﻿ Short Tautologies

# "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 negation of a conjunction). Also, ``Not(Not(P))" means the same as ``P'' (see negation). Hence, ``[Not(P)] or P'' is always true. Recall that "if P, then P" means the same as "[Not(P)] or P" (see conditional). 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 conditional). 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