Does the modal logic S5 have a binary sole sufficient operator?

Question

It is well-known that nand and nor are sole sufficient operators for classical logic.

I found a ternary operator earlier today that can express classical modal logic, given below:

$$ J(a,b,c) \;\;\text{is defined as}\;\; \square(a \barwedge b) \land (b \barwedge c) $$

With the addition of a truth value constant $\bot$, $\{\bot, J\}$ does generate all the classical connectives and the modal connectives $\square$ and $\lozenge$. However, without an explicit $\bot$, it is more difficult to see whether it does or not.

Here is my attempt to define a few connectives using $J$ and build my way back to $\bot$.

NOTE: an earlier version of this question erroneously said that $J(a,a,a)$ is equivalent to $\bot$.

• $J(a,a,a)$ is $\square(\lnot a)$.
• $J(\square(\lnot(a)), \square(\lnot(a)), \square(\lnot(a)))$ is $\square\lozenge(a)$ or $\lozenge a$ in S5.
• $J(\lozenge(a), \square(\lnot(a)), \lozenge(a))$ is $\top$ in S5.
• $\square\lnot(\top)$ is $\bot$.
• $J(\bot, b, c)$ is $b \barwedge c$.
• $b \barwedge b$ is $\lnot b$.
• $J(\lnot a, \lnot a, \bot)$ is $\square a$.

From there, any classical connective as well as $\square$ and $\lozenge$ can be made, so $J$ is a sole sufficient operator.

Let's focus on the logic S5, where the accessibility relation is an equivalence relation, for concreteness.

My gut says that a binary sole sufficient operator $B$ for S5 is probably impossible since we need two positions to behave non-modally, at least under some circumstances, to make a nand or a nor. I'm struggling to prove it though.

Is there a binary sole sufficient operator for S5?

Corrections:

1. $J(a,a,a)$ is NOT equivalent to bottom, it is equivalent to $\square(a \barwedge a) \land (a \barwedge a)$, which is equivalent to $\square \lnot a$ in S5.
2. Added explicit language saying that $\{J, \bot\}$ generates S5. This is hardly special though; $\{\square, \barwedge\}$ does too.

Answer 1

There is a paper that proves that it is impossible to get a three-valued basis for S5. Dugundji, James. Note on a Property of Matrices for Lewis and Langford's Calculi of Propositions. Journal of Symbolic Logic 5 (1940), no. 4, 150--151. jstor, doi: 10.2307/2268175

You may get some form of modal logic out of it, but it won't be S5, or any other of the Lewis systems.