For an explanation of the notation used in this article, see the explanation folllowing "(work in progress)".
Ok, this is what I propose for a trivalent system of
unary operators in lojban:
cai = (1,-1,-1)
sai = (1,0,0)
ru'e = (1,1,-1)
cu'i = (0,1,-1)
nai = (-1,0,1)
I think {cai}, {ru'e} and {nai} are the easiest to accept.
(-1,0,1) is the most obvious generalization of {nai} from
binary. (1,-1,-1) corresponds to the strongest assertion
(certainty or necessity, depending on what system we use
it on) so it has to be {cai}. (1,1,-1) is possibility or
a weak assertion, so I think {ru'e} fits well.
Now, (1,0,0) is also an assertion, but not as strong
as certainty, something like "this is how it is, but I
give no guarantees". I think {sai} can work for that.
And finally, {cu'i} is for neutral. (0,1,-1) is not
absolutely neutral, it is uncertainty with a bent towards
assertion, but it is the closest to neutral and we do need
it to generate others, so {cu'i} has to be it.
With those 5 it is possible to generate all 27 unary
operators, with at most three of them. For example,
(0,0,1) is {naisai}, (-1,1,-1) is {cu'icai}, (0,0,0)
is {sairu'ecu'i} (among several possibilities), etc.
Only 8 of the 27 need three basic functions, the rest
can be formed from just two.
The nice thing about this system is that it can be used
for different things. For example, for a strictly logical
system we just attach them to {ja'a}, using {ja'acai},
{ja'acu'inai}, etc. And {na}={ja'anai}, so some of them
can be shortened.
But they can also be used for evidentials, attaching them
to {ju'a} for example. Then again there might be some
shortcuts, like {ju'acai} might be {za'a} and {ju'asairu'e}
might be {ca'e}, etc, but we know that we can get all
27 of them from just the simplest, which is always (1,0,-1)
and doesn't take any modifier. We can use {la'a} as the
basis for the probability set, etc.
Would that work?
co'o mi'e xorxes
The three assertions:
cai = (1,-1,-1) = necessarily
sai = (1,0,0) = probably
ru'e = (1,1,-1) = possibly
are the three that differ minimally from the simple
assertion (1,0,-1).
(work in progress)
I don't get it. What are your axes?
This proposal is from a discussion of trivalent logic in the summer of 2000. See also the Aymara Language and the paper describing Aymara suffixes in terms of trivalent logic. A detailed description of trivalent logic begins at the section labeled "4.1" in Chapter IV: The Logical Suffixes of the Aymara Language of that paper.
In brief, trivalent logic uses, in addition to the standard two truth values of TRUE and FALSE, a third value (neither true nor false, both true and false, etc.), which might be called ina, after the Aymara, or norje'u in Lojban. True is represented by 1, norje'u by 0, and false by -1. The notation P = (p1, p2, p3) for a unary operator P means that P(1) =
p1, P(0) = p2, P(-1) = p3. It relates the truth value (one of 1, 0, or -1) of an operand to the truth value of a claim (P, in this case) about the operand. p1 represents the truth of P(x) when x is known to be true (1). p2 represents the truth value of P(x) when x is known to be neither-true-nor-false (0, or ina). p3 represents the truth value of P(x) if x is known to be false (-1).
The following truth tables are the return value for true, norje'u, and false, in that order. (The plus in parentheses means that it is plausible (for example) that the statement is true, and the minus, that it is plausible that the statement is false. The glosses are an elaboration of the glosses presented in the above paper.)
| (1, 1, 1) | tautology | sairu'e | |
| (1, 1, 0) | plausibility (+)/feasibility (+) | ru'esai, naisaicu'i | |
| (1, 1, -1) | possibility (+)/eventuality (+)/doubt (-) | ru'e, naicainai | |
| (1, 0, 1) | cu'isaicu'i | ||
| (1, 0, 0) | likelihood (+)/probability (+)/adversative (-)/favourable(+)? | sai | |
| (1, 0, -1) | true, irrefutable, reliable (+) | cu'icu'i, nainai, ja'ai | |
| (1, -1, 1) | falsifiable/determinant | cu'inairu'e, cu'icainai | |
| (1, -1, 0) | evidence (+)/no controversy? | naicu'inai | |
| (1, -1, -1) | certainty (+) | cai, nairu'enai | |
| (0, 1, 1) | plausibility (-)/feasibility (-) | saicu'i, nairu'esai, cainaisai | |
| (0, 1, 0) | contingency/symmetric doubt | cu'isai | |
| (0, 1, -1) | no evidence (-)/controversy (-) | cu'i | |
| (0, 0, 1) | likelihood (-)/adversative | naisai, ru'ecu'inai | |
| (0, 0, 0) | (abdiction/abduction/abdication(?)) unimportant/aoristic/apathy | sairu'ecu'i, caicu'isai, ru'ecu'isai | |
| (0, 0, -1) | unlikelihood? (-)/granted that/unfavourable (-) | ru'ecu'i, naisainai | |
| (0, -1, 1) | evidence (-) | cu'inai | |
| (0, -1, 0) | incontingent | cu'isainai | |
| (0, -1, -1) | unfeasibility (-) | caicu'i | |
| (-1, 1, 1) | doubt (+)/uncertainty/possibility (-)/eventuality (-) | nairu'e, cainai | |
| (-1, 1, 0) | no evidence (+)/controversy (+)/extortion (-) | naicu'i | |
| (-1, 1, -1) | unfalsifiable/total contingency/certainly contingent | cu'icai | |
| (-1, 0, 1) | false, negation, reliable (-) | nai | |
| (-1, 0, 0) | unlikelihood? (+)/improbability/unfavourable (+) | sainai, nairu'ecu'i, cainaicu'i | |
| (-1, 0, -1) | cu'icaicu'i | ||
| (-1, -1, 1) | impossibility, certainty (-), not eventual | ru'enai, naicai | |
| (-1, -1, 0) | unfeasibility (+) | naicaicu'i, ru'enaicu'i, ru'esainai | |
| (-1, -1, -1) | contradiction, paradox | sairu'enai, sainaicai, caicu'icai, ru'ecu'icai | |
I am not sure what '0'/'ina' is. Is it "either true/1 or false/-1 — but I don't know which", i.e. a kind of epistemic modality as suggested by the glosses for the operators and by the gloss 'maybe'? Or is it "neither true nor false (but rather, in between)", as in fuzzy logic, in which case, 'sort of' might be a better gloss? I don't deny that when reasoning, we may want to treat 'maybe' and 'sort of' alike, but in understanding the trivalent system, and in thinking about how to implement it in Lojban, we need to get a clearer handle on the nature of '0'/'ina'.
"Trivalent Logic" in Lojban
.i le logji cu logji lo se vamji be ci da i logji lo cibdzaselva'i i ta'unai le logji cu cibdzaselva'i logji
All very fascinating! Do you suppose the Aymara language was created artificially? (At least one really weird person — I think the author of the text this all is based on — believes that Aymara was created by the culture god = space man. pc)Have you ever heard it spoken, Jorge? (Nope, but I found the article fascinating too mi'e xorxes)I see they understand the difference between nibli, mukti, and rinka! But I still don't get your logic system. I need to study it more. Why aren't you using cai, cu'i, and nai? The 3rd value is "maybe". Why do we need a new word "ina"? xod
I used the word ina just in order to use the source culture's word; in English maybe will work and in Lojban norje'u. The system still needs work, but it promises to be a rich and interesting logic which allows reasoning about much more complex things than the standard logic. Sorry for the confusion, it was me who put up the above table. — Adam
But cu'i is the neutral term, which my reading of the Aymara link indicates is the meaning of the "third term of logic". --xod
cu'i was used in the above system to represent a unary operator, not a truth-value, though it no doubt could have been done differently. Also, I was looking for a word to parallel jetnu and jitfa. — Adam
In some contexts in might be na'i in Lojban. In Zen and the Art of Motorcycle Maintenance it is called mu (from Japanese).
Just a couple of add-ons:
pc >|8}
For a given unary operator, its opposite is the result of applying nai to it, or of multiplying it by -1. For example, the opposite of ru'e (1, 1, -1) is ru'enai (-1, -1, 1). Also, its antonym is the result of applying it to nai, or of reversing its order. For example, the antonym of ru'e (1, 1, -1) is nairu'e (-1, 1, 1).
Unary operators are commutative ([cairu'e]sai = cai[ru'esai]) and so the opposite of the antonym is the same as the antonym of the opposite ([nairu'e]nai = nai[ru'enai]), and the opposite-antonym of the opposite-antonym is the operator itself (nai[nairu'enai]nai = [nainai]ru'e[nainai] = ru'e). ru'e and cai are opposite antonyms.
It is possible to move nai around, but every operator that it crosses must be replaced by its opposite-antonym. cainai -> nairu'enai nai -> nairu'e, etc.
This works very similarly to the traditional logical operators of possibility and necessity, i.e. it is possible that not ... = it is not necessary that ..., etc. However, there are differences. It is possible that broda and it is possible that not broda in two-valued modal logic says nothing about the actual truth value of broda. However, (taking AND(x, y) to be min(x, y)) the truth-value of ge ru'e broda gi nairu'e broda is equivalent to the truth-value of cu'icai broda, and the statement claims that the truth-value of broda must necessarily be norje'u:
| broda | ge | ru'e | broda gi nairu'e broda |
| 1 | -1 | 1 | -1 |
| 0 | 1 | 1 | 1 |
| -1 | -1 | -1 | 1 |
The opposite and the antonym can be defined with any unary operator that nullifies itself when applied to itself, i.e. ja'ai, nai, cu'i, and naicu'inai. For example, the cu'i-opposite antonym of ru'esai (1, 1, 0) is cu'iru'esaicu'i = naisai (0, 0, 1), and so ru'esai cu'i = cu'i naisai.
It is relatively intuitive to understand what the nai-opposite antonym means ("it is not possible/probable/certain, etc. that not"), but it is less obvious what the cu'i-opposite antonym means. (Or for that matter what the cu'i-opposite or cu'i-antonym means) at least for me. Any suggestions?
There are 3**9 = 19,683 possible binary operators. A simple and relatively intuitive definition for AND is the minimum of the two predicates and for OR is the maximum of the two predicates:
| x | y | x AND y | |
| 1 | 1 | 1 | |
| 1 | 0 | 0 | |
| 1 | -1 | -1 | |
| 0 | 1 | 0 | |
| 0 | 0 | 0 | |
| 0 | -1 | -1 | |
| -1 | 1 | -1 | |
| -1 | 0 | -1 | |
| -1 | -1 | -1 | |
| x | y | x OR y | |
| 1 | 1 | 1 | |
| 1 | 0 | 1 | |
| 1 | -1 | 1 | |
| 0 | 1 | 1 | |
| 0 | 0 | 0 | |
| 0 | -1 | 0 | |
| -1 | 1 | 1 | |
| -1 | 0 | 0 | |
| -1 | -1 | -1 | |
-- Adam
This is all well and good if you want to represent degrees of trueness in language, but can you use it to prove things about propositions with values over the three-valued set? Like do you have the equivalent of many versions of modens ponen: In ordinary logic modus ponens says A -> B and you have A true, then B is true. And you also get the contrapositive, modus tollens: if you have not B, you have not A. But if A is probably true, is B possibly true? If B is possibly not true, is A probably not true? If A implies only probably B, does having A mean you have possibly B? (you need the truth/falsity operators in unary forms that you can attach to letters) And does not having B mean you possibly don't have A?
With just this rule and the rule that you can accept any tautology as true, you can prove all the theorems in ordinary propositional calculus (it's complete--- too simple for Godel's theorrem to apply (in fact he proved the "Completeness theorem" for propositional calculus), for that you need some numbers or sets, like if you make the propositions things like a is an element of the set B then you run into incompleteness (and I may add theorems that are actually worth proving!)). But Anyway with the modus ponens/tollens I proposed, can we prove all the theorems of this logic? Or is it, as I fear, inconsistent somewhere? Or incomplete? Or both? (It might be incomplete because two propositions that might have a difference in truth value between one of our three values and then you'd have to start reasoning about possibly probably and so on as someone did above with the 27 beautiful words that unfortunately sound like trig functions to me...
-Millie
Getting numbers in does not guarantee incompleteness, since Tarski analysis is complete, while Peano arithmentic is not. And you can get incompleteness without numbers (though Goedel's proof will not be directly useful) in the strong sense that no finite extension is complete (just this is not complete is a snap - leave out a rule or an axiom). Many of these systems may be incomplete in the sense that not all possible three-valued claims can be expressed in them, but at least some are complete in this sense also. pc
On creating three-value logic truth tables. For combining "true" & "maybe"
(or "fictive"): a novel about Teddy Roosevelt as a detective is not more
"true" for having a real person as a character. But an appearance of Sherlock
Holmes in a biography of Teddy Roosevelt seriously discredits the veracity of
the author... (mi'e maikl)