Hi everyone. la mukti asked me to weigh in on this. I’ve given it a good bit of thought, as it’s one of the two most serious problems in Lojban foundations as defined in the CLL.
la mukti's analysis is excellent; his simple unicorn sentences demonstrate the contradiction in action and the connection to Aristotelian logic explains how importation might have crept in. This association with Aristotle also provides an argument against importation. Modern logic has simply left Aristotle behind, as should, in my opinion, any conlang built on the developments in logic from the last century.
Furthermore, it's not quite right to say that the CLL simply chooses to use Aristotelian logic in this one case. This is because in Aristotelian logic there are no quantifiers as they are understood in predicate logic (or in Lojban). In fact this is one of the limitations of Aristotle's rules for reasoning: it ignored a lot of the inner structure of the statements involved and so could not account for the relationships between the objects involved in the statements. So importing ro here is not actually historical, but anachronistic. The CLL definition essentially creates a bizarre hybrid of Aristotelian and predicate logic which no one uses. Incompatibility with the classical negation theorem is one way this break is showing up. Aristotle would not have said that when moving the negation sign across bound variables you must flip the quantifier to preserve truth values because those things weren't part of his system at all.
There are only three choices here as I see it. We can use the standard semantics from predicate logic for the universal quantifier and keep the standard negation theorem; or we can keep importing ro and lose the negation theorem; or we can do nothing and allow an internal contradiction to lie in the foundations of Lojban. A strong argument in favor of importing ro would include an account of the way negation works in this new system. Although I am solidly in favor of non-importing ro, I will sketch out how to do that in a moment. But first I'd like to examine John's point.
The way that "All unicorns are white." is represented in predicate logic is with the formula $$ \forall x : [ U(x) \rightarrow W(x) ] $$. (The stuff between the dollar signs is LaTeX markup; if you can't read it you can plug it into an online renderer. \forall is the universal quantifier, x is the bound variable, \rightarrow is implication, and U and W are functions corresponding to 'x is a unicorn' and 'x is white' resp.). As John says, one way to translate this into Lojban is "ro da zo'u ganai da pavyseljirna gi da blabi". This is irrelevant to la mukti's construction, however. He did not use that Lojban sentence in his example, he used one that's formally equivalent to da with poi. The negation theorem is stated in its full generality in the CLL and not only on sentences of the form above. Indeed, using that implication form as a definition of "ro da poi X" is precisely what is needed to fit with the negation theorem and with predicate logic, and those are precisely the semantics that I am advocating.
It’s easy to see that these sentences are consistent with the negation theorem. Recall that a logical implication is a function of statements; it's truth value depends only on the truth value of the statements it acts on. An IF (...) THEN (...) statement is defined to be false when the first argument, called the antecedent, is true and the second argument, called the consequent, is false. All other pairs of arguments result in true.
Under our assumption that nothing satisfies pavyseljirna, "ro da zo'u ganai da pavyseljirna gi da blabi" is true because for every value of da, the antecedent is false. Therefore "naku ro da zo'u ganai da pavyseljirna gi da blabi" is false. According to the negation theorem "su'o da naku zo'u ganai da pavyseljirna gi da blabi" must also be false. This says that there must an object which falsifies the implication, and as I said in the last paragraph this can only happen when the antecedent is true and the consequent false. The antecedent claims that x is a unicorn, so a true antecedent would contradict our assumption about unicorns. Of course the particular functions we chose, unicorns and white, are not important; all statements of this form are consistent with the negation theorem.
So if we wanted to keep the importing semantics, how would negation have to work? We first rewrite "ro da poi P" in the importing sense as a formula in predicate logic to manipulate it symbolically, then translate it back into Lojban. This still uses the implication, but includes the additional restriction that something must satisfy P. We therefore represent "naku ro da poi P zo’u Q" as $$ \neg \forall x \exists y : P(y) \land [P(x) \rightarrow Q(x)]) $$. Applying the theorem to the formula, we get $$ \exists x \forall y : \neg (P(y) \land [P(x) \rightarrow Q(x)]) $$ which is equivalent by another elementary theorem to $$ \exists x \forall y : \neg P(y) \lor \neg (P(x) \rightarrow Q(x)) $$ which can be translated back into Lojban as “ro da su’o de zo’u de P inajanai ganai da P gi da Q”. Notice in particular that there are now two sumti involved. This is because in the importing sense there are really two different claims being made and each use their own variable. I played with this for about half an hour tonight and couldn’t find an equivalent form that resulted in more elegant Lojban; perhaps an importing advocate can do better.
That’s one of four cases; three others are treated similarly, and then negation dragging across unrestricted da operates according to the normal rules. Imagine trying to move naku around in an ordinary sentence under these rules!
I don’t know what pc said to John but it is simply not true that the Aristotelian sense of “All P are Q” is compatible with predicate logic. On page 54 of Hilbert and Ackermann’s classic _Principles of Mathematical Logic_ appears the following:
“According to Aristotle the sentence ‘All A is B’ is valid only when there are objects which are A. Our deviation from Aristotle in this respect is justified by the mathematical applications of logic, in which the Aristotelian interpretation would not be useful.”
Its possible that there is some confusion over an elementary theorem which states $$ \forall x : P(x) $$ implies $$ \exists x : P(x) $$. If we look closely at that we see that, in John’s words, the quantification there corresponds to Lojban’s unrestricted logical variables; restricted logical variables must first be rewritten as pure formulae, as I did above, before applying the theorem.
mi’e az
On Sunday, October 19, 2014 10:08:14 AM UTC-7, John Cowan wrote:
Alex Burka scripsit:
> Ok, so just to clarify what you were correcting, with importing {ro}
> you would say {ro broda cu brode} and {ro da poi broda cu brode} are
> the same thing and require {su'o da broda}, while {ro da ganai broda
> gi brode} is different and just requires a non-empty universe?
Right. The difference is between restricted and unrestricted quantification.
--
John Cowan http://www.ccil.org/~cowan co...@ccil.org
Lope de Vega: "It wonders me I can speak at all. Some caitiff rogue
did rudely yerk me on the knob, wherefrom my wits yet wander."
An Englishman: "Ay, belike a filchman to the nab'll leave you
crank for a spell." --Harry Turtledove, Ruled Britannia