T1.3-4

T1.3-4 - &lt;?xml version=&quot;1.0&quot;...

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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 are propositional functions . They are often denoted: P(x): x>3 While P(x) is a propositional function, P(9) or P(1) are propositions. P(9) is 9>3 and is T. P(1) is 1>3 and is F. In effect, the substitution of a number for x in the propositional function has the effect of binding the variable x to that number. 1

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We can do this with two 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 . 2
Predicates and propositional functions go beyond numbers. We can make them do anything we want. Let P(x): x murdered Nicole Then P(OJ) is a proposition, namely OJ murdered Nicole. To ensure that propositional functions “make sense” they are always defined relative to a UNIVERSE OF DISCOURSE (often denoted as U) . U is the collection of entities to which the variables in a propositional function are allowed to be bound. The Universe of Discourse, also called the “Domain of discourse,” is obligatory . 3

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Let the Universe U be numbers. Consider “For every number x, x>3.” Is it a proposition? Can we determine whether it is true or false? Logically we proceed this way.
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T1.3-4 - &lt;?xml version=&quot;1.0&quot;...

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