Up A Level
"And"
"And" of An "Or"
Contrapositive
"For All"
"If and Only If"
"If..., Then..."
"Not"
"Not" of An "And"
"Not" of An "If...Then"
"Not" of An "Or"
"Or"
"Or" of An "And"
Short Tautologies
"There exists"

# "P And Not(P)", as well as "[Not(P)] Or P"; also "If P, Then P"

Let P be a sentence which is true or false, but not both true and false. The sentence ``P and Not(P)'' is known as a contradiction. Regardless of whether P is true, ``P and Not(P)'' is always false. If P is true, Not(P) is false and the ``and'' of the two of them is false. If P is false, the ``and'' of the two of them is false. The table below summarizes these facts:

P and Not(P)
P Not(P) P and Not(P)
T F F
F T F
Of course, ``Not{P and Not(P)}'' must then be always true. Such sentences are called tautologies (sentences which are always true). Recall that ``Not{P and Not(P)}'' means the same as ``[Not(P)] or [Not(Not(P)]'' (see "not" of an "and" sentence). Also, ``Not(Not(P))" means the same as ``P'' (see the story of "not"). Hence, ``[Not(P)] or P'' is always true. Recall that "if P, then P" means the same as "[Not(P)] or P" (see the story of "if...then"). So, "if P, then P" is also always true and hence a tautology.

Second, consider any sentences, P and Q, each of which is true or false and neither of which is both true and false. Consider the sentence, ``(P and Not(P)) or Q''. This means exactly the same as Q, because ``P and Not(P))'' is always false. This principle is more formally explained by the truth table below: note that columns 2 and 5 have the same truth values.

[P and Not(P)] or Q
P Q Not(P) P and Not(P) [P and Not(P)] or Q
T T F F T
T F F F F
F T T F T
F F T F F

Third, let us continue with P and Q as above. The sentence ``if [P and Not(P)], then Q'' is always true, regardless of the truth values of P and Q. This is the principle that, from a contradiction, anything (and everything) follows as a logical conclusion. The table below explores the four possible cases, but the truth is simpler than that. In an ``if---then'' sentence, if the sentence in the ``if'' part is false, the entire ``if---then'' sentence is true. Since ``P and Not(P)'' is always false, making it the ``if'' part of an ''if---then'' sentence always produces a true sentence. This is like making a promise based on conditions that you know will never be satisfied.

If[P and Not(P)], then Q
P Q Not(P) P and Not(P) If [P and Not(P)], then Q
T T F F T
T F F F T
F T T F T
F F T F T

Fourth, continue with P and Q as above. The sentence ``if Q, then [P and Not(P)]'' means the same as ``Not(Q)''. This captures the principle of proof by contradiction. If some assumption such as ``Q'' implies a contradiction such as ``P and Not(P)'', then ``Q'' is false. Here the ``then'' part is always false; the only way for the entire ``if---then'' to be true is that ``Q'' is likewise false (see the story of "if---then"). The truth table below presents this more formally: note that columns 5 and 6 have the same truth values.

If Q, then [P and Not(P)]
P Q Not(P) P and Not(P) If Q, then [P and Not(P)] Not(Q)
T T F F F F
T F F F T T
F T T F F F
F F T F T T

Fifth, let us continue as above but play this time with ``{[Not(P)] or P} and Q''. Note that ``[Not(P) or P]'' is always true, and thus the truth of the entire ``and'' sentence is determined by Q. This sentence means the same as Q. Here is a truth table for this principle: note that columns 2 and 5 have the same truth values.

{[Not(P)] or P} and Q
P Q Not(P) [Not(P)] or P {[Not(P)] or P} and Q
T T F T T
T F F T F
F T T T T
F F T T F

Sixth, with P and Q as above, consider ``If {[Not(P)] or P}, then Q''. Note that the ``if'' part is always true. So the truth of the whole ``if---then'' depends only upon Q; if Q is false the promise is broken and if Q is true the promise is kept. This sentence means the same as Q, as the following truth table formalizes: note that columns 2 and 5 have the same truth values.

If {[Not(P)] or P}, then Q
P Q Not(P) [Not(P)] or P If {[Not(P)] or P}, then Q
T T F T T
T F F T F
F T T T T
F F T T F

Seventh, with P and Q as above, consider ``if Q, then {[Not(P)] or P}''. This time, the ``then'' part is always true which makes the entire ``if---then'' always true. It's like promising something that the universe always provides no matter what---it's any easy promise to keep! Here is a truth table for it:

If Q, then {[Not(P)] or P}
P Q Not(P) [Not(P)] or P If Q, then {[Not(P)] or P}
T T F T T
T F F T T
F T T T T
F F T T T

 Go to an overview of logic. Go to the home page for Tom Ramsey Go to the home page for the UHM Department of Mathematics Your comments and questions are welcome. Please email them ramsey@math.hawaii.edu.