I see that Wikipedia’s article on ternary logic references Aymara “a Bolivian language famous for using ternary rather than binary logic.” Aside from the vaunted adjective “famous,” I am skeptical of the claim that Aymara uses a ternary logic rather than binary, skeptical also of the presupposition that natural languages use a particular logic rather than another logic, and skeptical as well of the implicature that other languages use only binary logic, infamously or otherwise.

Much of the excitement over Aymara derives from a monograph written  in the 1980’s by an engineer and machine translation pioneer, Ivan Guzman de Rojas, who observed that Aymara indicated in its inflections the degree of certitude of its respective assertions. He takes these as logical operators, just as “not” can be taken in English as a negation operator: in English, “not” takes a true statement into a false one, and a false one into a true one. E.g.,

snow is white =>True;  snow is not white =>False

snow is green =>False; snow is not green =>True

Aymara, however, also uses an inflection that takes a true statement into a neither-true-nor-false statement. This shows, he claims, that Aymara uses a third truth value, neither-true-nor-false, which is used for uncertainty.

He says, further, that the ternary logic allows the Aymara people to derive logical conclusions that are not available in binary logic, and that the Aymara people think differently from people who are limited to binary logic.

It may already have occurred to the reader that English does have exactly such an operator, “might”:

snow will fall;  snow will not fall; snow might fall

that is, “might” takes an assertion or its negation into an uncertainty.

Does this mean that English has a third truth value? Well, yes and no. The presupposition that natural languages have truth values somehow associated with them structurally seems to me to fail. Languages can express any truth values that are useful. What’s important for a language is having the operators. English actually has many such: might, may, can, could, should and must. English can express beyond notions of uncertainty and probability. It can express intention: would and will. And there are some dialects that nuance these: might could, is used in the Carolinas.

All of these are really modal notions, and can easily be handled in a bivalent logic with a modal operator, either necessity or possibility, along with negation, perhaps with an epistemic modal, believe or know. So I don’t understand why Aymara should be described as different from English in this regard: the Aymara inflections are modal notions that can be expressed in modalities.

Unless Aymara can derive conclusions unavailable to bivalent systems. De Rojas shows that in a trivalent system, a set of premises that result in indeterminate conclusions, can produce a univocal answer. If this were so, de Rojas might not have proof that Aymara is trivalent, but he would have demonstrated that trivalence is more powerful in this regard than bivalence. But his analysis here seems flawed to me. If I am right, trivalence does no better than bivalence, and actually does a bit worse. (You can read it here if you scroll down to just past the mid point of his page.) Here are his premises:

1. If it is cloudy now, there was a full moon last night  =  p->q

2. It is not the case that if it rained, it is cloudy now   =  -(r->p)

3. It is not cloudy  =   -p

From these premises it cannot be determined whether there was a full moon last night or not, since the antecedent of (1) fails, and by the definition of implication, the sentence is true. But in a trivalent system, there is one set of circumstances that certainly satisfy the premises: it rained and there was a full moon. But in a trivalent system, the uncertain options also satisfy the premises. So there remains an uncertainty that there was a full moon and it rained. One can also conclude that it might have rained and there was a full moon. Also this can be concluded: it rained, and it is uncertain whether there was a full moon. Another: maybe it rained and maybe there was a full moon. It might have rained, and there was no full moon. All of those are satisfied.

So it seems to me that trivalence here has not helped at all. In the bivalent system, we’re left with two possibilities: it rained and either there was a full moon or not. The trivalent system has laid out all the uncertainties. (I should add that de Rojas has oddly changed the premises in his two examples — if he hadn’t added uncertainty to “it has rained” there would be fewer uncertainties in the conclusions.)

So I think this particular trivalent system is not ideal. Its costs are heavy, and I can’t see the benefit.

An interesting question arises as I looked at the whole issue. How should the bivalent results be appropriately described? Suppose more than one statement satisfies a set of premises, and they are contradictory. How should those statements or the premises be described? Are the premises contradictory? Lead to a contradiction? That description doesn’t seem right, although the statements taken together are contradictory. The natural response is to say that these two statements may each satisfy the premises: p v -p. But that’s a tautology and satisfies everything.  To describe them as uncertain assumes a notion of possibility and epistemic lack of knowledge, as if the mere propositional logic had entailed a modal notion.

Since modality is available as a description, I see no problem, although it is somehow odd that possibility would be intrinsically implicated in amodal statements. What I don’t see is using a complex trivalence in place of modality.

That’s not to imply that trivalence is useless. Trivalence is needed for sentences that are semantically anomalous and maybe for presupposition failures, if you can’t swallow Russell’s analysis. But it’s a different system of trivalence. In particular, there is no operator that can take a sentence into a semantically anomalous sentence: there are too many ways of anomaly. I’m not sure there’s even a class of anomalies. And there isn’t any single means of adding or transforming into a failed presupposition.

Without an operator, a trivalent system is much simpler than the one de Rojas constructs. The truth table for anomalous sentences would look like this

assertion                    t | f | 0

negation –                  f | t | 0

double negation – –   t | f | 0

as compared with a trivalent system with an uncertainty operator:

assertion                   t | f | 0

negation –                 f | t | 0

uncertain ?                0 | 0 | t

??                               t | t | 0

-?                               0 | 0 | f

?-                               0 | 0 | t


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