Homework 9-1

# Homework 9-1 - Let us take a look at our truth-tree rules...

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Let us take a look at our truth-tree rules so far: DEFINITION: Truth-Tree Construction Rules How about our final connective, negation? We pointed out and exploited the fact that a formula is true just in case its negation is false and vice versa, but how are we going to use this fact to deal with negation in truth-trees? Consider for a moment the fact that each branch of a truth-tree corresponds to a set of truth-value assignments, namely, the set of truth-value assignments that make all the formulae appearing on that branch true. This being the case, we thus know that if an atomic formula appears on a particular branch, then that formula must be true on each of the corresponding truth-value assignments, and hence that each such assignment assigns the value t to that atomic formula. Now consider again the fact that a formula is false just in case its negation is true—if the negation of an atomic formula appears on the branch, then we know that the truth-value assignments in question must make this negation true, and hence must assign the value f to the atomic formula. Here are all the truth-tree rules we need: There is just one thing to note about these rules: they must be used to extend every branch of the tree on which the formula in question is located. In other words, if you are applying a rule to a formula that already has more than one branch growing out of it, you must extend each such branch before you can check off the formula. Just to make sure this is absolutely clear, here is an example: There are a few other things we ought to mention regarding truth-trees. First, though it might be intuitive, we should spell out how to read a truth-value assignment from a truth-tree. Each branch of the tree corresponds to a set of truth-value assignments, so

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