Aymara, trivalence, competing satisfaction and modality

Originally published on Language and Philosophy, January 6, 2012

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.

language and logic, English and Aymara

Originally published on Language and Philosophy, January 5, 2012

Following up on the last post —

If English were incapable of expressing ternary logic, then Aymara could not be described in English. It does not seem too contentious, then, to conclude that any natural language, like English, Spanish (Rojas’ original) or Aymara, can express any known logic. The means will no doubt differ: analytic languages would express notions of uncertainty with word-morphemes (like “might”); agglutinative and inflectional languages should express them with affixal morphemes.

The notions seem to drive the languages, not the other way around. For example, German has a past tense conjugation for the equivalent of English “must.” English doesn’t. Instead, English uses “had to.” No one would conclude that English speakers are at a loss for the notion. Rather, the pervasive usefulness of the notion has compelled English speakers to formulate an analytical combination that has every mark of a modal expression. You can see how far English speakers have strayed from the analytical elements of those combinatorial elements: students regularly write “could of” and “would of” for “could have” and “would have.” “Could of” has become a virtual past tense inflection. English isn’t missing anything.

On the other hand, English speakers don’t use the subjunctive much at all, and standard speakers don’t use the Inner City English aspectual be. Both notions are useful, and neither is easy to accomplish without a functional morpheme, and English has such morphemes available, yet English users of the standard prestige dialect don’t use them (if that doesn’t kill the Sapir-Whorf hypothesis what would?). So maybe, as useful as they are, they are not crucial, or not as crucial as past tense modals.

It is claimed that Aymara speakers treat the past as metaphorically in front of them, the future, behind them. And the presence of uncertainty markers is brought to bear as an explanation: what you see is clear and certain like the past; the future is uncertain like the unseen behind your back.

I find the explanation unsatisfying. We all view the future as uncertain and we all see the visual field as more certain than the unseen. The explanation implies that English speakers have greater certainty about the future simply because we don’t regularly temper statements of the future with “maybe.” Well, maybe. Or maybe we just use a different metaphor.

I am less excited by such differences as I am puzzled by them. The future and the visual field are strongly linked together. Sight is the sense we most rely on. Our goals in the visual field are before us, not behind us. We walk towards our goals so we can see them, not walk backwards where we’d stumble towards it, not knowing if it is still even there.

Human understanding and reasoning are motivated by our interests. Cognitive scientists have observed that people who have lost intention or desire, lose the ability to plan and reason things through and simply can’t make sense of the world any more. And computers that don’t have such a direction, don’t learn much. So it seems that treating the future as in front of us is not a metonymy or a metaphor, but a mere factual generalization.

Whenever we are motivated by any appetite, we set ourselves to work, which means looking at the means. The past may be more certain, but it is not anywhere near as present and pressing. When I am hungry, I don’t remember the dinner I ate last night. I can’t even remember what I ate last night.

The past, in that sense, is not certain at all. It’s a haze of incomplete memories. Its certainty is something of an abstraction. The future of today’s dinner presses on me immediately. What am I hungry for? What would I like and where will I get it and what do I have to do to get it? Whatever I do, it’ll all be in front of my face, with open eyes. We set the future always in front of us. Wherever you go, you don’t get there until the future comes, and you get there facing it, whether it’s getting to the kitchen or the refrigerator or to the corner of the street or to your friend’s house or a concert hall.

So why do the Aymara gesture behind them when they talk about the future? Reichenbach pointed out that linguistic time is not a mere line extending on either side of the speaker’s present point. Linguistic time includes the event discussed and the point of reference relative to the speaker’s present. So why can’t a speaker project his reference point into the future-front to express the conviction that the projected future will be accomplished?

Just as English speakers use “may” for “might” or “have to” for “must” (with no relation to the auxilliary “have” or the possession “have”), it has to be established that the Aymara affixes always retain their uncertainty meanings. And the same with gestures.

In any case, for me the puzzle exceeds the phenomenal fact.

Aymara

Originally published on Language and Philosophy, January 4, 2012

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

Lukasiewicz, bivalence and the future

Originally published on Language and Philosophy, April 7, 2011

Just now looking at David Foster Wallace’s Fate, Time and Language, I’m puzzled by Lukasiewicz’ argument, quoted in his text, that statements about the future cannot be true or false at the moment when they are stated. It seems obvious to me that any statement about the future must be true or false, it’s just that we don’t know their truth value at the moment (except for necessary truths and inconsistent statements which may be deemed false and if contradictory, plainly false).
~K(p) does not imply (p) or (~p).
Not knowing the truth value of a statement means that the epistemic certainty of it has a degree of probability <1. But that doesn’t imply that the proposition itself has a certainty <1. The proposition itself has a probability of either 1 or 0. Why would anyone conflate the epistemic with the realis assertive?

Am I missing something? The probability of a belief for a determinist depends on the known circumstances. Those known circumstances often do not suffice for certainty.

The issue for Lukaseiwicz lies in the way we speak about possibility. If I say, “I will be at your place tonight,” even I can’t say for sure that I really will get there — I could get run over, I could get distracted by a friend. So we venture to say that it’s possible I’ll get there, and, likewise, it’s possible that I won’t. Using P for “possible” and T for “I’ll get there tonight”

P(T) & P~(T)

When the future arrives, we’ll know which of the conjuncts is true. If we’re not determinists, there’s no problem. But if we’re determinists, then one of these conjuncts is necessarily true, and the other necessarily false: necessity is interchangeable with “not possibly not,” and “necessarily not” is interchangeable with “not possibly”:

N(T) = ~P~(T)

N~(T) = ~P(T)

but if T is true, then the statement before the future arrived, added to our knowledge of necessity now in the future, yields a contradiction

N(T) & P~(T) =

~P~(T) & P~(T) =

N(T) & ~N(T)

and if T turns out to be false

N~(T) & P(T) =

N~(T) & ~N~(T) =

~P(T) & P(T)

Now, if we are not determinists, there’s no problem: the future isn’t necessary, so the truth value at the future doesn’t contradict any assertion in the past. So non determinists can assert that propositions about the future have distinct possibilities. But if we buy into determinism, we can’t assign probabilities to propositions about the future. So Lukawiewicz offered to abandon bivalence: statements about the future are neither true nor false, but somewhere in between.

But all that’s ignoring the epistemic context of our assertions of possibility. The correct formulation of our assertions, if we are determinists-in-ignorance is:

B(P(T) & P~(T))

“I believe that possibly T and possibly not T” or alternatively

B(P(T)) & B(P~(T))

“I believe possibly T and I believe possibly not T”

Believing possibly T or possibly not T is in no way inconsistent with T or ~T or N(T) or N~(T).

B(P(T) & P~(T)) & N(T)

is consistent, as is

B(P(T) & P~(T)) & N~(T)

A simpler formulation uses the anepistemic mode

~K(T)

“I don’t know T for sure” which itself implies

~K~(T)

“I don’t know ~T for sure” and therefore

~K(T) & ~K(~T)

(I’m leaving out for the moment the possibility that ~K(T) can mean “I don’t know of T,” which allows for three possibilities: I don’t know that T is true, I don’t know that T is false, I don’t know of T at all)

These are also consistent with either of T or ~T or their modal necessary versions. There are no contradictions here:

~K(T) & ~K(~T) & N(T)

~K(T) & ~K(~T) & N~(T)

The implication is that “I might not be there tonight” means both that I don’t know whether I’ll be there or not — it means the exact same as “I might be there tonight.”
Elsewhere I’ve given the evidence of the equivalence:
???I might go but I will
???I might go and I will
???I might go but I won’t
???I might go and I won’t
???I might not go but I will
???I might not go and I will
???I might not go but I won’t
???I might not go and I won’t
Unless the speaker has had a change of mind mid-utterance, these sentences are semantically incoherent. It is uncontroversial that the consequent conjuncts assert certainty over intention, so, presumably the incoherence lies in the uncertainty of the antecedent conjunct. Since the same certainties clash with the negation or without, the implication is that “might” and “might not” bear the same uncertainty: “I might not go” implies “I might go” and both can be cashed out as the anepistemic

~K(G)

~K(~G)

~K(G) & ~K(~G)

but not  ~K(G & ~G) unless we’re very contrary, since we all know

K~(G&~G)

and we know that we know it, too.