logic+review+notes+exam+2 - Second Exam Review Notes...

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Second Exam Review Notes: Derivations in sentential logic: In this section we look at a technique designed to allow us to prove that valid arguments are in fact valid. Notice that all the arguments we deal with in this section are valid. Getting started: We use the following convention for writing out arguments: premises are written before a slash; each premise is separated by a semi-colon (if there is more than one that is); the conclusion is written after a slash. Each premise and the conclusion is a (single) formula in sentential logic. That is, a premise is not an argument in itself- it is simply asserting something. The same is true of the conclusion. Example 1: P Q; P / Q This has two premises: (1) P Q, and (2) P. And the conclusion is Q. Example 2: P & Q ; Q R ; R S/ T v S R, (3) R S. The conclusion is T v S. All derivations begin with us writing out the premises on separate lines. Each premise is annotated with ‘Pr’ written to its right (short for ‘premise). On a line immediately beneath the premises we write that we are going to show the conclusion by writing (surprisingly!) ‘SHOW’ and then whatever the conclusion is for the specific argument. We would start off the derivation in example 1 as follows: (1) P Q Pr (2) P Pr (3) SHOW: Q We would start off the derivation in example 2 like this: (1) P& Q Pr (2) Q R Pr (3) R S Pr (4) SHOW: T v S The idea is that we show that the conclusion follows from the premise(s). This is the same as showing that if the premises are true then the conclusion must be true. So, notice that for the purposes of doing the derivation (and thereby showing the validity of the arguments for which we do such) we assume that the premises are true.
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But how do we show that the conclusion follows from the premise(s)? We do it by employing a number of rules. The rules represent arguments that are valid (to show that these rules represent valid arguments we would have to use some other method besides derivation, such as using truth tables). Each rule is represented in a general form using upper case letters (or a single letter in the case of the double negation rule). Each letter stands for any arbitrary formula in sentential logic. Consider the rule arrow-out (this is represented as O in our derivations). This actually covers two rules, which are written as follows: A B A B A___ ~B__ B ~A Take the first instance (the one on the left). This is short for an argument of the following form: A B ; A / B (more accurately, it stands for an infinite number of arguments that may be obtained by replacing A and B by arbitrary formulas in sentential logic). The rule asserts that any argument that we get from substituting formulas into the letters A and B is a valid argument. So, for example, the rule would allow us to conclude that the following arguments are
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This note was uploaded on 09/15/2011 for the course PHILOSOPHY 201 taught by Professor Morgan during the Spring '08 term at Rutgers.

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logic+review+notes+exam+2 - Second Exam Review Notes...

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