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Symbolic Logic
--------------
Symbolic logic is the method of representing logical expressions
through the use of symbols and variables, rather than in ordinary
language. This has the benefit of removing the ambiguity that
normally accompanies ordinary languages, such as English, and
allows easier operation.
Basic Notation
--------------
p -> q: if p then q
p q p -> q
- - ------
0 0 1
1 0 0
0 1 1
1 1 1
p <-> q: p if and only if q XNOR
p q p <-> q
- - -------
0 0 1
1 0 0
0 1 0
1 1 1
~p: not p
p ^ q: p and q
p q p ^ q
- - -------
0 0 0
1 0 0
0 1 0
1 1 1
p v q: p or q
p q p v q
- - -------
0 0 0
1 0 1
0 1 1
1 1 1
~p ^ ~q: neither p nor q
Example
-------
p = I have a college degree.
~p = I don't have a college degree.
q = I am lazy.
~q = I am not lazy.
If I have a college degree, then I am not lazy p -> ~q
I don't have a college degree
Therefore, I am lazy.
In symbolic form: ((p -> ~q) ^ ~p) -> q
p q ~p ~q p -> ~q (p -> ~q) ^ ~p ((p -> ~q) ^ ~p) -> q
- - -- -- ------- -------------- ---------------------
0 0 1 1 1 1 0
1 0 0 1 1 0 1
0 1 1 0 1 1 1
1 1 0 0 0 1 1
Since the last column has a 0 the argument is invalid.
Equivalent Statements
---------------------
Example:
If it is Tuesday, then we will study math and physics.
p = Tuesday
q = math
r = physics
p -> (q ^ r) ≡ ~p -> ~(q ^ r)
(If it is not Tuesday, then we will not study math and physics.)
De Morgan's Law
---------------
~(p ^ q) ≡ (~p) v (~q)
and,
~(p v q) ≡ (~p) ^ (~q)
Example:
Write an equivalent statement to:
A. If the temperature is over 80 F, then the A/C will come on.
B. The temperature is not over 80 F, or the A/C will come on.
C. It is false that the temperature is over 80 F and the A/C
will not come on.
p = over 80 F
q = the A/C will come on.
A: p -> q
B: ~p ^ q
C: ~(p ^ ~q) ≡ (~p) v (q) de Morgan
p q ~p ~q p -> q ~p ^ q ~(p ^ ~q) (~p) v (q)
- - -- -- ------- ------ --------- ----------
0 0 1 1 1 0 1 1
1 0 0 1 0 0 0 0
0 1 1 0 1 1 1 1
1 1 0 0 1 0 1 1
Therefore, A and C are equivalent.