Math With Base 3 Truth Values

Question

I was wondering if there is any work towards a system of mathematics were truth values could take on 3 values. Traditionally, we view every statement as either true or false. Computers are built in binary but in theory there could be a ternary computer https://hackaday.com/2016/12/16/building-the-first-ternary-microprocessor/. Could we do something similar in mathematics and what would it look like? Also, is there anyone working towards this? Thanks in advance!

Answer 1

Sure, this can be and has been done.

A third truth value $u$ may be thought of in different ways: as ”neither true nor false“ (a truth value gap) or as ”both true and false“ (a truth value glut), or possibly something different altogether.

Two common examples of three-valued logics which assume a third truth value $u$ include strong Kleene 3-valued logic $K_3$, and $LP$ .

Both of these logics define the following truth tables for the standard connectives:

Note that the particular semantics defined here is a conservative extension of classical logic: Classical input values yield classical output values.

The interpretation of $u$ is where $K3$ and $LP$ differ: $K_3$ treats $u$ as a truth value gap; $LP$ treats $u$ as a truth value glut. Although the semantics of the object language connectives are the same between the two logics, the different interpretations associated with the truth value $u$ give rise to a different meta logic, resulting in some logical inferences being valid in $K_3$ but not in $LP$ or vice versa. For instance, $\nvDash_{K_3} p ∧ ¬p → q$, and $p → q, p \nvDash_{LP} q$ (that is, $LP$ rejects modus ponens!).

A reasonable alternative to the above semantics is to change the truth tables for $\to$ and $\leftrightarrow$ such that in the interpretation where $φ$ and $ψ$ are both $u$ (middle cell), the implication and the biimplication is $1$ rather than $u$. This modification is known as Lukasiewicz logic, $L_3$.

And of course, there are a number of other possible three-valued logics with different truth tables and logical laws.

If you would like to read up more, I recommend

Priest, Graham. 2008. An introduction to non-classical logic: From if to is. Cambridge University Press.

This is a rather accessibly written overview of non-standard logics that includes a chapter on many-valued logics.