Propositional logic does not allow us to go
from statements like "All men are mortal" and
"Socrates is a man" to a conclusion like
"Socrates is mortal."
For that we need predicate logic.
Statements involving variables are not propositions so long
as the variable is free. Only when the variable is
bound do such statements take on a truth value.
x > 3
is neither true nor false if no value has been assigned to
x. Only when a value has been assigned to the
variable does the predicate (here, "is greater
than 3") become true or false of that value.
From the textbook:
"Quantification expresses the extent to which a predicate is
true over a range of elements.
We will focus on two types of quantification here: universal
quantification, which tells us that a predicate is true for
every element under consideration, and existential
quantification, which tells us that there is one or more
element under consideration for which the predicate is true."
The universal quantifier asserts that some predicate
holds for all elements of some set, e.g.:
"For every integer m, there exists an integer
n such that n = m + 1."
It is represented by the symbol ∀.
Universe of discourse = swans
Let W(s) be the statement "s is
white."
Then, ∀s W(s) means
"All swans are white."
A universal quantifier is rendered false by a single
counter-example.
The existentital quantifier asserts that
a predicate is true of some member of a
set, e.g.:
"There exists a real number x such that
y * x = y."
It is represented by the symbol ∃.
Universe of discourse = swans
Let B(s) be the statement "s is
black."
Then, ∃s B(s) means
"Some swan is black."
An existentital quantifier is made true by a single
example.
∀x > 0 (x3 > 0)
is the same as:
∀x(x > 0 → x3
> 0)
∃x > 0 (x2 = 2)
is the same as:
∃x(x > 0 ∧ x2 = 2)
1. c; 2. b; 3. c; 4. d; 5. c;