Propositional logic · Definition

Truth table

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Before you start you need to know

A truth table organizes all possible values of the propositions and the result of a formula in each case.

Definition

Let \(P_1,\ldots,P_n\) be atomic propositions. A truth table records the value of a formula for every assignment of them.

\[\left|\mathrm{Asig}(P_1,\ldots,P_n)\right|=2^n\]

With \(n\) atomic propositions there are \(2^n\) assignments, and each one occupies a row. The final column gives the formula's value in that row.

For \(P\to Q\) with two propositions there are 4 rows: \(T,T\), \(T,F\), \(F,T\), \(F,F\). The formula is false only in the row \(T,F\).

A common mistake is to omit some assignment, for example writing only 3 rows for two propositions; the table must cover all \(2^n\) combinations.

Think about it

How many rows of assignments does a formula with three atomic propositions require?

Eight, because there are 2³ possible assignments.

Frederic P. Schuller, Lectures on the Geometrical Anatomy of Theoretical Physics · Lecture 1, Introduction / Logic of Propositions and Predicates, YouTube, 2015.