Download Notes Predicate Logic

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.
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NOTES ON PREDICATE LOGIC
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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)).
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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)