Recipe

Test an argument with a truth table

Use when: an argument is written in symbols and uses few enough letters to list every row

Goal: a verdict, valid or invalid, and a countermodel row if it is invalid

Taught in The truth table test

1. 1

   Collect every sentence letter from all the premises and the conclusion. Write out all 2^k rows, all-T first, with the rightmost letter alternating fastest.

   Example

   For P \to Q,\ Q \therefore P the letters are P and Q, so there are 4 rows: TT, TF, FT, FF.

   Why: one shared set of rows is the only way to see which values occur together.

2. 2

   Work out every premise and the conclusion on every row, inside out as in Chapter 12.

   Example

   Row FT: P \to Q is T (false antecedent), Q is T, and the conclusion P is F.

   Why: the verdict compares these columns row by row.

3. 3

   Scan across each row. Keep it only if every premise column shows T.

   Example

   Rows TT and FT have both premises true. Rows TF and FF each have a false premise, so drop them.

   Why: validity only promises something when the premises are all true.

4. 4

   Look at the conclusion column on the marked rows only.

   Example

   On row FT the conclusion P is F, so the argument is invalid. Row TT, where everything is T, changes nothing.

   • T on every marked row — the argument is valid; write A_1, \dots, A_n \models C.
   • F on at least one marked row — the argument is invalid; that row is a countermodel.

   Why: an argument is invalid when, and only when, some valuation makes every premise T and the conclusion F.