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NOTES ON PREDICATE LOGIC DAMIEN PITMAN Definition: A universal set U or simply a universe is a collection of all values under consideration. Definition: Given an open sentence P( x ), the statement that asserts the truth of P( x ) for each and every value of x in a universe U is denoted ∀ x, P( x ). The statement ∀ x, P( x ) is reffered to as a universally quantified statement. In English, it is common to use the phrases for all, for each, for any, for every, or even simply for to indicate universal quantification. Definition: Given an open sentence P( x ), the statement that asserts the truth of P( x ) for at least one element of the universe is denoted ∃ x, P( x ). The statement ∃ x, P( x ) is referred to as an existentially quantified statement. In English, it is common to write for some, there exists, or even simply there is to indicate existential quantification. Theorem: For any open sentence P( x ), (∀ x, P( x )) ⇒ (∃ x, P( x )) ¬(∀ x, P( x )) ≡ (∃ x, ¬ P( x )) ¬(∃ x, P( x )) ≡ (∀ x, ¬ P( x )). Comment: When two or more variables share the same quantifier and universe, it is common to write the quantifier and universe only once. To be specific, for any statement P( x, y) and any universes U, ∀ x, y, P( x, y) ≡ ∀ x, ∀y, P( x, y), ∃ x, y, P( x, y) ≡ ∃ x, ∃y, P( x, y). However, distinct quantifiers do not commute. ∀ x, ∃y, P( x, y) 6≡ ∃ x, ∀y, P( x, y). Quantifiers are applied following a left to right precedence. That is, each quantifier applies to the statement to its right. Thus ∀ x, ∃y, P( x, y) asserts that for each x, it is true that there exists a y, which may depend on x, for which P( x, y) is true. On the other hand ∃ x, ∀y, P( x, y) asserts that there is at least one special x for which P( x, y) is true regardless of which y is under consideration. 1 NOTES ON PREDICATE LOGIC 2 Definition: An open sentence with one or more variables is quantified iff each variable has been either universally or existentially quantified. Comment: It is common to quantify over a set other than the universal set. This is accomplished by explicitly specifying the domains of the variables following the quantification. To be precise, ∀ x ∈ S, ∃y ∈ T, P( x, y) ≡ ∀ x, x ∈ S ⇒ (∃y, y ∈ T ∧ P( x, y)). Theorem: For any open sentence P( x, y), ¬(∀ x, ∃y, P( x, y)) ≡ (∃ x, ∀y, ¬ P( x, y)), ¬(∃ x, ∀y, P( x, y)) ≡ (∀ x, ∃y, ¬ P( x, y)). NOTES ON PREDICATE LOGIC 3 Examples Example: To say every rational number is real, we write ∀ x ∈ Q, x ∈ R. Comment: The statement above is equivalent to ∀ x, x ∈ Q ⇒ x ∈ R. However, it is quite common to drop the quantification when we state/write theorems. In this case, the theorem could be written: If x is rational, then x is real. In general, a univeral quantification asserts that if an element is within an understood universe or within a specified domain, then the statement that follows is true. Propositional logic doesn’t work as well with existential quantification. Example: To say some real number is irrational, we write ∃ x ∈ R, x ∈ / Q. Comment: In propositional logic, we could write ¬( x ∈ R ⇒ x ∈ Q), or in English: It is not true that if x is real, then x is rational. Since this is awkward, we almost always use existential quantification. Comment: Negation becomes just a little delicate when we use this notation. The bottom line is that the domains of quantification do not change following negation. Example: The negations of the last two examples follow. These are false! ¬(∀ x ∈ Q, x ∈ R) ≡ ∃ x ∈ Q, x ∈ /R ¬(∃ x ∈ R, x ∈ / Q) ≡ ∀ x ∈ R, x ∈ Q. Example: Write the following statement in English as simply as possible. Then disprove it. ∃ x ∈ R+ , ∀y ∈ R+ , x > y. Theorem: There is no least positive real number. Proof: In symbols, the theorem asserts: ¬(∃ x ∈ R+ , ∀y ∈ R+ , x ≤ y). This statement is equivalent to: ∀ x ∈ R+ , ∃y ∈ R+ , x > y. But, this is clearly true since given any x ∈ R+ , we can choose y = x/2 and we have y ∈ R+ and y < x. Corollary: There is no least positive rational number. Example: Where does the argument above break down if we were to try to prove that there is no least positive integer? Theorem: (Triangle Inequality) ∀ x ∈ R, ∀y ∈ R, | x + y| ≤ | x | + |y|. Example: (Archimedean Property) ∀ a, b ∈ N, ∃n ∈ N, an > b. Example: (Division Algorithm)