Tautologies and Contradictions: Always True and Never True

Some statements are true because of their structure, not their content.

"Either it will rain tomorrow or it won't rain tomorrow."

That's true. And it tells you absolutely nothing about the weather.

It's true not because you checked the forecast. It's true because of how the words fit together. The structure guarantees truth regardless of what's happening in the world.

This is a tautology — a statement that's true by logical form alone.

Structure Versus Content

Most statements make claims about reality:

• "It's raining" — check the window
• "2 + 2 = 4" — do the math
• "Paris is in France" — look at a map

These statements could be false. They're true because of facts about the world.

But "It's raining or it's not raining" can't be false. No matter what the weather is doing, one of those options must hold. The statement is guaranteed true before you look outside.

Tautologies are true by structure. Contradictions are false by structure.

And that makes them profoundly useless — and profoundly important.

Tautologies: Always True

A tautology is a statement that's true for every possible combination of truth values.

P ∨ ¬P is the classic example: "P or not-P."

No matter what P is, the disjunction is true. If P is true, the left side is true. If P is false, the right side is true. You can't escape.

This is called the law of excluded middle — there's no third option. A proposition is either true or its negation is true.

More Tautologies

"If P implies Q, and P is true, then Q is true."

This is modus ponens: ((P → Q) ∧ P) → Q

Build the truth table:

The final column is all T. Every row. Modus ponens is a tautology — it's always valid regardless of what P and Q represent.

Why Tautologies Matter

Tautologies say nothing about the world — but they reveal what's guaranteed by logic itself.

Every valid argument form is a tautology. If you can show that "premises imply conclusion" is a tautology, you've proven that the argument structure is valid. The conclusion must follow from the premises, regardless of content.

This is the foundation of formal proof. Mathematicians spend their careers building chains of tautologies. Each step is guaranteed true by structure. String enough of them together, and you've proven theorems about prime numbers, geometry, infinity.

Contradictions: Always False

A contradiction is a statement that's false for every possible combination of truth values.

P ∧ ¬P is the classic: "P and not-P."

No matter what P is, the conjunction is false. If P is true, ¬P is false, so the "and" fails. If P is false, the left side fails. You can't satisfy both simultaneously.

This is the law of non-contradiction — a statement and its negation can't both be true.

More Contradictions

"P is true and P is false."

(P ∧ ¬P) is a contradiction.

"If P then Q, and P is true, but Q is false."

(P → Q) ∧ P ∧ ¬Q is a contradiction.

Check the truth table:

All F. This is impossible to satisfy. If you assume "P implies Q" and "P is true," you're committed to Q being true. Claiming ¬Q contradicts that.

Why Contradictions Matter

If you can derive a contradiction from a set of assumptions, you've proven that at least one assumption is false.

Proof by contradiction works because contradictions are impossible.

Suppose you want to prove Q. Assume ¬Q and see what happens. If you can derive P ∧ ¬P — a contradiction — you've shown that assuming ¬Q leads to impossibility. Therefore, Q must be true.

This technique is everywhere in mathematics. Euclid used it to prove there are infinitely many primes. Cantor used it to prove some infinities are bigger than others.

Contradictions are the boundary lines of logical possibility. Cross them, and you're in impossible territory.

Contingent Statements: Sometimes True

Most statements are neither tautologies nor contradictions.

"It's raining" is contingent — sometimes true, sometimes false. It depends on the weather.

P ∧ Q is contingent:

Sometimes true (first row), sometimes false (other rows). The truth value depends on P and Q.

Contingent statements make claims about the world. Tautologies and contradictions don't.

Recognizing Tautologies

How do you know if a statement is a tautology?

Build the truth table. If the final column is all T, it's a tautology.

Some tautologies are obvious:

• P ∨ ¬P (law of excluded middle)
• ¬(P ∧ ¬P) (law of non-contradiction)
• P → P (everything implies itself)

Some are subtle:

• (P → Q) ↔ (¬Q → ¬P) (contrapositive equivalence)
• (P ∧ Q) → P (conjunction elimination)
• ¬(P ∧ Q) ↔ (¬P ∨ ¬Q) (De Morgan's law)

These are all tautologies. Their truth doesn't depend on P or Q — the structure guarantees it.

Recognizing Contradictions

If the truth table's final column is all F, it's a contradiction.

Some contradictions are obvious:

• P ∧ ¬P
• ¬(P ∨ ¬P)

Some are subtle:

• (P → Q) ∧ (Q → R) ∧ P ∧ ¬R
• (P ↔ Q) ∧ (P ∧ ¬Q)

These can't be satisfied. They describe impossible situations.

Logical Equivalence

Two statements are logically equivalent if they have the same truth value in every row of their truth tables.

P → Q is logically equivalent to ¬P ∨ Q.

Check it:

Columns 3 and 5 match. These statements are equivalent — they say the same thing in different forms.

If two statements are logically equivalent, their biconditional is a tautology.

(P → Q) ↔ (¬P ∨ Q) is always true.

Satisfiability

A statement is satisfiable if there's at least one row where it's true.

Tautologies are satisfiable (all rows true). Contingent statements are satisfiable (some rows true). Contradictions are not satisfiable (no rows true).

Satisfiability is important in computer science. SAT solvers try to find truth assignments that satisfy logical formulas. If no assignment works, the formula is contradictory.

Vacuous Truth

Tautologies include statements that feel weird.

"If the moon is made of cheese, then 2 + 2 = 5" is false, right?

Actually, in classical logic, this conditional is true. The antecedent is false (the moon isn't made of cheese), so the conditional is vacuously true.

P → Q is true whenever P is false. The conditional only makes a claim about what happens when P is true. If P is false, the conditional can't be violated.

"All my unicorns are blue" is true. I have no unicorns, so the statement can't be falsified.

This is vacuous truth — truth by default when the premise doesn't apply.

Tautologies in Natural Language

"You can't have your cake and eat it too" is sometimes called a tautology — but that's a different sense of the word.

In everyday usage, "tautology" means a statement that's redundant or circular: "Free gift!" (gifts are free by definition).

In logic, tautology has a precise technical meaning: a statement true by logical form alone, regardless of content.

The two meanings overlap but aren't identical. Logical tautologies are structurally guaranteed. Linguistic tautologies are redundant because of word meanings.

Why This Matters

Tautologies and contradictions are the boundary cases of logic.

Tautologies are the statements logic guarantees. Every valid inference rule is a tautology. If you can reduce an argument to a tautology, you've proven it's valid.

Contradictions are the statements logic forbids. If your assumptions lead to a contradiction, at least one assumption must be false. Proof by contradiction leverages this to establish truths.

Contingent statements are everything in between. They're the claims about the world that could go either way. Science, history, daily life — all contingent.

Logic doesn't tell you which contingent statements are true. It tells you what must be true, what must be false, and what relationships hold between statements.

The Power of Structure

Here's the key insight: logic operates on structure, not content.

You don't need to know what P and Q represent to know that P ∨ ¬P is true. The form guarantees it.

You don't need to check the premises to know that modus ponens is valid. If the premises are true, the conclusion must be true — that's guaranteed by the argument's structure.

This is what makes formal logic possible. Strip away content. Represent statements as variables. Analyze the structure. The patterns you find apply universally, regardless of what the variables represent.

Tautologies are truths of structure. Learn to recognize them, and you can see the skeleton underneath all rigorous reasoning.

Part 6 of the Logic series.

Previous: Logical Equivalence: Different Statements Same Meaning Next: Arguments and Validity: When Conclusions Follow