Binary Truth Functions

Theorem

There are $16$ distinct binary truth functions:

• Two constant operations:
  • $\map {f_\F} {p, q} = \F$
  • $\map {f_\T} {p, q} = \T$

• Two projections:
  • $\map {\pr_1} {p, q} = p$
  • $\map {\pr_2} {p, q} = q$

• Two negated projections:
  • $\map {\overline {\pr_1} } {p, q} = \neg p$
  • $\map {\overline {\pr_2} } {p, q} = \neg q$

• The conjunction: $p \land q$
• The disjunction: $p \lor q$

• Two conditionals:
  • $p \implies q$
  • $q \implies p$

• The biconditional: $p \iff q$
• The exclusive or: $\map \neg {p \iff q}$

• Two negated conditionals:
  • $\map \neg {p \implies q}$
  • $\map \neg {q \implies p}$

• The NAND: $p \uparrow q$
• The NOR: $p \downarrow q$

Proof

From Count of Truth Functions there are $2^{\paren {2^2} } = 16$ distinct truth functions on $2$ variables.

These can be depicted in a truth table as follows:

$\begin{array}{|r|cccc|} \hline p & \T & \T & \F & \F \\ q & \T & \F & \T & \F \\ \hline \map {f_\T} {p, q} & \T & \T & \T & \T \\ p \lor q & \T & \T & \T & \F \\ p \impliedby q & \T & \T & \F & \T \\ \map {\pr_1} {p, q} & \T & \T & \F & \F \\ p \implies q & \T & \F & \T & \T \\ \map {\pr_2} {p, q} & \T & \F & \T & \F \\ p \iff q & \T & \F & \F & \T \\ p \land q & \T & \F & \F & \F \\ p \uparrow q & \F & \T & \T & \T \\ \map \neg {p \iff q} & \F & \T & \T & \F \\ \map {\overline {\pr_2} } {p, q} & \F & \T & \F & \T \\ \map \neg {p \implies q} & \F & \T & \F & \F \\ \map {\overline {\pr_1} } {p, q} & \F & \F & \T & \T \\ \map \neg {p \impliedby q} & \F & \F & \T & \F \\ p \downarrow q & \F & \F & \F & \T \\ \map {f_\F} {p, q} & \F & \F & \F & \F \\ \hline \end{array}$

That accounts for all $16$ of them.

$\blacksquare$

Sources

• 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.5$: Further Logical Constants
• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $3$ Truth-Tables
• 1993: M. Ben-Ari: Mathematical Logic for Computer Science ... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.1$: Boolean operators
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.4.1$: Figure $2.5$