History: existential requantification

Source of version: 1 (current)

From ((noda)): If ''da'' is already bound, ''noda'' does not have the meaning discussed above. ''da prami noda'' is true, for example. ''mi'e ((jezrax))'' ''da prami noda'' may be true, but as far as I can tell it means the same as ''da prami node'', i.e. that there is someone who doesn't love anybody. (I suppose there might be some such unloving soul...). The way I understand it, when a variable is re-bound, it is restricted to the same previous restriction, but since in this example the first time ''da'' was used it was unrestricted, (except by an implicit restriction to people maybe, or to entities capable of loving) then the second time the same (un)restriction applies. --((xorxes)) ''Why do you think the variable is re-bound the second time? Isn't the noda referring to the same exact da at the beginning?'' It is re-bound because {no} is a quantifier, a quantifier implies a binding. You're saying: "at least one x loves no x", even in English there is a re-binding. The way to show most clearly how the quantifiers work in a given sentence is by bringing them to the prenex. In this case: ''su'oda node zo'u da prami de'', "there is at least one x, such that for no y, x loves y". Using the same variable twice is not ambiguous in the non-prenex form, but you have to use two variables in the prenex form. --((xorxes)) * This contradicts ((the Book)), chapter 16, section 10: "A variable may have a quantifier placed in front of it even though it has already been quantified explicitly or implicitly by a previous appearance." The examples and discussion make it clear that ''da'' is not rebound by a quantifier. See also the mentions of scope in the same chapter. ''mi'e ((jezrax))'' ** Yes, in my opinion the Book is wrong there. This was discussed on the list sometime last year. It is not difficult to come up with examples that lead to contradictions. For example, with normal quantifications, ''su'oda naku'' is equivalent to ''naku roda''. But what happens if you allow this weird second quantification. Is ''su'oda naku cida'' equivalent to ''naku roda cida''? The Book's proposal is just not workable, it doesn't make sense in logical terms. --((xorxes)) ** Can you point out the list discussion? *** I tried, but I couldn't find it. But don't worry too much, at least I don't remember anything very revealing being said. --((xorxes)) ''So you're saying the only way to use one variable would be ''su'oda noda zo'u'', which would be a contradiction? There is at least one X that doesn't love itself.--((xod))'' That's ''da naku prami da'', or in prenex form ''su'oda naku zo'u da prami da''. "For some x, it is not the case that x loves x". You can also bring the ''naku'' to the beginning, changing the quantifier, and you get ''roda na prami da'', "it is not the case that every x loves itself", with the same meaning. ''su'oda noda zo'u'' doesn't make much sense, which quantifier is the valid one? The last one? Notice BTW that in ''da naku prami da'' only the first ''da'' has an implicit quantifier ''su'o'', the second one is already bound by the first quantifier. --((xorxes)) '' Now what about ''2da prami 1da''. Why doesn't the second ''da'' come from the initial set of two? --((xod))'' Because there is no initial set of two. Quantifiers do not determine sets. Consider this case: John loves Mary Mary loves herself Paul loves himself Jane loves Paul Is it true that ''2da prami 1da''? --((xorxes)) 2da prami 1da '' looks to me like the first two sentences and nothing else. In other words: There are two things such that they love one of the things. If the second da could be interpreted as some third person, what's stopping ''da prami da'' from meaning that John loves Mary? Then each variable would have a scope of nothing. --((xod))'' In ''da prami da'', the second ''da'' is not rebound, it is within the same binding as the first. Rebinding the same variable is anomalous, and it requires a second quantifier. ''da prami da'' can only mean "someone loves themself". (''da prami de'' can't mean that John loves Mary either. If John loves Mary, then ''da prami de'' is true, but it doesn't mean that John loves Mary.) --((xorxes)) ''But the implicit quantification of the second is su'o. So you're saying it's OK to rebind a variable with the same quantification as the first? How about ''3da prami 3da'' --((xod))''? Nooooooooooooooooooooooooooooooooooo! A bound variable has only one quantification, no matter how many times the variable appears. ''da'' has an implicit ''su'o'' only if it has not been previously bound. If it has been bound, it doesn't add any new quantifier, it would not make sense! --((xorxes)) ''I've already offered a reasonable interpretation of a sentence which contains multiply quantified variables. Why do you say it doesn't make sense? --((xod))'' It is not enough to offer a reasonable interpretation of one sentence. You have to show that it doesn't break the system as a whole. Tell me if you disagree with any of these equivalences: da prami noda = naku naku su'oda prami noda = naku roda naku prami noda = naku roda naku prami naku su'oda = naku roda prami su'oda If you agree that they are all equivalent, how do you read the last one? If you don't, then are there special rules of manipulation of negations when these multiple quantifications are in place? --((xorxes)) I don't understand your point. Your transformations all look correct to me, and the first and last sentences seem to be equivalent in meaning, exactly as I would expect. By the Book's formula, ''da prami noda'' means "there exists something that doesn't love itself" (literally, that loves zero of itself), and ''naku roda prami da'' means "it is false that everything loves itself." So if you're trying to point out a mistake in the Book, I'm not seeing it. ''mi'e ((jezrax))'' How do you figure that ''su'o da'' in the last sentence refers each time to "itself" and not to any other member of the "set" "defined" by ''roda''? I would have said that by the Book rules the last sentence meant that "it is false that everyone loves someone", but you seem to read it as ''naku roda prami da''. I'll give you a simpler case: naku su'oda prami cida = noda prami cida Are those two equivalent? Where are the ''cida'' taken from? --((xorxes)) Tricky. Per the Book, the ''ci da'' must be chosen from among the ''su'oda'' or the ''noda'', and there aren't guaranteed to be enough of them to choose from. The Book does not weigh in on whether ''pamai'' that makes the bridi false, or ''remai'' is a semantic error, or ''cimai'' is to be interpreted some other way. ''pamai'' If we assume that an impossible requantification makes the bridi false, then ''naku su'oda prami cida'' is true thanks to the negation and ''noda prami cida'' is false, implying that the ''naku'' rules lifted directly from classical logic do not apply in this case. ''remai'' If we call it a semantic error, the sentences are equivalent: they're both erroneous. ''cimai'' Other interpretations are possible. Xorxes wants to sidestep requantification by defining it to rebind the variable, so that ''su'oda prami cida'' means ''su'odaxipa prami cidaxire''. This set of sentences avoids the impossible requantification. (''vei ga me'i re gi za'u re'' means "<2 or >2", or in other words "not equal to 2", the inversion of the quantifier 2.) There should be no question of calling this a semantic error: reda prami su'oda reda prami naku noda naku vei ga me'i re gi za'u re da prami noda Aside, I don't know how to go about ((inverting ji'i)). ''mi'e ((jezrax))'' ---- I have come up with a way to transform a bridi with requantification to prenex form. This preserves ((the Book))'s semantics and makes the mathematical meaning clear. Consider the Book's example 14.2: ci da poi prenu cu se ralju pa da Pulling ''da'' into the prenex: ci da poi prenu zo'u da se ralju pa da But this is not fully-prenexed form, because there is a quantifier in the predicate. To pull this quantifier out, we have to equate it to another variable, say ''de''. The new variable has the same scope as ''da'', so the two are joined in a termset. ci da poi prenu ku'o ce'e pa de po'u da zo'u da se ralju de This is now a mathematical expression with no room for ambiguity outside the meanings of the words ''prenu'' and ''ralju'', and it means what the Book says it means. The exact rules for using ''naku'' follow by implication. ''mi'e ((jezrax))'' If we remove ''ce'e'' we have a well formed logical expression: ci da poi prenu ku'o pa de po'u da zo'u da se ralju de Which means that for exactly three persons x, there is for each x exactly one y (= x itself), such that x se ralju y. This is not what the Book pretends. How does ''ce'e'' change this? --((xorxes)) ''ce'e'' puts them in the same logical scope; ''da'' and ''de'' are defined "simultaneously", if you like. See chapter 16 section 7. It is not "there exist 3 x such that for each x there exists one y=x such that...", it is "there exist 3 x and 1 y=x such that...." I have never in mathematics seen variables in the same scope which depend on each other, but it doesn't seem to cause a problem unless there are reciprocal dependencies, which can't arise in this case. ''mi'e ((jezrax))'' I can't make sense of "there exist 3 x and 1 y=x such that...." In any case, do the negation rules break down in the presence of this secondary quantification? For example: su'oda prami su'oda naku roda prami noda Those two mean different things? --((xorxes)) ''pa de po'u da'' (or maybe ''pa de po'u pa da'') can be considered ill-defined. Here's how to expand it even further and make it clear, even to a stickler like me, by defining quantification more explicitly. "There exist two x such that..." means "there exists one x1 and one x2, x1 not equal to x2, such that....". Numbers are exact. re da broda re da zo'u da broda pa daxipa ce'e pa daxire poi na du daxipa zo'u daxipa .e daxire broda *''The third line is always false. Suppose ko'a and ko'e are the only two things that broda. Then it is not the case that there is only one x and only one y different from x such that x and y broda. There are two: x can be ko'a (and then y is ko'e) and x can be ko'e (and then y is ko'a). The way to do it is with ''su'o da'' instead of ''pada'' and then a third condition that there is ''no de poi de du daxipa .a daxire'', such that... ''--((xorxes)) ** You're right; the variables can bind in either permutation. But I think I made my point that the construction can be put on a formal basis. ''mi'e ((jezrax))'' ----- I'm not sure you have. Let's do an easier case, let's consider ''reda blanu'' and ''reda prami su'oda'': re da blanu re da zo'u da blanu su'o da su'o de poi na du da ku'o ro di poi na du da a de zo'u da e de enai di blanu Now for the weird one: re da prami su'o da su'o da su'o de poi na du da ku'o ro di poi na du da a de ku'o daxipa poi du da a de zo'u da e de enai di prami daxipa Is that correct? (Doing ''pada'' instead of ''su'oda'' will be a lot harder, but let's make sure we're on the same page so far.) --((xorxes)) ----- Similarly for three (it gets long real fast): ci da broda ci da zo'u da broda pa daxipa ce'e pa daxire poi na du daxipa ku'o ce'e pa daxici poi na du daxipa .e daxire zo'u daxipa .e daxire .e daxici broda *''Same objection. Plus, the restriction on ''daxici'' would have to be ''poi na du daxipa .a daxire'', otherwise you would be allowing ''daxici'' to be one but not both of the others. That's a minor point though, the major flaw is the one pointed out above.''--((xorxes)) ** ''daxici na du daxipa .e daxire'' expands to ''daxici na du daxipa .ije daxici na du daxire'', so I think I got that detail correct. ''mi'e ((jezrax))'' ***''na'' has scope over the entire bridi, so ''ko'a e ko'e na broda'' expands to ''naku zo'u ge ko'a broda gi ko'a brode'', which in turn can be transformed into ''ganai ko'a broda ginai ko'e broda'', which is ''ko'a na broda ija ko'e na broda''. --((xorxes)) *** You are right again. ''mi'e ((jezrax))'' Here's the same treatment for the requantification example above. Deep breath... hope I caught all the typos. ci da poi prenu cu se ralju pa da ci da poi prenu ku'o ce'e pa de po'u da zo'u da se ralju de pa daxipa ce'e pa daxire poi na du daxipa ku'o ce'e pa daxici poi na du daxipa .e daxire ku'o ce'e pa de poi du daxipa .a daxire .a daxici zo'u daxipa .e daxire .e daxici se ralju de The above shows how to reduce all exact positive numbers to ''pa''. The numbers can be completely removed by rewriting ''pa'' like this (there are a lot of equivalent ways to put this into Lojban; I picked one). "There exists exactly one x such that p(x)" means "There exists at least one x such that p(x) and for all y not equal to x, not p(y)." pa da broda pa da zo'u broda da rode poi na du da zo'u da broda .ije de na broda The question of how termsets interact with ''naku'' is independent of that. I will have to work through it to figure it out; it may take me a while. I wouldn't be surprised if ((pc)) knows off the top of his head. ''mi'e ((jezrax))'' *In any case, the conclusion is that you can no longer move ''naku'' following the usual rules when there are requantified variables around. To me, that alone is enough to discard this way of interpreting a second quantification. --((xorxes)) ** To me, the conclusion that the book's definition of the meaning of requantification is well-defined, and not confused as it might appear, is enough reason to accept it. ''mi'e ((jezrax))'' *** De Morgan's transformation rules won't apply in the presence of requantification. Are you really willing to give up so much for the sake of a not very useful requantification rule? --((xorxes)) ---- If a concept causes this much uncertainty, maybe it's time for reform? * I don't think it causes uncertainty, merely disagreement. Trying to reform won't solve that, only make it worse! ''mi'e ((jezrax))''