"Tautology -- it's what it is!"
What do the following statements have in common?
- "It rains when it rains."
- "All natural numbers are either less than zero, greater than zero, or equal to zero."
- "All humans are mortal. Socrates is human. Therefore, Socrates is mortal."
The answer is that all of these are examples of a most useful tool in logic: the tautology. The ancient greeks knew of tautology; it obviously comes from the greek word ταυτολογος (tautologos, 'repeated saying') Most of what we know of greek logic comes from Aristotle's attempts to lay down the fundamental laws of logic. As logic deals mainly with deducing truth from previously known truth, statement formats that cannot possibly be false provide good justification for our arguments.
A statement which is always false is a contradiction. The negation of a tautology is, then, always a contradiction.
Uncovering Tautology:
Method I: Truth tables
Symbolic logic abstracts away all the words in a statement, leaving only the grammatical structure of the argument. On the surface, this is what the logician is concerned with: determining whether the underlying format of the argument is syntatically valid. So we use variables to stand for the various propositional functions in an argument.
We leave in the grammatical connectors, but we represent them with special symbols so that it looks like we're doing work. Typically ∧ represents and, meaning both at the same time; ∨ for or, 'either one or both'; → for implication, 'if this, then that'; ↔ for equivalence, 'this is the same as that'; and ¬ for negation, 'not this'.
You computer science people will notice symbolic logic uses the inclusive or instead of the exclusive or. There was a big fight over which 'or' was more useful -- inclusive or won. See Lewis Carroll. XOR is still sometimes used, as either (hah!) ∨ or a circled ∨ sign.
Others may notice that there's an alternative notation for implication and equivalence. Some people use ⇒ and ⇔ -- these mean that, within their logical system, the implication or equivalence is always true. Some also use ≡ for equivalence, but ≡ has the same truth table as ↔.
Gah, enough tangents!
Thankfully, it's really easy to tell when an expression is a tautology if you have its truth table handy. Let's take one of the more trivial ones, transitivity:
Transitivity: [(A → B) ∧ (B → C)] → (A → C)
Premises | 1 | 2 | 3 | 4 | 5 |
A | B | C | A → B | B → C | 1 ∧ 2 | A → C | 4 → 5 |
--------------------------------------------------------
T | T | T | T | T | T | T | T |
T | T | F | T | F | F | F | T |
T | F | T | F | T | F | T | T |
T | F | F | F | T | F | F | T |
F | T | T | T | T | T | T | T |
F | T | F | T | F | F | T | T |
F | F | T | T | T | F | T | T |
F | F | F | T | T | F | T | T |
As you can see, the final column will be all true, which matches our definition of tautology. No matter what the premises are, the expression will always be logically valid.
Discovering Tautology:
Method II: Using equivalences
The elder gods of logic have already laid down several tautologies. Among them:
So using these and other tautologies with substitution, you can usually simplfy formulas down to a simple truth or falsehood.
Fudging Tautology:
Method III: Using the Principle of Duality
One of the cooler things about tautologies is that whenever you discover one, you discover two. Take any tautology, reduce it to variables, ¬'s, ∨'s, ∧'s, and then do the following two steps:
- Wherever you see a ∨, replace it with an ∧, and vice versa.
- Wherever you see a T (true), replace it with a F (false), and vice versa.
And like that, you get another tautology -- the dual tautology.
This writeup uses HTML Symbols for logical operators.