7.1 The start of Proofs From now on the arguments we are working with are all VALID. There are 18 Rules of Inference (see the last 2 pages in Course Packet, or front of txt book). Each of these rules is itself a Valid argument. Remember that the lower case p,q,r, and s are statement variables, so can stand for any statement. These 18 rules are kind of like formula you will be using in the proof itself. In 7.1 we are going over the first 4 of the 18. These are the first 4/8 Rules of Implication. With these rules, you need to have every statement above the line to get the one statement below the line. Modus ponens (MP) If p, then q You have p Therefore, you have q p q p q If it s raining, then the ground gets wet It is raining Therefore the ground is wet. R W R W I.e. W K ~H J (U v B) (K F) W ~H U v B K J K F For a horseshoe statement, having the matching p (left side) gives you the q (right side) by itself. It does NOT work with matching q s. This is one of the 2 main ways to break a horseshoe statement down. Once you identify the main connective horseshoe, remember that you want to also find the matching left half of that horseshoe statement (the p ) by itself. Then using MP, you can get the right half (the q ) by itself. Modus Tollens (MT) If p, then q You don t have q (not q) Therefore you don t have p (not p) p q ~q ~p If it s raining, then the ground gets wet. R W The ground isn t wet ~W Therefore it s not raining. ~R I.e. I J ~O ~Y (I S) ~(H v B) ~J Y H v B ~I O ~(I S)
For a horseshoe statement, the opposite of the q (right side) gives you the opposite of the p (left side). It doesn t matter where the tilde is as long as one of them has one and the other one doesn t. MT only works this one way. Opposite p s gives you nothing. MT is the other of the 2 ways to break a horseshoe statement down. Once you identify the main connective horseshoe, remember that you want to also find the opposite of the RIGHT half of that horseshoe statement (the not-q) by itself. Then using MT, you can get the opposite of the LEFT half (the not-p) by itself. Hypothetical Syllogism (HS) If p, then q p q or q I.e. r Ex ~Y (K L) If q, then r q r p q M ~Y Therefore, if p then r p r p r M (K L) If it s raining, then the ground gets wet. If the ground gets wet, then the flowers will bloom Therefore, if it s raining, then the flowers will bloom. R W W F R F If you have 2 horseshoe statements with a match along either diagonal, the matching statements get crossed out and you re left with the p on the left, and the q on the right. Disjunctive syllogism (DS) Either you have p or q You don t have p (not-p) Therefore you re left with q p v q or p v q ~p ~q q p Either it s raining, or it s snowing R v S It s not raining ~R Therefore it is snowing S I.e. U v W M v ~T (I v G) v ~F ~U T ~(I v G) W M ~F For a wedge statement, having the opposite of one side will give you what s left on the other side. This is the only way to break down a wedge statement. Once you identify the main connective wedge, remember that you want to also find the opposite of either the
RIGHT half OR LEFT half of that wedge statement (the not-p, or not-q) by itself. Then using DS, you can get what s left on the other side by itself. Exercises on the rules (fill in blank & state the rule): 1. L ~Y 2. J ~F 3. (U F) ~D 4. ~I v H ~L M ~F ~D H 1. Y (MT) 2. M J (HS) 3. U F (MP) 4. I (DS) Proofs: We will be given a set of premises and the conclusion. We will be proving that we can get from the premises to the conclusion step by step using the rules of inference. We can use the given premises in any order and any number of times. Any line that we prove becomes another premise that can be used. For now we only have 4 of the 18 rules of inference to work with: Ex. For this first example (below), we are given 3 premises and on the same line as the last premise, is also given the conclusion. You will copy the proof down exactly as it is written here. Make sure you number the lines and put the conclusion on the SAME line as the last premise. Now, we are trying to eventually get to ~R. We are finished the proof when the last line is ~R. We have 4 possible rules to work with. See if you can fit 2 lines of the proof to any of the 4 rules. Look at lines 1 and 3. Those two fit the MP rule because you have a p q, and also the matching p by itself. You will now supply the 3 rd line (conclusion) of that MP and write it down on line 4. On the same line, put the line # s used and the rule. Now you can use line 4 with line 2. Those two lines fit the MT rule because you have p q, and the opposite of the q by itself. That will give you the ~R you are looking for. Then you re done. A. 1. M 2. R H 3. M ~H / ~R 4. ~H 1,3 MP 5. ~R 2,4 MT
B. 1. J v ~T 2. T 3. J (Z ~T) / ~Z 4. J 1,2 DS (with wedge: opposite of q leaves me with p) 5. Z ~T 3,4 MP (with horseshoe: matching p gives me q) 6. ~Z 2,5 MT (with horseshoe: opposite of q gives me opposite of p) Do the next 2 in class: C. 1. T ~J 2. O (M T) 3. (M ~J) R 4. O / R D. (from textbook) 1. ~A [A v (T R)] 2. ~R [R v (A R)] 3. (T v D) ~R 4. T v D / D When you look at a statement, after identifying the main connective, you know that everything on the left is p, and everything on the right is q. Right now we can only break down statements at their main connectives. Thus when you have a statement like #1 in the above proof: ~A [A v (T R)] The only way to break this down (right now) is to have the matching p (~A), or to have the OPPOSITE of q ( ~[A v (T R)] ). That is because this statement is a p q statement. Having a T by itself won t help you until you can get the T R by itself. Then MP will give you R. 1. ~A [A v (T R)] 2. ~A 3. T / R 4. A v (T R) 1,2 MP (you half the entire left half of line 1 by itself on line 2.) 5. T R 2,4 DS (You have the opposite of one half of your wedge by itself) 6. R 3,6 MP (You have the entire left half of line 5 on line 3) Note that the main connectives for the larger statements are in blue. That completely separates the p side from the q side.
See if you can spot the Incorrect examples of the 4 rules: 1. ~U v ~F 2. Y ~G 3. ~R E 4. W G ~F ~Y ~R E G U DS G MT E MP W E HS The only correct one out of the four is #3. MP is matching the p or left half of a horseshoe statement to get the q right half by itself. #1 is incorrect because you would need an F to get ~U with DS, or you d need U to get ~F. #2 is incorrect because for MT you would need the OPPOSITE of q or G by itself to get the opposite of p or ~Y. For MP you would need a Y to get ~G. #4 is incorrect because you don t have a match on the diagonal.