05-predlang

# 05-predlang - Going beyond propositional logic Consider the...

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Going beyond propositional logic Consider the following statements: p : Ling took CS245 q : Ling passed CS245 r : Ling failed CS245 Taken literally, these are all atomic statements, and formally they have no relationship with each other. But we know the following: If Ling took CS245, and Ling did not pass CS245, then Ling failed CS245. So we could add p ∧ ¬ q r to our system. 1

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But what about Lucy, Larry, Lance, Lisa, . ..? We could write p Lucy : Lucy took CS245 q Lucy : Lucy passed CS245 r Lucy : Lucy failed CS245 p Larry : Larry took CS245 q Larry : Larry passed CS245 r Larry : Larry failed CS245 etc. And then we have p Lucy ∧ ¬ q Lucy r Lucy , p Larry ∧ ¬ q Larry r Larry , etc. 2
So for each student s , we will need the premise p s ∧ ¬ q s r s . We may need other premises as well, e.g., ¬ ( q s r s ) . How many premises is that? Do we even know how many students there are? What if there are infinitely many? How many atoms must we invent?? 3

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How can we economize the presentation? Let P ( * ) denote the statement, “ * took CS245.” Let Q ( * ) denote the statement, “ * passed CS245.” Let R ( * ) denote the statement, “ * failed CS245.” P , Q , and R may be viewed as functions taking some value * and returning something that functions as an atom. We call them predicate symbols . Hence, for example, P (Lucy) denotes the atomic statement, “Lucy took CS245.” Then our premises become P (Lucy) ∧ ¬ Q (Lucy) R (Lucy) , P (Larry) ∧ ¬ Q (Larry) R (Larry) , etc. 4
We still haven’t managed to reduce the number of premises we must adopt in order to accommodate reasoning about passing and failing into our system. Trivially, we could reduce all of the premises to one, as follows: ( P (Lucy) ∧¬ Q (Lucy) R (Lucy)) ( P (Larry) ∧¬ Q (Larry) R (Larry)) ∧··· But this doesn’t really reduce anything. To achieve economy, let us introduce another element of syntax into the language: the variable . We typically use the letters x , y , z , etc., to denote variables. Variables act as placeholders for values. Hence, for example, if a variable x is taken to stand for Lucy, then P ( x ) is the same as P (Lucy) . On the other hand, if we take x to mean Larry, then P ( x ) means P (Larry) . 5

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So we would like to say that P ( x ) ∧ ¬ Q ( x ) R ( x ) is a premise, no matter what value we take x to represent. 6
Universal statements The statement P ( x ) ∧ ¬ Q ( x ) R ( x ) on its own doesn’t really mean anything. It says, “If x took CS245, but did not pass it, then x failed CS245.” But what is x ? We wish to say that the statement is a premise, independent of our choice of x . For this, we introduce some additional syntax and write x ( P ( x ) ∧ ¬ Q ( x ) R ( x )) , which says that P ( x ) ∧ ¬ Q ( x ) R ( x ) holds for every choice of x . 7

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With the universal statement x ( P ( x ) ∧ ¬ Q ( x ) R ( x )) , as a premise, we may now say, as a consequence, P (Lucy) ∧ ¬ Q (Lucy) R (Lucy) , P (Larry) ∧ ¬ Q (Larry) R (Larry) , P (17) ∧ ¬ Q (17) R (17) , by letting x represent, respectively, Lucy , Larry , and 17 .
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## This note was uploaded on 12/08/2011 for the course CS 246 taught by Professor Wormer during the Spring '08 term at Waterloo.

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05-predlang - Going beyond propositional logic Consider the...

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