Tautology in Math

Written by

Malcolm McKinsey

January 12, 2023

Fact-checked by

Paul Mazzola

Tautology definition

A tautology in math (and logic) is a compound statement (premise and conclusion) that always produces truth. No matter what the individual parts are, the result is a true statement; a tautology is always true. The opposite of a tautology is a contradiction or a fallacy, which is "always false".

Logic symbols in math

Tautologies are typically found in the branch of mathematics called logic. They use their own special symbols:

• $\wedge$ means "and"

• $=$ signifies is "equivalent to"

• $\neg$ indicates "negation"

• $\sim$ shows "not"

• $\vee$ means "or"

• $\to$ means "implies" or "if, then"

p, ~p, and q all signify the statements, with pp generally reserved for the first one and either ~p or q for the second statement

You can "translate" tautologies from ordinary language into mathematical expressions. To translate the compound statement, "I will give you $5 or I will not give you $5," we could write:

The two statements match the two parts, with the connector symbolized by $\vee$:

• $p$ takes the place of "I will give you $5"

• $\vee$ means the word "or"

• $\sim p$ takes the place of "I will not give you $5"

We can determine the two conditions of this statement (either I give you $5 or I don't) and see that both produce valid answers:

1. I give you $5, so the first statement is true and the second is false, producing a true statement.

2. I do not give you $5, so the first statement is false and the second is true. This again produces a true statement.

Truth table

Constructing a truth table helps make the definition of a tautology more clear. A truth table tests the various parts of any logic statement, including compound statements.

The first part of the compound statement, the premise, is symbolized in the first column. Logical connectors (words that tie the two statements together) are words like or, and, if. They provide conditions like sequence, reason and purpose, opposition and/or unexpected result, and so forth.

The conclusion or second statement, following the logical connector, is symbolized in the second column. The third column of the truth table shows the relationship between the two statements as either true, T, or false, F.

If every result in the third column is T, True, then the compound statement is a tautology. Here is a simple truth table built from the compound statement, "It will either snow today or it will not snow today."

The two statements together will always be true, so before we subject it to a truth table, know that it is a tautology.

No matter what, the compound statement always leads to a true result, so the statement is a tautology.

Tautology math examples

Our examples, "I will give you $5 or I will not give you $5," and "It will either snow today or it will not snow today," are very simple.

What about a logic statement that is a bit more complicated? In fact, what if we did not have even the English words, but started with just the symbols?

We hope you can "decode" that without the words, but just in case the pure logic evades you, it essentially says this:

The truth table for this must have columns for p, for q, for (p ∧ q), and a fourth column for (p ∧ q) → p.

No matter what we find with Propositions p, q, and (p ∧ q), we end up with truth, so this is a tautology. If even one of the final column's findings was false, then we would not have a tautology.