Quantifiers Flashcards
a declarative statement with variables, whose truth can be found by assigning “values” to the variables.
Predicate
The statement “x is taller than y.” is an example of
a predicate on x and y
The statement “x is divisible by 2.” is an example of
a predicate on x
Denoted using function notation, usually with capital letters
Predicates
Used to join propositions and predicates to form more complex statements
connectives
Used to denote that a predicate is always true.
Universal Quantifier (∀)
Used to denote that a predicate is sometimes true.
Existential Quantifier (∃)
¬∀x(P(x)) ≡
∃x(¬P(x))
¬∃x(P(x)) ≡
∀x(¬P(x))
“Let
P(x) ≡ “x is hot.”
Q(x) ≡ “x is cold.”
R(x) ≡ “x is expensive.”
Translate the following statement into quantifiers.
Something is hot.”
∃x(P(x))
“Let
P(x) ≡ “x is hot.”
Q(x) ≡ “x is cold.”
R(x) ≡ “x is expensive.”
Translate the following statement into quantifiers.
All things are expensive.”
∀x(R(x))
“Let
P(x) ≡ “x is hot.”
Q(x) ≡ “x is cold.”
R(x) ≡ “x is expensive.”
Translate the following statement into quantifiers.
Nothing is hot and cold.”
¬∃x(P(x) ∧ Q(x))
“Let
P(x) ≡ “x is hot.”
Q(x) ≡ “x is cold.”
R(x) ≡ “x is expensive.”
Translate the following statement into quantifiers.
Hot things are not cold.”
∀x(P(x) → ¬Q(x))
“Let
P(x) ≡ “x is hot.”
Q(x) ≡ “x is cold.”
R(x) ≡ “x is expensive.”
Translate the following statement into quantifiers.
If coffee is hot, then it is expensive.”
P(coffee) → R(coffee)
What is the justification for this statement?
Universal Instantiation on 1.
What is the justification for this statement?
Existential instantiation on 1.
The key words for universal quantifiers are
arbitrary and any
The key words for existential quantifiers are
fixed and some
takes a quantified statement and removes the quantifer.
(turns the statement into English)
Instantiation
takes a unquantified statement and introduces the quantifer.
(turns the English into logic)
Generalization
What is the justification for this statement?
Universal Generalization on 1.
What is the justification for this statement?
Existential Generalization on 1.
___ values can be turned into or defined in terms of any fixed or arbitrary value.
Arbitrary
___ values cannot be redefined.
Fixed
Suppose “Some men are doctors” and “Some doctors are tall”. Is it the case that “Some men are tall”?
No
Suppose “Some men are doctors” and “All doctors are tall”. Is it the case that “Some men are tall”?
Yes
Suppose “All men are doctors” and “Some doctors are tall”. Is it the case that “Some men are tall”?
No
