[lojban] RECORD: Quantifier Scope, 1999

[pycyn@aol.com]

Note: Lojban is a logical language because its basic grammar is first order 
predicate 
logic, not because it satisfies someone's (anyone's) definition of the 
English word 
"logical".  So, in particular,
The scope of quantifiers that occur in Lojban sentences is determined by the 
order of their first appearance, left to right.  This means that {ro da prami 
de} "Everybody loves somebody" expands to "a loves somebody and b loves 
somebody and c loves
somebody....," possibly a different somebody for each everybody.  But {de se 
prami ro da} means "Somebody is loved by everybody" and expands to "a is 
loved by everybody or b is...,"  that there is a single somebody that 
everybody loves.  
This appears to violate the symmetry of {se} conversion, but is, in fact not 
a case of {se} conversion.  The problem is that for speaking convenience we 
shave shortened the basic sentence down somewhat, the first from {ro da de 
zo'u da prami de}, the second from {de ro da zo'u de se prami da}.  Clearly 
in addition to {se} conversion there has also be reordering of the 
quantifiers here, so the paradox disappears.  Note that {ro da de zo'u de se 
prami da}, a simple {se} conversion, still means the same as the first 
sentence.
If you think that prenex reordering ought to always work just as much as {se} 
conversion, I can only report that for most kinds of quantifiers, it just 
doesn't,  if the variables bound by the quantifiers occur in the same bridi.  
It does however work with strings of all universals or all particulars (and 
in a few other really weird cases). It happens that, given second order 
logic, we can define quantifiers that are independent of one another, and, 
since all second order logic can be mirrored somehow in Lojban, we can 
introduce these -- not exactly as quantifiers usually (the mirror distorts a 
bit). The case that started this run through this point,  {ci gerku cu batci 
re nanmu} ordinarily means that each of the three dogs bit some two men, that 
there are between 2 and 6 dog-bit men involved.  To get the "symmetrical" 
case, three dogs each biting two men, resulting in two men each with bites 
from three different dogs, we need only use {le ci gerku cu batchi le re 
nanmu, the quantifiers are now all universal and, so, reorderable.  A 
reorderable case for {ro da prami
de} is harder to come up with, since it is a bit harder to imagine what is 
wanted, but
maybe {ro lo prenu cu prami le pa prenu} would do -- though it says a bit 
more than may strictly be needed.

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