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# May_20 [Compatibility Mode] - Wednesday May 20 So Today...

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Wednesday, May 20 So. Today, we'll talk about - - We'll probably do a little more Natural Deduction - 9.6 Rules of Replacement - 9.7, 9.8 More Natural Deduction - 9.9 Proof of Invalidity - 9.10 inconsistency - 9.11, 9.12 Indirect Proof of Validity and Short Truth Table - Introduction to Inductive Arguments

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9.6 Rules of Replacement In addition to the Rules of Inference, there are 10 Rules of Replacement. Some comments about Rules of Replacement: - You can only apply Rules of Inference to whole lines – for example, you can only do Addition to a whole line, you can't do it to just part of a line. Rules of Replacement, on the other hand, can be applied to part of a line. - Also, because both sides of a Rule of Replacement are logically equivalent, you can go back and forth between both sides of a Rule of Replacement as needed. You can never go from the conclusion of a Rule of Inference to its premises. - It is easy to forget that Simplification and Addition are Rules of Inference
9.7, 9.8 More Natural Deduction Like I said last Friday, you have to do Rules of Inference exactly as written. Rules of Inference comprise a precise system that is relatively complete but that also contains a minimum number of rules (more or less), and including (p v q); ~q; therefore p in addition to Disjunctive Syllogism would clutter things up. I will probably try and trip you up on this on a test at some point – be wary.

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9.9 Proof of Invalidity So, you know how to prove validity with Natural Deduction. However, you may have wondered at some point or another how to prove invalidity – if we don't have a way to prove invalidity, we'll just end up adding lines to our proofs until we give up. What we can do is ASSUME THE CONCLUSION TO BE FALSE. This is how proofs are normally done. An argument is shown invalid if there is at least one row of its truth table that has all true premises and a false conclusion. We can assume the conclusion is false, and then see if we can get the premises all to be true while keeping the truth values of all the letters we're using constant. (And I should throw an example in here).
9.10 Inconsistency A deductive argument that is not valid, is invalid. So, if we're attempting to prove invalidity by assuming the conclusion false and the premises true, and we cannot do so, the argument is valid.

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