*Contents+Home»,Bookshelves»,Philosophy»,Introduction to Logic and Critical Thinking (van Cleave)»-2: Formal Methods..The LibreTexts libraries arePowered by MindTouchand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UCDavis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grantnumbers 1246120, 1525057, and 1413739. Unless otherwise noted, LibreTexts content is licensed byCC BY-NC-SA 3.0.Legal. Have questions or comments? For moreinformationcontact us at [email protected]or check out ourstatus page at .®Last updated:Mar 10, 2021Back to top2.10: Tautologies, Contradictions, and Contingent State…2.12: How to Construct ProofsWas this article helpful?YesNo2.11: Proofs and the Eight Valid Forms of Inference2.10: Tautologies, Contradictions, and Contingent State…2.12: How to Construct ProofsDonateText Author(s):Matthew Van CleaveProfessor (Philosophy) atLansing Community CollegeAlthough truth tables are our only formal method of deciding whether an argument is valid or invalid in propositional logic, there is another formal method ofproving that an argument is valid: the method of proof. Although you cannot construct a proof to show that an argument is invalid, you can construct proofs toshow that an argument is valid. The reason proofs are helpful, is that they allow us to show that certain arguments are valid much more efficiently than do truthtables. For example, consider the following argument:1. (R v S)⊃(T⊃K)2. ~K3. R v S /∴~T(Note: in this section I will be writing the conclusion of the argument to the right of the last premise—in this case premise 3. As before, the conclusion we aretrying to derive is denoted by the “therefore” sign, “∴”.) We could attempt to prove this argument is valid with a truth table, but the truth table would be 16rows long because there are four different atomic propositions that occur in this argument, R, S, T, and K. If there were 5 or 6 different atomic propositions, thetruth table would be 32 or 64 lines long! However, as we will soon see, we could also prove this argument is valid with only two additional lines. That seems amuch more efficient way of establishing that this argument is valid. We will do this a little later—after we have introduced the 8 valid forms of inference that youwill need in order to do proofs. Each line of the proof will be justified by citing one of these rules, with the last line of the proof being the conclusion that we aretrying to ultimately establish. I will introduce the 8 valid forms of inference in groups, starting with the rules that utilize the horseshoe and negation.The first of the 8 forms of inference is “modus ponens” which is Latin for “way that affirms.” Modus ponens has the following form:1. p⊃q2. p3.∴qWhat this form says, in words, is that if we have asserted a conditional statement (p⊃q) and we have also asserted the antecedent of that conditionalstatement (p), then we are entitled to infer the consequent of that conditional statement (q). For example, if I asserted the conditional, “if it is raining, then theground is wet” and I also asserted “it is raining” (the antecedent of that conditional) then I (or anyone else, for that matter) am entitled to assert theconsequent of the conditional, “the ground is wet.”
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