Recipe

Check a prediction with a partial truth table

Use when: you have guessed that one sentence is a tautology, a contradiction, or a contingency, and want to confirm or refute the guess without building every row

Goal: either one row that breaks the guess, or a clash showing that no such row exists

Taught in Predict, then check with partial truth tables

1. 1

   Write down your guess and the value a breaking row would give the sentence: F for a tautology, T for a contradiction.

   Example

   Guessing that P \to (Q \to P) is a tautology means a breaking row would make it F.

   Why: a guess is only worth checking if you know what would break it.

2. 2

   Give the main connective that value and fill in whatever the characteristic tables force, working inward.

   Example

   P \to (Q \to P) F forces P T and Q \to P F; then Q \to P F forces Q T and P F.

   • a step forces one combination — write it down and keep going.
   • a step allows several combinations — try each one in turn.

   Why: each connective's table says which values of its parts produce the value you assumed.

3. 3

   Check whether any letter has been given both T and F.

   • every option ends in a clash — no breaking row can exist, so the guess (tautology or contradiction) is confirmed.
   • some option finishes with no clash — that option is a breaking row; the guess is wrong, and the row shows why.

   Why: a clash means no row has the assumed value, which is what "true on every row" or "false on every row" requires.