I looked over Arthur Prior's formal logic tonight.  Let V indicate a functioral variable of one argument.  Prior's book indicates that from the following all 2-valued propositions can get derived.  Actually, 2-5 in the following can get derived form 6. under substitution for V, substitution for variables, and detachment.

1. 1

2. C V 1 C V C00.

3. C V 1 C V C11.

4. C V 0 C V C10.

5. C V 1 C V C01.

6. C V 1 C V 0 V p.

I read about definitions getting expressed via a functioral variable " V " in two-valued logic last year and realized that definitions could get expressed in n-valued or infinite-valued logic via " V ".  For example, we could express the definition of alternation (disjunction) in three-valued logic as follows:

Definition of Alternation: C V Apq V CCpqq.

I wondered if there existed a way to axiomatize a multi-valued logic via a protothetical axiom set such as the one given above by Prior.  Since truth tables work for every single finite-valued logic, it turns out that every single finite-valued logic has a protothetical axiom scheme.  Note that axiom 6 encompasses all truth values of two-valued logic.

For example of a prothetical axiom set of a multi-valued logic, three-valued Lukasiewicz logic has the following table for C and N:

C  0  1  2  N
0   2  2  2  2
1   1  2  2  1
2* 0  1  2  0

Thus, a protothetical axiom set for Lukasiewicz 3-valued logic is the following (compare the last axiom with axiom 6. of the above).

1. 1

2. C V 2 C V C00.

3. C V 2 C V C01.

4. C V 2 C V C02.

5. C V 2 C V N0.

6. C V 1 C V C10.

7. C V 2 C V C11.

8. C V 2 C V C12.

9. C V 1 C V N1.

10. C V 0 C V C20.

11. C V 1 C V C21.

12. C V 2 C V C22.

13. C V 0 C V N0.

14. C V 2 C V 1 C V 0 V p.