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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 $\vdash$E$\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.

Doug Spoonwood
  • 11,211

1 Answers1

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Yes, Bochvar's 3-valued logic works out this way, since it has the same set of tautologies as two-valued logic does, and it contains an equivalence, E-connective. Consequently, due to truth tables, we can in principle show that the set of tautologies for the fragment of Bochvar's 3-valued logic which has just the E-connective, has the same set of tautologies as 2-valued logic with just the E-connective.

Doug Spoonwood
  • 11,211