Homework 10

Homework 10 - Consider the following, fairly simple...

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Consider the following, fairly simple argument: * The cat is purring. * The dog is wagging his tail. * If the cat is purring, then she is happy. * If the dog's wagging his tail, then he is happy. * * The cat is happy and the dog is happy. Here's the symbolization: * P * W * (P → C) * (W → D) * * It's pretty clear that this argument is valid just by a cursory examination, but in order to demonstrate this using a truth-table, you'd have to go through the whole cumbersome process of producing the sixteen row truth-table for the argument. By using the deductive method, on the other hand, we can accomplish the same goal in just three steps: by applying syntactically formulated rules of inference that correspond to valid patterns of inference, we derive the conclusion of the argument in a series of three moves, and thereby establish its validity. Of course, the claim that derivability and validity amount to the same thing is a substantive claim, and one that deserves an in-depth discussion and proof. Indeed, we will provide just such a discussion and proof in a later chapter. If you take a good look at this derivation and compare it to the symbolization of the argument above, most of what is going on should be pretty obvious, but we'll go through it explicitly in any case. The first thing to note about this derivation is its basic structure—it consists of seven lines, each numbered on the left. Following the line number, each line contains a formula of sentential logic. To the right of each formula we find the justification for that formula's presence in the derivation. You'll note that the formulae appearing on the first four lines of the derivation are just the premises of the argument whose validity we wish to demonstrate. Quite sensibly, then, our justification for including these formulae is that they are indeed premises. The formula on the last line of the derivation, as one might expect, is the conclusion of the argument. You'll note here that the justification for the conclusion is not that it is the conclusion, but rather the—so far somewhat cryptic—expression ‘&I: 5, 6’.
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This expression is actually an explanation of how we derived this formula from the earlier lines in the derivation. The two numbers refer to line numbers, and &I is the rule of inference we applied to the formulae appearing on those two lines in order to infer the formula we are justifying. Here is the derivation again with a little color thrown in to help make this obvious: At this point, you are probably wondering what these mysterious rules of inference are, how many we have, and things like that. We will begin introducing the rules of inference in just a moment, but first let us make sure we have covered everything you need to do before you can start using the rules of inference. The first thing we ought to mention here is that our way of representing a derivation is
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Homework 10 - Consider the following, fairly simple...

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