Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 0 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.