3.4 Tautologies, Contradictions, and Contingencies

Logical Constants

Tautologies, Contradictions, and Contingencies

Every well-formed formula (wff) in propositional logic falls into exactly one of three categories based on its truth table output. Learning to classify formulas this way is one of the core skills in this unit.

Tautology: A wff that is true under every possible assignment of truth values to its atomic components. When you build the truth table, the main column (the column for the whole formula) contains only Ts.

• Example: $P \vee \neg P$ is a tautology. If $P$ is true, the disjunction is true. If $P$ is false, $\neg P$ is true, so the disjunction is still true. There's no way to make it false.

Contradiction: A wff that is false under every possible assignment of truth values to its atomic components. The main column of its truth table contains only Fs.

• Example: $P \wedge \neg P$ is a contradiction. For the conjunction to be true, both conjuncts would need to be true at the same time. But $P$ and $\neg P$ can never both be true, so the result is always false.

Contingency: A wff that is neither a tautology nor a contradiction. Its truth value depends on which truth values you assign to its atomic components. The main column will have a mix of Ts and Fs.

• Example: $P \vee Q$ is a contingency. It's true when at least one of $P$ or $Q$ is true, but false when both are false. Because the output varies, it's contingent.

The key takeaway: classification is determined entirely by the pattern in the main column of the truth table. All Ts = tautology. All Fs = contradiction. Mixed = contingency.

Truth and Falsity

Logical Truth and Logical Falsity

"Logical truth" is just another name for tautology, and "logical falsity" is another name for contradiction. The important thing to understand is why these formulas always come out the same way: it's because of their logical structure, not because of what the atomic sentences happen to mean.

A logically true statement is true under all possible interpretations. Consider this less obvious example:

$((P \rightarrow Q) \vee (Q \rightarrow P))$

This might look like it could be false, but build the truth table and you'll find it's true in every row. The reason: for any pair of truth values, at least one of the two conditionals will come out true, which is enough to make the disjunction true.

A logically false statement is false under all possible interpretations. For example:

$(P \wedge Q) \wedge (\neg P \vee \neg Q)$

The left conjunct says both $P$ and $Q$ are true. The right conjunct (by De Morgan's equivalence) says at least one of them is false. No assignment can satisfy both demands at once, so the conjunction is always false.

Logical Relationships

Logical Equivalence and Validity

These two concepts connect tautologies to broader logical reasoning.

Logical equivalence holds between two wffs when they have the same truth value under every possible truth value assignment. A reliable test: construct the biconditional $wff_1 \leftrightarrow wff_2$ and check whether it's a tautology. If it is, the two formulas are logically equivalent.

• Example: $P \rightarrow Q$ and $\neg Q \rightarrow \neg P$ (the contrapositive) are logically equivalent. Build truth tables for both and you'll see their main columns match row for row. Their biconditional is a tautology.

Validity concerns arguments rather than individual formulas. An argument is valid if and only if there is no possible truth value assignment where all the premises are true and the conclusion is false.

To test validity with a truth table:

1. Identify the premises and the conclusion.
2. Build a truth table that includes columns for each premise and for the conclusion.
3. Look at every row where all premises are true.
4. Check whether the conclusion is also true in each of those rows.
5. If the conclusion is true in every such row, the argument is valid. If even one row has all true premises and a false conclusion, the argument is invalid.

• Example: Premise 1: $P \rightarrow Q$. Premise 2: $P$. Conclusion: $Q$. This is modus ponens. In the truth table, the only row where both premises are true is the row where $P$ is true and $Q$ is true. The conclusion $Q$ is true there, so the argument is valid.

Notice the connection: an argument is valid exactly when the conditional "if (all premises conjoined) then (conclusion)" is a tautology. This ties validity directly back to the concept of logical truth.