Recipe

Classify a sentence as a tautology, a contradiction, or a contingency

Use when: you are given one sentence of sentential logic and asked whether it is always true, always false, or sometimes true and sometimes false

Goal: one of the three labels, backed by rows of the truth table that settle it

Taught in Classifying with truth tables

1. 1

   Bracket the sentence and find the connective that holds the whole thing together. Its column is the only one you will classify.

   Example

   In (P \to Q) \vee (Q \to P), the \vee between the two bracketed parts is the main connective.

   Why: every other column describes a part of the sentence, and a part can have a different label from the whole.

2. 2

   With n different letters, write all 2^n rows: all T first, rightmost letter alternating fastest.

   Example

   P and Q give 2^2 = 4 rows: TT, TF, FT, FF.

   Why: the three labels are claims about every row, so no row can be left out.

3. 3

   Compute each smaller part before the part that contains it, and finish with the main connective's column.

   ⚡ Shortcut

   For a conditional, fill only the rows where its antecedent is T. Every other row is T automatically, because a conditional is false only when a true antecedent meets a false consequent.

   Why: a connective's value on a row depends only on the values of its immediate parts on that row.

4. 4

   Look down the main connective's column and apply the definitions.

   • every entry is T — a tautology.
   • every entry is F — a contradiction.
   • at least one T and at least one F — a contingency; you could have stopped the moment both appeared.

   Why: these are the definitions of the three kinds, so the column settles the question with nothing left to check.