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Unformatted text preview: Predicate Logic: First-order Quantifiers Periklis A. Papakonstantinou ? University of Toronto Quantifiers and the order of quantification as it appears in definitions, statements and proofs is a very important issue. In this handout we will focus on these concepts with emphasis both on the intuition and on how to read definitions and how to read and write proofs involving quantification. “For all, there exists” vs “there exists, for all” As we have already seen the quantifiers of Predicate (or First-order Logic) are two: (i) there exists ( ∃ ) and (ii) for every ( ∀ ). Sometimes instead of “there exists an x ” we may write: “there is x ”, “there is at least one x ”, “for some x ” etc. All these English expressions correspond to the same technical notion of “there exists”. Similarly, instead of “for every x ” we could have written “for each x ”, “for all x ”, “every x ” etc. Pick a random individual from the streets. With significant probability she/he will under- stand the statement: “every human being that breaths is alive”. Furthermore, given the current knowledge for the world, she will understand that this is a true statement. How do we evaluate the truth value of such a statement, in everyday life. Well, we check this fact in our mind for an arbitrary person that breaths we conclude that she is alive and then we infer that every person who breaths is alive. What we really did in our mind is to check the statement “ x breaths then x is alive” for every x . So random individuals from the streets are capable of evaluating logical sentences (aka first-order sentences) involving one quantifier. What happens if instead of just having: “every human” we had more than one quantifiers? Here is an example: (*) “for every person there exists another person such that the second person is the mother of the first” (not that it really matters but this roughly stands for “every person has a mother”) Again, if you pick a random individual from the streets she would have definitely agreed that this is a true statement. How one concludes that (*) is true? In other words, that the truth value of (*) is true? We first pick one person and then we find her/his mother, then we first pick another person and then we find a mother for this person too, and so on. We carry on this procedure for every person. Observe that in this procedure first we deal with the leftmost/first quantifier - which in this case is ∀- and then we deal with the next one. Also, intuitively speaking, whatever is the effect of the first quantifier determines what we are going to do with the next one. ? Please notify the author regarding any typos, remarks, suggestions [email protected] . 2 Periklis Papakonstantinou (ECE 190, Fall 2006) We can certainly evaluate truth values for such real-world related sentences. Therefore, we can evaluate the truth value, in the same intuitive way, for mathematical/formal statements as well.as well....
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- Fall '06
- Logic, Predicate logic, Quantification, Universal quantification, Periklis A. Papakonstantinou, log log2 nn