Learning Objectives
Interpreting Truth Tables
- Tautology
- Contradiction
- Contingency
- Logically Equivalent
A tautology is a proposition that is always true, regardless of the truth values of the propositional variables it contains. A proposition that is always false is called a contradiction. A proposition that is neither a tautology nor a contradiction is called a contingency.
Examples
An example of a tautology is: I am going to take Math for Liberal Arts this semester or I’ m not going to take Math for Liberal Arts this semester. This statement is always true so it is a tautology.
An example of a self-contradition is: I will get an A in this class and I will not get an A in this class. This statement is always false so it is a self-contradiction.
Caution: Don’t make the mistake that every statement is either a tautology or a self-contradiction. We have seen many examples of truth tables which have a mixture of trues and falses in the final column. These statements are sometimes true and sometimes not true. These are called a contingency.
Examples
Complete a truth table to classify the statement as tautology, self-contradiction or contingency.
a. p 
(~p)
b. (p q) 
(~p ~q)
Solution:
a. This is a tautology
b. This is a self-contradiction
Logically Equivalent Statements
When the truth values for two statements are identical, the statements are said to be logically equivalent. That is, both compositions of the same simple statements have the same meaning.
Common Equivalent Statements
~(~p) 
p the Double Negative Law
pq qp the Commutative Law for conjunction
pq qp the Commutative Law for disjunction
(pq)r p(qr) the Associative Law for conjunction
(pq)r p(qr) the Associative Law for disjunction
~(pq) (~p)(~q) DeMorgan’s Laws (more on this one in the next section)
~(pq) (~p)(~q)
p(qr) (pq)(pr) the Distributive Laws
p(qr) (pq)(pr)
pp p Absorption Laws
pp p
For an example of logically equivalent, lets look at the forms of the conditional that we’ve learned. We can see that the original conditional and the contrapositive are logically equivalent, and that the converse and inverse are logically equivalent.
| Conditional | Converse | Inverse | Contrapositive | ||
|---|---|---|---|---|---|
| p | q | [latex]p\rightarrow{q}[/latex] | [latex]q{\rightarrow}p[/latex] | [latex]\sim{p}\rightarrow\sim{q}[/latex] | [latex]\sim{q}\rightarrow\sim{p}[/latex] |
| T | T | T | T | T | T |
| T | F | F | T | T | F |
| F | T | T | F | F | T |
| F | F | T | T | T | T |
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