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# Deductive Logical Proofs Study guide 101

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This 7 page Study Guide was uploaded by a.cole.f on Saturday January 23, 2016. The Study Guide belongs to 101 at University of Cincinnati taught by in Winter 2016. Since its upload, it has received 47 views. For similar materials see Introduction to Logic in PHIL-Philosophy at University of Cincinnati.

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Date Created: 01/23/16

Logical Derivations / Proofs Basic Tips Watch your logical connective symbols (i.e., the →, v (wedge), and • (dot) being the basic ones) and know what rules may be used with them. Acommon mistake students make is that they try to use Modus Ponens (M.P.) on disjunctions (like P v E), forgetting that such a rule only applies to if-then statements (like P → E, “if P then E”). If you get lost during the proof process, always be sure to refer back to the provided conclusion – what are you trying to get? The simpler the better – and, oftentimes, the simpler way is the correct way. Your professor will not have you use transposition on a large statement like [~B • ~(C → ~D)] → M; this is just too complicated. You will probably see complex statements like this, but you can usually break them apart by applying your rules to the simpler premises. Some examples I will supply you with will help illustrate this point. If you see multiple possibilities for applying your rules, but aren't sure if it'll be useful to you, then do it anyway. Exhaust all your options if you do not see any way to move on, and hopefully something will “fall into place.” Remember that some rules require two statements to use, while others require just one. Consider, for example, the Rule of Implication known as Modus Ponens. This rule requires both an if-then statement and the antecedent (the “if” part of the if-then statement), so this rule requires two lines. Rules like simplification only require one line. Basic Guidelines for Solving Proofs Here is a rough, general outline for solving any proof: 1) Identify your conclusion; this is what you are trying to get. Take note of what symbols and letters are in the conclusion. This will frequently give you hints as to what you should do with the statements you have. 2) Identify the premises that have letters which are in the conclusion. It is not always the case that you have to start off manipulating these statements, but they will ultimately be what you use to get to the conclusion. 3) Find a statement that you see you can manipulate with a rule. You may or may not get a result that you can immediately use, but this helps you get over the biggest hurdle of the proof process: starting it. Once you start, things tend to fall into place, and you subsequently build up some momentum. 4) Do not forget the conclusion. If you end up getting lost somewhere during the proof process, keep in mind what you set out to solve for in the first place. Sometimes, students actually get their conclusion but continue on with the proof process, forgetting their goal due to the frustration of the proof process. 5) Try to identify where even complex statements fit the form of the rules. You will, by and large, need to look past all of the letters and find the general form of the rules hiding behind them. For example: [~P • (M v ~N)] → {~[Q v (C • M)]} can simply be thought as P → Q 6) This is because everything within parentheses or brackets can be thought of as one large statement consisting of other statements. It is a “complex” statement, so to speak. You will not always encounter statements which perfectly or ideally fit the form of your rules! Using the Rules of Implication and Replacement There are some things to keep in mind when employing your rules. Here is a (relatively) short list of important aspects of applying your rules to the proof process: Rules of Inference / Implication I) Modus Ponens requires two statements to use: P → Q and the antecedent, P Consider the following example: (M • N) → [X v (M • N)] Modus Ponens states that P → Q P / therefore Q Can I use Modus Ponens to get [X v (M • N)], which is, in this case, our Q? I cannot, and that is because I do not have the antecedent (M • N) in a separate statement. Observe, if I had: (M • N) → [X v (M • N)] M • N then I could use Modus Ponens. Note that our P, in this case, is (M • N).As noted above, you can treat anything within parentheses as one statement; here I refer to (M • N) as P because that is one of the letters your rules use. In other words, I am trying to remain consistent with your rule sheet. If I were given M • N (M • N) → [X v (M • N)] I could still use Modus Ponens, even though I was provided the necessary statements in a different order. II) Modus Tollens is much the same as Modus Ponens, in terms of how you may use it. You still need two separate statements: P → Q ~ Q If you have these two statements (in any order, as demonstrated above with Modus Ponens), then you can infer ~ P III) Hypothetical Syllogism can be thought of as a connecting rule. It is much like the transitive property in math, which states that ifA= B and B = C, thenA= C. If you see multiple if-then statements, then it is likely that you may need to use a hypothetical syllogism. If you see the same symbols from your conclusion in multiple if-then statements, then it is even more likely that you'll need to use the hypothetical syllogism! So if I have (M • N) → S S → [L• (N v ~P)] then I can easily get (M • N) → [L• (N v ~P)] If you use the same symbols I used above for my math example (A, B, and C), then the H.S. would look like this: A → B B → C thereforeA→ C IV) Disjunctive Syllogism, like Modus Ponens, Modus Tollens, and the Hypothetical Syllogism, requires two individual statements. First you need a disjunction such as M v N, then you need another statement which states either ~M or ~N, so you can cancel out the appropriate letter from the disjunction M v N. In a problem, you might see something much like M v (E • Q) ~M You could use D.S. to make life easier and eliminate the M from the disjunction, resulting in E • Q If you were given ~(E • Q), then you could use the D.S. to cancel out (E • Q) and just be left with M. V) Constructive Dilemma is a tricky one for many, and I am not sure if your professor wants you to know it. If so, then remember these rough ideas which may help you pick out when to use C.D.: When you have something of the form (P → Q) • (R → S) and you are also given (or you get by using your rules) a statement like P v R then you can infer R v S Note that the P v R is a disjunction of the first two terms in the conjunction of if-then statement; R v S, being the last two terms of the conjunction of the if-then statement, follows by means of C.D. So if you ever see a dot (•) connecting two if-then statements, then see if there is a disjunction anywhere consisting of either the first two or last two terms in the conjunction. For example: [(M • N) → S] • [(M • D) → L] S v L You see that both S and L are in the conjunction of if-then statements; even though these are the last two terms of either if-then statement, you can still use C.D. to infer (M • N) v (M • D) To make things worse, consider if the above statements looked like this: [(M • N) → S] • [(M • N) → L] S v L You see that M • N occurs twice. Nevertheless, you can still use C.D. and get: (M • N) v (M • N) VI) Conjunction is a very powerful tool that you may use often. For any two statements stated individually, you can conjoin those two statements in the form of a conjunction. Observe: (M • N) → S P v M F Q v (S • N) using conjunction I could get (P v M) • F or I could get F • (P v M) Although it would result in a complicated statement, I could even get (P v M) • [Q v (S • N)] That last example is particularly complex, and you will not need make such conjunctions in your class; I only did this for demonstration purposes: you can join any individual statements by using conjunction! VII) Simplification only requires one line, and it is also very basic yet very powerful. Consider that last example above. If I had the statement (P v M) • [Q v (S • N)] I could simplify it by removing either one of the statements on either side of the dot. If I wanted the first statement, I could simplify the above down to P v M VIII) Addition, finally, is particularly useful. It essentially allows you to add anything to a statement, resulting in a disjunction between the original statement and the statement you added. For instance P v M What if I needed to get (P v M) v L? I could easily do so by adding Lto the statement P v M: (P v M) v L Rules of Replacement Key Ideas: * Rules of replacement may be used “forwards or backwards.” So for the statement ~(P • Q) I can use De Morgan's Theorem and get logically equivalent statement of ~P v ~Q But what if I started with ~ P v ~Q? I can run De Morgan's Theorem “backwards” and get ~(P • Q) * Rules of replacement do not function like your rules of inference. When using a rule of replacement, you are not inferring anything, but rather writing the same thing in a different way. *Anegation sign in a rule of replacement means that you must add a negation to the statement when you use the rule. This may not be very clear, so I shall show you what I mean: Consider the rule of replacement transposition. It says that for any statement of the general form P → Q one can write the logically equivalent statement ~Q → ~P But what if I have the following statement that I need to transpose: (M • N) → ~X How would it look when I use the transposition rule? Well, the rule states that I can “flip” this if-then statement, and in doing so I must add a negation to either term. So, if I use transposition correctly, then I should get this: ~~X → ~(M • N) Notice the ~~X. Even though X was already negated in the original statement, the rule states that I must add another negation to the term when I flip it across the arrow in the if-then statement. Of course, by the rule of replacement known as Double Negation, I can simply write this as X → ~(M • N) So consider material implication. It states that one can rewrite an in-then statement as a disjunction, but you add a negation to the first term, P. So, taking the above example... (M • N) → ~X By means of material implication, I get ~(M • N) v ~X If I originally had ~(M • N) → ~X then what would it look like after applying material implication to it? Something like this: ~~(M • N) v ~X or (M • N) v ~X by means of Double Negation. Here's another example. For the statement [X • (Z • M)] v ~ M I could run the material implication “backwards” and get the following: ~[X • (Z • M)] → ~ M I still had to add the negation symbol while going backwards!

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