Cactus Language • Semantics 8

Posted on November 1, 2025 by Jon Awbrey

The 16 boolean functions on two variables F^{(2)} : \mathbb{B}^2 \to \mathbb{B} are shown in the following Table.

\text{Boolean Functions on Two Variables}
Boolean Functions on Two Variables

As before, all boolean functions on proper subsets of the current variables are subsumed in the Table at hand.  In particular, we have the following inclusions.

  • The constant function 0 ~:~ \mathbb{B}^2 \to \mathbb{B} appears under the name F_{0}^{(2)}.
  • The constant function 1 ~:~ \mathbb{B}^2 \to \mathbb{B} appears under the name F_{15}^{(2)}.
  • The function expressing the assertion of the first variable is F_{12}^{(2)}.
  • The function expressing the negation of the first variable is F_{3}^{(2)}.
  • The function expressing the assertion of the second variable is F_{10}^{(2)}.
  • The function expressing the negation of the second variable is F_{5}^{(2)}.

Next come the functions on two variables whose output values change depending on changes in both input variables.  Notable among them are the following examples.

  • The logical conjunction is given by the function F_{8}^{(2)} (x, y) ~=~ x \cdot y.
  • The logical disjunction is given by the function F_{14}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{)(} ~y~ \texttt{))}.

Functions expressing the conditionals, implications, or if‑then statements appear as follows.

  • [x \Rightarrow y] ~=~ F_{11}^{(2)} (x, y) ~=~ \texttt{(} ~x~ \texttt{(} ~y~ \texttt{))} ~=~ [\mathrm{not}~ x ~\mathrm{without}~ y].
  • [x \Leftarrow y] ~=~ F_{13}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{)} ~y~ \texttt{)} ~=~ [\mathrm{not}~ y ~\mathrm{without}~ x].

The function expressing the biconditional, equivalence, or if‑and‑only‑if statement appears in the following form.

  • [x \Leftrightarrow y] ~=~ [x = y] ~=~ F_{9}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{,} ~y~ \texttt{))}.

Finally, the boolean function expressing the exclusive disjunction, inequivalence, or not equals statement, algebraically associated with the binary sum operation, and geometrically associated with the symmetric difference of sets, appears as follows.

  • [x \neq y] ~=~ [x + y] ~=~ F_{6}^{(2)} (x, y) ~=~ \texttt{(} ~x~ \texttt{,} ~y~ \texttt{)}.

Resources

cc: Academia.edu • BlueSky • Laws of Form • Mathstodon • Research Gate
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

This entry was posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization and tagged , , , , , , , , , , , , , , , , . Bookmark the permalink.

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