Examples 6.3

# Examples 6.3 - Examples 6.3 Truth Tables for Propositions...

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Examples 6.3 Truth Tables for Propositions So, 6.2 gave us truth functions for all of the operators we use. Now, we’re going to implement those functions on whole propositions with multiple operators. We then stand in a position to note similarities when comparing propositions. We do this by constructing so-called truth tables . Truth tables will be just that: tables with truth values everywhere. Like: (A B) · R T T T T T F T F T T F F F T T F T F F F T F F F What’s most important for any truth table is that we specify how many lines there are going to be. In the above example, there are 8. There have to be so many, in this case, because we’re trying to create every possibility of combination of truth values. So, obviously, the more components, the more truth value combination possibilities. The normal function for figuring out how many lines is: Lines = 2ⁿ (where n = number of components). So, above we have 3 components (propositions). 2 to the 3 rd is 8—so we have 8 lines. Number of components Number of lines in a truth table 1 2 2 4 3 8 4 16 5 32 . . . . . . (I swear to god I won’t make them longer than 5—because that would be a massive truth table.) So, let’s fill a truth table in for this proposition: (A v T) · ~ T

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First, we know we have two different components, so we will have 4 lines in our truth table. Now go to the first component, and divide the number of lines by 2—that will give us half Ts and half Fs, which we want. (A v T) · ~ T T T F F Now, we divide that number (2, in this case) by 2, so we can stick a T or a F by each of the pairs of Ts and Fs—again, we want this because we’re trying to generate every possibility of true/false combinations. Then we stick in Ts and Fs accordingly:
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## This note was uploaded on 05/04/2008 for the course PHIL 2203 taught by Professor Barrett during the Spring '08 term at Arkansas.

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Examples 6.3 - Examples 6.3 Truth Tables for Propositions...

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