Unformatted text preview: w we have: Lewis Carroll Example ༉ The ﬁrst two are premises and the third is the conclusion. “All lions are ﬁerce.” 2. “Some lions do not drink coﬀee.” 3. “Some ﬁerce creatures do not drink coﬀee.” 1. ༉ Let P(x), Q(x), and R(x) be the propositional functions “x is a lion,” “x is ﬁerce,” and “x drinks coﬀee,” respectively. 1. ∀x (P(x)→ Q(x)) 2. ∃x (P(x) ∧ ¬R(x)) 3. ∃x (Q(x) ∧ ¬R(x)) Nested Quan$ﬁers ༉ Nested quantiﬁers are often necessary to express the meaning of sentences in English as well as important concepts in computer science and mathematics. Example: “Every real number has an inverse” is ∀x ∃y (x + y = 0) where the domains of x and y are the real numbers. ༉ We can also think of nested propositional functions: ∀x ∃y(x + y = 0) can be viewed as ∀x Q(x) where Q(x) is ∃y P(x, y) where P(x, y) is (x + y = 0) Thinking of Nested Quan$ﬁca$on ༉ To evaluate ∀ x ∀ y P(x, y), loop through the values of x : ༉ At each step, loop through the values for y. ༉ If for some pair of x and y, P(x, y) is false, ∀x ∀y P(x, y) is false and both the outer and inner loop terminate. ༉ To evaluate ∀ x ∃ y P(x, y), loop through the values of x: ༉
༉ At each step, loop through the values for y. If no y is found such that P(x, y) is true, the outer loop terminates as ∀x ∃y P(x, y) has been shown to be false. Order of Quan$ﬁers Examples: 1. Let P(x, y) be the statement “x + y = y + x.” Assume that domain is the real numbers. Then ∀x ∀y P(x, y) and ∀y ∀x P(x, y) have the same truth value. 2. Let Q(x, y) be...
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 Spring '14
 M.Nojoumian
 Computer Science, Logic, ∀x, Quan

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