# Do There Exist Normal Multi-Valued Interpretations for the Equivlaential Calculus?

Suppose we define a propositional calculus, just by its (object language) theorem set and its rules of inference. For example, suppose we define the C-N propositional calculus by the set of theorems deducible from

1. CCpCqrCCpqCpr (self-distribution)
2. CpCqp (simplifcation)
3. CCNpNqCqp

under the rules of inference of C-detachment "From $\vdash$C$\alpha$$\beta, as well as \vdash$$\alpha$, we may infer $\vdash$$\beta," and uniform substitution. If we do this, at least some logical systems can admit of different matrices, since the above C-N propositional calculus satisfies both the two-valued matrix: C| 1 0| N ------------ 1| 1 0| 0 0| 1 1| 1 As well as Slupecki's three-valued matrix (and other multi-valued matrices too): C| 1 .5 0| N ------------------ 1| 1 .5 0| 0 .5| 1 1 1| 1 0| 1 1 1| 1 Slupecki's matrix qualifies as a normal 3-valued matrix in the sense that if the atomic formulas take on the values {0, 1}, then the valuation of Cpq, denoted v(Cpq) \epsilon {0, 1} and v(Np) \epsilon {0, 1}. What I've read indicates that the equivalential calculus can get axiomizated by these two axioms 1. EEpqEqp "commutation" 2. EEEpqrEpEqr "association" with rules of inference of uniform substitution, and E-detachment "From \vdashE\alpha$$\beta$, as well as $\vdash$$\alpha, we may infer \vdash$$\beta$." But, I've only see authors refer to a two-valued matrix such as:

E|  0  1
--------
0|  1  0
1|  0  1

when talking about the equivalential calculus.

I feel inclined to believe that we can't have a normal 3-valued matrix which satisfies these two axioms of the equivalential calculus and still has E-detachment as a valid rule of inference, nor will any odd-valued matrix work. But, could we have a 4, 6, or an n-valued (normal) matrix where n does not equal 2? Could we have an odd-valued matrix which satisfies those axioms? If not, how do we disprove it?

As I understand things, the equivalential calculus has what gets called the two-property, which means that a formula F (formulas only involving E) qualifies as a theorem iff every lower case letter appears in F an even number of times.

-