# Four valued bit code (4VBC)

4VBC is a tabular decision procedure for propositional, modal, and deontic logic. Unlike standard truth tables constructed upon T and F or 1 and 0, 4VBC is built from the 2-bit code 00, 10, 01, and 11.

The code is defined: 00 Void; 01 True; 10 False; 11 Not void.

The easiest way to understand 4VBC is to see how its values are applied to proof tables. The table for binary conjunction comes out:

p AND Not-p

01 01 01

10 00 01

01 00 10

10 10 10

Instead of false the second and fourth rows of the 4VBC proof table gives void as its result. This, however, is not especially exciting.

What is novel regarding 4VBC is the way it defines alternative modal and deontic operators by combining any two of its four values. In addition to p and not p there are a further fourteen propositional forms; four of which are modal operators.

01 Necessarily p

00

10 Not possibily p

00

01 Possibly p is the case

11

10 Not necessarily p

11

It is a simple move, one that makes full use of the two bit code while extending a semantic vocabularly beyond the usual limits of truth tables. To see how the code is used the proof table for necessarily p then p comes out:

[]p IMP p

01 11 01

00 11 10

4VBC takes the standard operations IMP, AND, OR, XOR, EQV, NOR etc and applies them to the two bit code.

Any table that results in the value 11 on every row is a 4VBC theorem. The values 01 and 10 ensure the 4VBC counterpart to theorems proved by standard truth tables are also proved theorems by the 4VBC method; but it is the application to modal logic which makes 4VBC interesting. The method proves the following modal axioms to be theorems K, S1, S2, S3, S4, T, D, C, GL, while the Necessitation rule N proves to be logically true. It is noteworthy that 4VBC finds modal system S5 invalid; a system not universally accepted.

4VBC also incorporates a version of predicate logic however this is still under development. Those working in the area of four valued logics, modal logics and who are looking for ways to compute the concepts of ought, forbidden and permitted will find 4VBC a useful tool.

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