I'm thinking about this question about modal logic.

I'm wondering whether a sole sufficient operator needs to be an abbreviation of existing functions or not.

More concretely, consider an equational theory $E$ in which the sole function is a binary function symbol $\to$ consisting of all equations $\varphi(\vec{x}) = \psi(\vec{x})$ such that $\varphi(\vec{x}) \leftrightarrow \psi(\vec{x})$ is a theorem of classical propositional calculus.

$\to$ is certainly a sole sufficient operator for $E$.

I'm wondering whether $\barwedge$ is a sole sufficient operator for $E$.

$x \to y$ is certainly equivalent to $x \barwedge (y \barwedge y)$, but $\barwedge$ is much more powerful.

The $\to$-fragment of classical propositional logic is not capable of expressing an outright contradiction, but with $\barwedge$ we can express it as $a \barwedge (a \barwedge a)$. This observation proves that $\barwedge$ is not an abbreviation for some expression containing only $\to$ as functions.

However, the fact that $a \barwedge (a \barwedge a)$ is a contradiction in the ur-semantics is not strictly speaking a problem. We can take our original theory $E$, go through, and replace all occurrences of $\to$ with its $\barwedge$-encoding and get a new theory $E^\barwedge$. $E^\barwedge$ is merely not a complete theory.

So I guess my question is:

  1. Can you call $\barwedge$ a sole sufficient operator?
  2. If (1), is there a way to distinguish $\barwedge$ from SSO's that are definable in terms of the existing functions? If not (1), what do you call the type of thing that $\barwedge$ is?


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