Hi everyone. la mukti asked me to weigh in on t= his. I=E2=80=99ve given it a good bit of thought, as it=E2=80=99s one= of the two most serious problems in Lojban foundations as defined in the C= LL.

la mukti's analysis is excellent; his simple u= nicorn 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 importati= on. Modern logic has simply left Aristotle behind, as should, in my o= pinion, any conlang built on the developments in logic from the last centur= y.

Furthermore, it's not quite right to say that t= he CLL simply chooses to use Aristotelian logic in this one case. Thi= s is because in Aristotelian logic there are no quantifiers as they are und= erstood 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 in= ner structure of the statements involved and so could not account for the r= elationships between the objects involved in the statements. So impor= ting ro here is not actually historical, but anachronistic. The CLL d= efinition essentially creates a bizarre hybrid of Aristotelian and predicat= e logic which no one uses. Incompatibility with the classical negatio= n theorem is one way this break is showing up. Aristotle would not ha= ve said that when moving the negation sign across bound variables you must = flip the quantifier to preserve truth values because those things weren't p= art of his system at all.

There are only three cho= ices here as I see it. We can use the standard semantics from predica= te logic for the universal quantifier and keep the standard negation theore= m; 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 Lo= jban. A strong argument in favor of importing ro would include an acc= ount of the way negation works in this new system. Although I am soli= dly in favor of non-importing ro, I will sketch out how to do that in a mom= ent. But first I'd like to examine John's point.

The way that "All unicorns are white." is represented in predicate log= ic 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 yo= u 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 funct= ions corresponding to 'x is a unicorn' and 'x is white' resp.). As Jo= hn says, one way to translate this into Lojban is "ro da zo'u ganai da pavy= seljirna 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 th= e 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 a= dvocating.

It=E2=80=99s easy to see that these sen= tences are consistent with the negation theorem. Recall that a logica= l 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 antec= edent, is true and the second argument, called the consequent, is false. &n= bsp;All other pairs of arguments result in true.

U= nder 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 ant= ecedent 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 i= n the last paragraph this can only happen when the antecedent is true and t= he consequent false. The antecedent claims that x is a unicorn, so a = true antecedent would contradict our assumption about unicorns. Of co= urse the particular functions we chose, unicorns and white, are not importa= nt; all statements of this form are consistent with the negation theorem.

So if we wanted to keep the importing semantics, ho= w 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 symbolical= ly, then translate it back into Lojban. This still uses the implicati= on, but includes the additional restriction that something must satisfy P. = We therefore represent "naku ro da poi P zo=E2=80=99u Q" as $$ \neg \= forall x \exists y : P(y) \land [P(x) \rightarrow Q(x)]) $$. Applying= the theorem to the formula, we get $$ \exists x \forall y : \neg (P(y) \la= nd [P(x) \rightarrow Q(x)]) $$ which is equivalent by another elementary th= eorem to $$ \exists x \forall y : \neg P(y) \lor \neg (P(x) \rightarrow Q(x= )) $$ which can be translated back into Lojban as =E2=80=9Cro da su=E2=80= =99o de zo=E2=80=99u de P inajanai ganai da P gi da Q=E2=80=9D. Notic= e in particular that there are now two sumti involved. This is becaus= e in the importing sense there are really two different claims being made a= nd each use their own variable. I played with this for about half an = hour tonight and couldn=E2=80=99t find an equivalent form that resulted in = more elegant Lojban; perhaps an importing advocate can do better.

That=E2=80=99s one of four cases; three others are treated = similarly, and then negation dragging across unrestricted da operates accor= ding to the normal rules. Imagine trying to move naku around in an or= dinary sentence under these rules!

I don=E2=80=99t= know what pc said to John but it is simply not true that the Aristotelian = sense of =E2=80=9CAll P are Q=E2=80=9D is compatible with predicate logic. = On page 54 of Hilbert and Ackermann=E2=80=99s classic _Principles of = Mathematical Logic_ appears the following:

=E2=80= =9CAccording to Aristotle the sentence =E2=80=98All A is B=E2=80=99 is vali= d only when there are objects which are A. Our deviation from Aristot= le in this respect is justified by the mathematical applications of logic, = in which the Aristotelian interpretation would not be useful.=E2=80=9D

Its possible that there is some confusion over an elem= entary theorem which states $$ \forall x : P(x) $$ implies $$ \exists x : P= (x) $$. If we look closely at that we see that, in John=E2=80=99s wor= ds, the quantification there corresponds to Lojban=E2=80=99s unrestricted l= ogical variables; restricted logical variables must first be rewritten as p= ure formulae, as I did above, before applying the theorem.

mi=E2=80=99e az

On Sunday, October 19, 201= 4 10:08:14 AM UTC-7, John Cowan wrote:

Alex Burka scripsit:

> Ok, so just to clarify what you were correcting, with importing {r= o}
> you would say {ro broda cu brode} and {ro da poi broda cu brode} a= re
> the same thing and require {su'o da broda}, while {ro da ganai bro= da
> gi brode} is different and just requires a non-empty universe?

Right. The difference is between restricted and unrestricted quan= tification.

--=20
John Cowan ht= tp://www.ccil.org/~cowan co...@ccil.org
Lope de Vega: "It wonders me I can speak at all. Some caitiff rog= ue
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

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