Lesson_Ten_and_Test - LESSON 10 Quantificational Logic Part...

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Sections 20 & 21: Proving Validity & Using the Method , Barker, pp. 121-130. ***Read the sections, then the following additional comments. Note that these comments and assignments refer to two sections in Barker. In order to do proofs in quantificational logic, we must follow the indirect method. In general, the use of the method in quantificational proofs follows the identical strategy as in truth-functional proofs. Once again this is what we do: 1. Include the negation of your intended conclusion as another premise. 2. Apply the rules of deduction until you reach a contradiction. 3. You have then shown that your intended conclusion is true since its negation leads to a contradiction . But before you can apply the truth-functional rules, you have to prepare the arguments in certain ways. There are two preliminary steps: 1. You need to reverse negative quantifiers. You do so by means of the equivalences we mentioned in the earlier section, the “slide-through” rules. So, when you set out to do your proof, you go through and apply the rules of equivalence in order to eliminate all negative quantifiers. 2. But you are still not ready to complete the proof. The second thing you need to do is to drop the quantifiers and supply names for the variables. You cannot apply the rules of truth-functional deduction to a logical relationship as long as it is governed by a quantifier. What we need to do for all of our quantified expressions is to supply names as constants. For example, if it is true that Everybody knows how to ride a bicycle.
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