Implications in LogicDate: 10/29/2002 at 18:25:26 From: Kristy Subject: Implications in Logic I don't understand the first four rules to do with implications: Modus Ponens, Modus Tollens, Hypothetical Syllogism, and Disjunctive Syllogism. Can you explain them step by step, please? Date: 10/29/2002 at 20:04:51 From: Doctor Achilles Subject: Re: Implications in Logic Hi Kristy, Thanks for writing to Dr. Math. First, just so we're on the same page, here are the symbols I use: (P -> Q) Means: "If P, then Q." This sentence is true unless P is true and Q is false. (P ^ Q) Means: "P and Q." This sentence is true if and only if P and Q are both true. (P v Q) Means: "P or Q." This sentence is true if P is true, and it is also true if Q is true. The only way it can be false is if both P and Q are false. ~P Means: "Not P." This is true if P is false. Modus Ponens: The rule for this is: If you have: (P -> Q) And you also have: P Then you can conclude: Q This follows directly from the definition of (P -> Q) Which is "If P is true, then Q is true." So, if I know that if P is true, then Q must be true, AND I know that P is true, then I can validly conclude that Q is true also. Modus Tollens: This is a little trickier. The rule here is: If you have: (P -> Q) And you have: ~Q Then you can conclude: ~P Here's why: We know first of all that "If P is true, then Q is true." And we know that Q is false. With Q false, is it possible for P to be true? No! Because if P were true, then Q would have to be true. So if Q is false, then P has to be false also. Hypothetical Syllogism: The rule here is: If you have: (P -> Q) And you have: (Q -> R) Then you can conclude: (P -> R) Here's why: We know that "if P is true, then Q is true." And we know that "if Q is true, then R is true." But we don't know ANYTHING about whether any of the letters are actually true or not. Let's assume (or hypothesize) for a second that P is true. Then, by modus ponens, Q is true. And then by modus ponens again, R is true. So: IF we assume P is true, THEN we conclude R is true. Since we didn't KNOW P was true, we cannot take R home with us, but we can say that "If P was true, then R would be true." This is equivalent to saying "If P, then R" or (P -> R). Disjunctive Syllogism: The rule here is: If you have: (P v Q) And you have: ~P Then you can conclude: Q [This also works if you have (P v Q) and ~Q, you can conclude P.] Here's why: We know first of all that "P or Q is true." We also know that P is false. If P or Q is true, and P is false, then Q has no choice but to be true. So we can conclude that Q is true. I hope this helps. If you have other questions or you'd like to talk about this some more, please write back. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ |
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