I've been reading a lot about trivalent logics to help with understanding Peter Vranas' work on a three-valued imperative logic. One thing in particular that I'm puzzled by is how we should understand negation in a trivalent logic. In this post I'm going to write some thoughts on this problem.
Consider that there are only six out of the 27 possible unary truth functions that are non-degenerate. By non-degenerate I mean that they do not reduce the number of possible outputs, given an arbitrary input. These are as follows:
It's a little simpler to tell what's going on here when you graph the relationship between the truth values. Graphically, the idea is that if each truth function is a relation between 3 truth values, then these six are the only relations where there is only one incoming edge and one outgoing edge for each truth value. In the case of identity and 1.1-3, some of these are the same edge.
Kleene, Priest, and others interpret 1.1 as negation (depending on how we interpret the truth values FALSE and UNKNOWN) but I'm not sure this is right. For one thing, this is what lets us preserve double negation elimination as a theorem/rule and with it DeMorgan's laws, etc. I think for a genuinely trivalent logic we should interpret either 2.1 or 2.2 as negation.
We can start to see the effects of this interpretation by looking at the binary truth functions. Since there are almost 20,000 of them in a trivalent logic, we're going to have to rely on symmetries to help make sense of things.
This is a collection of the characteristic truth tables for 24 binary truth functions in trivalent logic, divided into six groups of four, based on symmetries between them. Starting with the leftmost set of four (labeled '1'), the top left 3x3 square of truth values is the characteristic truth table for conjunction. Moving counterclockwise in that set of four, we see the characteristic truth tables for conjunction, nonimplication, sheffer stroke (nand), and converse nonimplication. In the fourth collection we have peirce's arrow (nor), converse implication, disjunction, and implication.
Likewise, the relationship between the unary function 1.1 and the binary functions in the first and fourth collections is as we would expect. Negating one or the other inputs rotates around the collection, and a negation in front of the function moves us to the other collection (and, importantly, back again).
What about the other collections of binary truth functions? These are truth functions that we can produce by applying the unary functions in 2.1 and 2.2. These, in some sense, rotate us through the pair of characteristic truth tables in (5 and 6) and (2 and 3), respectively.
OK, so what does all this show us? Honestly, I'm not sure. But I think that these truth tables show that there is a kind of symmetry that would give us two unary truth functions that capture many, and perhaps all, of the properties of negation that we care about. But instead of negation, in a trivalent logic we have left-handed negation and right-handed negation, or something like that.
These "handed" negation functions would have some interesting properties. First of all, they would cancel each other out:
For any proposition P, LRP = RLP = P.
Also, they would obey a triple negation elimination rule:
For any proposition P, LLLP = RRRP = P.
This would imply a modification to the DeMorgan's Laws for operator duality. Duals would no longer be defined in the typical way, and instead each operator would have two intermediate stages (not represented in the truth tables above) that it would have to pass through on the way to its classical dual.
I'm going to have to spend some more time on figuring out what these intermediate steps are, but it seems plausible to me that this is a more thoroughgoing trivalent logic than the traditional interpretation.