| The other day on the main list (I think), Cowan actually committed himself to a distinction between universal quatifiers with existential import and those without. While details are fuzzy, it seems that quantifiers that govern variables explicitly, do not have existential import, those that attach directly to {lo} or a bridi do, the models being symbolic logic and traditional logic respectively. xorxes immediately countered that that couldn't be right beause of the way the quantifers worked with negation: {ro broda naku brode} ought to be equivalent to {naku su'o broda cu brode}, but, while the first has existential import, the second does not. From this xorxes concluded that the first did not really have existential import either. He then went on to hold that, since universals don't have existential import, the {lo broda}, which is {lo ro broda} could work sometimes even if there were no brodas (I forget what that was supposed to help with or be aproblem for). Point one. In the modern logic on which Lojban is built, the universal quantifier has existential import: from "for all x, Fx" it is always possible to infer (in two steps, usually) "for some x, Fx." regardless of how complex F is. Historically, the universal quantifier of symbolic logic is the A proposition of traditional logic with an expanded predicate and atrophied subject -- all the content is shifted to the predicate and the subject becomes a tautological one (so unmentioned). The apparent lack of existential import comes from taking the predication in the antecedent of a conditional following the quantifier as though it were the subject rather than part of the predicate (so its potential emptiness is a result of the conditional, not the quantifier). This came about partly for practical reasons, partly because the third generation of logicians never learned or early forgot their history. The final stage of amnesia came when, in the 1950s, someone decided that this process of loss of existential import ought to be carried all the way and developed a (very boring) system in which the inference mentioned at the beginning of this paragraph did not hold. That is not the present system, however. So, two. xorxes' inference is quite right under the present rules, so, if the distinction between is to be maintained, the rules need changing to conform. Trying to change as little as possible, it is clear that the negation of a sentence with existential import, is one that does not have and conversely, and, under Cowan's suggestion, those with have {ro (lo) broda} and those without have {ro da}. So, passing a negation around not only changes {ro} to {su'o} and conversely, but also {Q (lo) broda} to {Q da broda} or some such and conversely. So {ro broda naku brode} becomes {naku su'o da broda je brode} (or something like that according to taste, but with a variable for sure), which , as the negation of a sentence without existential import, has it. and, indeed, amounts to {no broda cu brode}, "none of the brodas are brode," as would have been expected. Which brings up three, the traditional problem of O, the negative particular, usually rendered "Some S are not P." The "some" seems to require existential import, but the internals of the system seem to require that it lack existential import -- it is entailed by the negative universal, of course. which lacks existential import in the original system. What to do? Aristotle wisely always said "not all S is P" for this case, the "some" version is later, when the problem was misunderstood or forgotten. But, whatever Aristotle did, what does Lojban do? {su'o da broda naku brode} has the same conflict as "Some S is not P" -- the variable says there doesn't have to be an S (even without the buried conditional, say), the {su'o} says there does, and pushing the negation through says "No" again. There are just two possibilities before we push on to a more complex solution. We can say that {su'o da} does not mean the same think when something inside it is negative, so that it does not have existential import then. Or we can invent a new quantifier (only one is needed, as opposed to every other solution I have thought of, which take more) which is always existentially free, whether before a variable or not (and make {su'o} always existential). The second alternative seem the wiser one, although it complicates the rules just used quite a bit. Fourth. The next step, if the above line does not work, is three new quantifiers -- the one mentioned above plus new universals (we might get by with two, but I haven't worked out all the details of that approach). Finally, the question of existential import seems to central to go unsolved. To be sure, not all the langauges behind Lojban have it as explicitly as English, though other langauges do (and still others deal with the same problems in totally unrelated ways). Even if we went all the way to restricting lack of import (the least common case, probably) to cases with explicit conditionals, we still have the problem of {ro brada na brode} which does not shaked down to {su'o broda naku brode}. Even the Indians, who didn't do quantifiers rejected this move, acknowledging that the absence of any brodas justified the first and falsified the second. |