Definition:Fundamental Truth Table
Definition
The fundamental truth tables are the characteristic truth tables for:
as follows:
Logical Not
The characteristic truth table of the negation operator $\neg p$ is as follows:
- $\begin{array}{|c||c|} \hline
p & \neg p \\ \hline \F & \T \\ \T & \F \\ \hline \end{array}$
Conjunction
The characteristic truth table of the logical conjunction operator $p \land q$ is as follows:
- $\begin{array}{|cc||c|} \hline
p & q & p \land q \\ \hline \F & \F & \F \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$
Disjunction
The characteristic truth table of the logical disjunction operator $p \lor q$ is as follows:
- $\begin{array}{|cc||c|} \hline
p & q & p \lor q \\ \hline \F & \F & \F \\ \F & \T & \T \\ \T & \F & \T \\ \T & \T & \T \\ \hline \end{array}$
Conditional
The characteristic truth table of the conditional (implication) operator $p \implies q$ is as follows:
- $\begin {array} {|cc||c|} \hline
p & q & p \implies q \\ \hline \F & \F & \T \\ \F & \T & \T \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end {array}$
Biconditional
The characteristic truth table of the biconditional operator $p \iff q$ is as follows:
- $\begin{array}{|cc||c|} \hline
p & q & p \iff q \\ \hline \F & \F & \T \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$
Historical Note
The separate categorization of the characteristic truth tables for these five statement forms appears in 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences.
This may be idiosyncratic.
Sources
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.13$: Symbolism of sentential calculus