existential requantification
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.--la 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? --la 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. --la 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 --la 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? --la 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
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