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Unformatted text preview: vacuous truth
We've already separated implication from quanti cation, so we can make sense of
P x ( ) A Q(x) It's true, except when P (x) is true and Q(x) is false. In particular, an implication is always true when the antecedent is false. For example, if your eyes wander to the consequent in V PR
x ;x 2 2 x +2=0Ax>x+5 . . . you could jump to the conclusion that the implication is false. Vacuous truth works because there are no counterexamples. Another way of thinking about this is that the empty set is a subset of every other set. All employees earning over 80 trillion dollars are female. All employees earning over 80 trillion dollars are male. All employees earning over 80 trillion dollars have mauve eyeballs and breathe ammonia. slide 2 equivalence
Suppose Al quits. Now consider the statement: Every male employees earns between 25,000 and 45,000. Is the statement true? What about its converse? Employee Gender Salary
Betty Carlos Doug Ellen Flo Gwen female male male female female female 500 40,000 30,000 50,000 20,000 95,000 An employee is male if, and only if, that employee earns 25,000{45,000. This is a double implication, P A Q and Q A P , or P D Q. Thought of as sets, they are equal (mutual subsets). slide 3 more equivalence
How do you feel about Break it into two implications:
x x V PR
x ;x 2 2
x x + 2 = 0 D x > x + 5: V PR V PR ;x + 2 = 0 A x > x + 5: 2 ; x > x + 5 A x 2x + 2 = 0:
2 2 The truth values are the same. English phrases: P is necessary and sucient for Q. P is true exactly when Q is true. P implies Q, and conversely. slide 4 symbolic idiom
Some expressions for restricting domains are more common than others. \Every V P x D D that is a P is also a Q." Usually Vx P D; P (x) A Q(x). Less common P; Q(x). What about Vx P D; P (x) Q(x) ( means \and")? \Some that is a is also a ." Usually W P W P . What about W P ( )A
D P Q x x D P Q x D; P x D; P x Q x ( )? ( ) Q(x). Less common slide 5 ...
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This note was uploaded on 10/13/2011 for the course COMPUTER S CSC 165 taught by Professor Dannyheap during the Fall '10 term at University of Toronto Toronto.
 Fall '10
 DannyHeap
 Computer Science

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