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Lec1.3-4 - L1.3-4 Notes Logic(continued The expression x>3...

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L1.3-4 Notes: Logic (continued) The expression x>3 is not a proposition. It contains a free variable. However, if we specify that x is 1, that is, if we bind x to the value 1, then x>3 becomes a proposition. It’s false in this case. Expressions like x>3, containing an unbound or free variable, are called propositional functions . We often denote them like this: P(x): x>3 or sometimes we simply say “Let P(x) denote x>3.” While P(x) is a propositional function, P(9) or P(1) are propositions. P(9) is 9>3 and is True. P(1) is 1>3 and is False. In effect, the substitution of a number for x in the propositional function has the effect of binding the variable x to that number. We can do this with two (or more) variables: Let Q(x,y) be x>y. It is a propositional function in two variables. Q(3,2) is a proposition. It has a truth value. Q(x,2) is a propositional function of one variable. In the above discussion P and Q (by themselves) are called predicates . x and y are variables . Predicates and propositional functions go beyond numbers. We can make them do anything we want. For instance: Let P(x): x murdered Nicole Then P(OJ) is a proposition, namely OJ murdered Nicole. In this case notice that P(3) wouldn’t make sense, any more than OJ>3 would make sense in the earlier cases. Thus we note that predicates and their propositional function require a UNIVERSE OF DISCOURSE (often denoted as U) . U is the collection of entitities which the variables in a propositional function are allowed to be bound to. When you have predicates, propositional functions and variables, there must be a universe of discourse. It may be implicit. It may consist of all possible entitities in the universe. But whatever it is, it must be there, and it is your right to demand to know what the universe of discourse is when shown a predicate used as a propositional function. Let’s move on. Let the Universe U be numbers. Consider the statement: 1
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“For every number x, x=3.” Is it a proposition? Can we determine whether it is true or false? The answer is yes, we can. It is a proposition and it is False in this case. We have succeeded in binding the variable by use of the words “For every.” Logically we proceed this way. Let P(x): x>3 We write 2200 xP(x), meaning 2200 x(x>3), meaning in words, for every number x, x>3. More accurately, 2200 x means “For every element in the Universe.” 2200 xP(x) IS TRUE IF AND ONLY IF P(x) IS TRUE FOR EVERY SUBSTITUTION FOR x FROM U. Consider in the same vein the statement: “There is a number x such that x>3.” Is that a proposition? Of course. It’s True. Letting P(x): x>3, we denote this expression as 5 xP(x) or 5 x(x>3) meaning, there is an element in the Universe such that it is larger than 3. 5 xP(x) IS TRUE IF AND ONLY IF P(x) IS TRUE FOR AT LEAST ONE SUBSTITUTION FOR x FROM U.
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