is six. On the other hand, the statement would come out false if it was true that Timmy is five AND true that he is six, or if it were false that he is five AND false that he is six. Intuitively, this is the correct result. 2. Truth Tables for 3-Letter Compound Statements: Next, we will look at propositions containing THREE distinct statements. We ve seen that statements with only ONE letter get truth tables with TWO lines, and TWO letters gets a table with FOUR lines, like this: Negation: 1 Letter A A T F F T Conjunction: 2 Letters A B A B T T T T F F F T F F F F When there are THREE letters, there will be EIGHT lines. The basic formula is that there will be 2 n lines, where n=the number of statement letters. For, instance, we ll try: Warren will go to the party if and only if both Belinda and Tina go to the party Symbolized as: W (B T) ; And its truth table will look like this: W B T W (B T) T T T? T T F? T F T? T F F? F T T? F T F? F F T? F F F? 5

6 Let s determine the truth values which go in place of the question marks. We begin by writing out all 8 lines with the truth values for the three different statements filled in: W (B T) T T T T T F T F T T F F F T T F T F F F T F F F Next, let s fill in the truth values for the disjunction. The disjunction will ONLY be true when BOTH conjuncts are true. So, our table should look like this: W (B T) T T T T T T F F T F F T T F F F F T T T F T F F F F F T F F F F Next, let s fill in the final truth values for the bi-conditional. Bi-conditionals are ONLY true whenever the statements on either side of the bi-conditional have the SAME truth value. So, here, we should be comparing the letters underneath the W with the green letters underneath the. Here is the answer: W (B T) T T T T T T F T F F T F F F T T F F F F F F T T T F T T F F F T F F T F T F F F 6

7 This statement is true on the first line, and the last 3 lines. In other words, the entire proposition is true if: Line 1: W=true, B=true, T=true Line 6: W=false, B=true, T=false Line 7: W=false, B=false, T=true Line 8: W=false, B=false, T=false 3. Classifying Statements: In the truth table we just did, the red letters are an assortment of true and false. When this happens, we say that the proposition is contingent. But, sometimes, the column will be ALL T s, or ALL F s. When the truth table for the proposition is ALL T s, we say that the proposition is tautologous. When the truth table is ALL F s, we say that it is self-contradictory. So: Column Under the Main Operator All True All False At Least One True & At Least One False Statement Classification Tautologous (Logically True) Self-Contradictory (Logically False) Contingent Tautologous statements are true, but this truth depends ONLY on the FORM of the statement, and not at all on the content so they are true in a really uninteresting and uninformative way. They are trivially true. Self-Contradictory statements are always false, and their falsehood also depends upon their FORM. Since their FORM is what makes them false, they can NEVER be true, no matter WHAT the content is. Contingent statements have a form that allows for them to be EITHER true OR false, depending on whether the statements that make up its content are true or false. All of the statements we have examined so far have been contingent ones. (To see this, look at their truth tables, and notice that there are both T s and F s in the column under the main operator). Let s try one that is NOT contingent: Someone says: If you know, then you just know Symbolized as: K K 7

8 K K T T F F And, one step later: K K T T T F T F This is a tautology. The statement is true in all possible scenarios. It cannot be false. Try another: Dude It both is, and it isn t. Symbolized as: I I I I T T F F First, let s do the negation: I I T FT F TF The truth values for the negation are in green. Now, let s do the conjunction. To do that, we should compare the letters under the R and the green letters under the : I I TF FT FF TF Since conjunctions are ONLY true when BOTH conjuncts are true, there is no way that the conjunction can ever be true. So, if someone tells you that something is AND it isn t, they are claiming something impossible. Since all of the letters under the main operator (in red) are FALSE, this statement is self-contradictory. 8