In his monograph on the trivalent logic of Aymara, Ivan Guzman de Rojas sets out to show that a trivalent logic can reach conclusions unavailable to bivalent logic. I want to tease out the import and accuracy of this extraordinary claim, and try to understand its significance for modal logic.

Consider two circumstances: p) there’s smoke; q) there’s fire.

And consider these two premises:

X) if there’s smoke, there’s fire;

¬p) there’s no smoke.

From these two premises in a bivalent logic using the standard definition of implication, you can conclude that (X) is true, but no conclusion can be reached as to whether (q) holds or not. Using a Rojas matrix:

p T | T | F | F

q T | F | T | F

X T | F | T | T

¬p F | F | T | T

The third and fourth columns reflect that the two cases in which the premises are both true, the circumstance described in (p), that there is smoke, clearly is false, but the circumstance described by (q), that there is fire, can hold or not.

Before looking at the trivalent deduction, I’d like to consider what the bivalent deduction means for bivalency. There’s a question just how to describe the consequences of the deduction. Can one deduce from the premises: (q v ¬q)? Of course, but that’s a tautology, and would be true even if (q) were provable from the premises. So, for example, from the premises it can still be concluded that (p v ¬p), even though we know from the premises that (p) is false and (¬p) holds.

Another way to express this would be (q & ¬q), a contradiction, which is always false. But that was so before the premises were stated. Similarly, we can conclude from the premises that(p & ¬p), even though the premises state (¬p).

So what is deduced from these premises with respect to (q)?

Using common sense English, the deduction sounds something like this: either of (q) or its negation could be true; (q) and (¬q) satisfy the combined premises; if (q) is true, then the premises are both true but also if (¬q) is true, both premises are still true. Let’s tentatively describe this with a modal notion of “possibly” notated as ◇. Can (◇q & ◇¬q) be concluded from the premises? Well, the premises don’t mention any modal notions, but in the meta-language of our reasoning over the premises, let’s allow modal notions. Is (◇q & ◇¬q) a tautology or a contradiction? Neither. And if we restrict the world of possibilities to the circumstances relative to the premises in our argument, then (◇p & ◇¬p) seems unequivocally false in our meta-language, since we deduced — prove — that (p) is true, and the world of possibilities are restricted only to those defined by the premises.

If this seems a leap, we could use a different modality, epistemic modality, to render the meta-language function explicit: K(◇q & ◇¬q) where “K” means “know” and to “know” means to have understood a set of sentences and made deductions. So we know that q is possible and not-q is also possible, but we also know that p is not possible. So, if we substitute p for q, K(◇p & ◇¬p) turns out obviously false, since we know (p) is true.

In plainer English, these premises and the circumstances together lay out a set of possible circumstances of what we can believe could hold.

So now how about trivalence?

Here’s a truth table in a trivalent system

p T | T | T | F | F | F | 0 | 0 | 0

q T | F | 0 | T | F | 0 | T | F | 0

X T | F | 0 | T | T | T | T | 0 | 0

¬p F | F | F | T | T | T | 0 | 0 | 0

The only cases in which the circumstances and the premises are all satsified are the fourth, fifth and sixth columns. The result is that we still know that p is false, but we now have three evaluations of (q): T, F and 0.

On the face of it, the deductive consequences are the same as the bivalent system, except that it is expressed with more truth values. It’s instructive for understanding the bivalent system in which we were left with two inconclusive “conclusions.” The common sense response to that was, we are uncertain which holds, q or not-q, based on the premises. The trivalent system has made this explicit, though somewhat redundantly, and so at the expense of elegance: the trivalent system allows three uncertainties. We are uncertain that q is true and uncertain that q is false and we either are certain that q is uncertain, or maybe we are uncertain of that as well, depending on what we mean by 0.

Let’s clean up the last option. If we treat all these deductions in the meta-language, we’ve got to say that from the premises we know that q is possible, we know ¬q is possible and we know that q is uncertain. Let’s use “~” to designate “uncertain” just as “¬” designates negation. So based on our knowledge of the premises and our deductive ability, we deduce in our metalanguage K(◇q & ◇¬q & ◇~q), or, in other words, we know that all three circumstances are possible given the world of possibilities given by the premises.

If in a trivalent system the third truth value means “neither true nor false,” then in our example, trivalence has simply added another possibility in our epistemic world of possibilities. If the third truth value means “uncertain,” then in our example, trivalence has simply added a redundancy.

There’s a difference between “uncertain” and “neither true nor false.” The former is a genuine epistemic value — “we don’t know q.” The latter is not modal — “q is neither true nor false.” Looking at the meaning of the premises, we want to be careful about this difference, since it could be significant. Remember that “q” is short for “there’s fire.” Now, in any circumstance, there’s either fire or not. It could be that we don’t know which, or it could be that, given some set of premises, we can’t conclude which, but nevertheless there’s either fire or not, and, if we have any deductive ability, we know that there’s either fire or not.

I conclude from this fact of the world — not a fact of logic, but of reality — that this use of a third truth value is intrinsically epistemic. It means not “neither true nor false” but rather “uncertain whether true or false.” If that’s so, then this use of trivalence, and its “~” operator, are redundant.

Not all uses of trivalence are redundant. Sentences that are semantically anomalous can be usefully designated as neither true nor false. “Obama stopped beating Michele” under the assumption that he never beat her, would be such a candidate for 0. It is true that either Obama beats her or not, but to have stopped beating her is to add the presuppositional premise that he once did. So there are three possible facts of the world: he used to beat her and doesn’t now; did and still does; never did. The first makes the sentence true, the second false; and the third is neither. It’s the presupposition that is either so or not; the sentence itself contains it as an assumption.

Since there is no operator that takes a sentence and turns it into a semantic anomaly, the kind of trivalence that Rojas proposed won’t work. So I find a dilemma for this kind of trivalence: it is either epistemically modal and redundant (interpreting 0 as “uncertain”) or it can’t have an operator.

Just to be comprehensive, let’s try a couple more premise sets. Suppose we replace ¬p with ~p (uncertain p).

p T | T | T | F | F | F | 0 | 0 | 0

q T | F | 0 | T | F | 0 | T | F | 0

X T | F | 0 | T | T | T | T | 0 | 0

~p 0 | 0 | 0 | 0 | 0 | 0 | T | T | T

Remember the common sense meaning: if there’s smoke, there’s fire; but we don’t know whether there’s smoke. Now according to Rojas, the Aymara would conclude the seventh column — that if we aren’t sure that there’s smoke, we still conclude that there’s fire. This is patently wrong, just for the simplest common sense. If we aren’t sure whether there’s smoke, the most confident conclusion is that it’s uncertain whether there is also fire. Now if it turns out that there’s fire, then of course, that would satisfy the premises, but that is not a certainty of the premises. The only certain conclusion is that fire is uncertain.

So what’s the mistake in the matrix? In a trivalent system, not only must all the T values be considered with respect to the conditional, but also the 0 values too. So all three last columns are possible worlds of deduction based on the premises. And those last three include each of q is true, q is false, and q is uncertain (or to insist stubbornly on a nonmodal view, neither true nor false). And that is the intuitive, obvious, common sense conclusion — the certain, unequivocal conclusion that the lack of smoke doesn’t say a damned thing about fire at all; could be fire, could be no fire but definitely fire is uncertain.

If the Aymara conclude that there’s fire from “if there’s smoke then there’s fire; and there’s no smoke” all the worse for them. They wouldn’t be the only people who have trouble understanding the conditional (it’s commonly mistaken for iff). But I’d seriously wonder how they built Tihuanaco, Puma Puncu and all those astonishing and inexplicable wall structures. So I’m confident that the Aymara don’t use such a deductive method, and are using epistemic modality much like English:

p is either T or F

uncertain of p is K(◇p & ◇¬p).

I don’t think all the above exhaust the problems with a trivalent system interpreted as “neither true nor false” with an operator “neither truly nor falsely x.” But I’d like to work them out in epistemic modality before judging.

Tags: bivalence, logic, modality, trivalence

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