tech:logic matters

["John E. Clifford" <pcliffje@CRL.COM>]

Although I do not really think I am being obscure and I know I am
trying not to be, a number of people, whom I have every reason to
believe as intelligent, well-educated and well-intentioned as I am,
claim not to understand my point and illustrate this claim by citing
things I have said as saying things I did not mean them to say (and
cannot now read them as saying).  So, I'll try again, noting in passing
that the point here is not a big one and not one that seems to affect
expressibility in Lojban nor one that is at all controversial in Logic,
so I am puzzled why this todo has gone on for now into its second
year.

First, language points.  English has four words that indicate
affirmative universals, their basic patterns being
        Each S is P
        Every S is P
        Any S is P
        All S are P.
The four differ in a variety of ways in English usage, but the only
one even slightly relevant to this discussion is that the first two
forms ("each" and  "ever-each" historically) have as part of their
meaning (semantic, not merely pragmatic) that the universal claim
implies that there are Ss (i.e., Some S is P).  "Any" clearly does not
have this feature at all and "all" is not clearly either way (the best
guess is that it has lacks the feature semantically but picks it up
pragmatically, by long association -- see below).  Some loose
examples in English may have contributed to the misunderstanding
here (using "all" when "every" or "any" would have been clearer).

Logic, in the broad sense, has one expression of this sort, but uses it
in two idioms, a basic and a derived form.  The basic form associates
one predicate with another, as above, reproducing in all respects
        Every S is P
except that a long (and largely inexplicable) habit has logicians
saying "All S is P" for this form.  In particular, this form entails
Some S is P.  This form gets symbolized in a variety of ways (even
within the already confusing variety in Logic); for now the two
forms SaP and (Ax:Sx)xPx will cover most of what we need to say.
Thus these forms entail SiP and (Ex:Sx)xPx respectively.

These two quantifiers form, in traditional logic, one side of a square
of quantifier expressions
                Affirmative                     Negative
Universal       SaP/(Ax:Sx)xPx          Sep/(0xSx)xPx
                Every S is P                    No S is P

Particular      SiP/(Ex:Sx)xPx          SoP/(OxSx)xPx
                Some S is P                     Some S isn't P

Among the traditional relations among these forms are the following
Contradictories: the negation of one form is equivalent to the form
diagonally opposite it.
Contraries: the two universal forms cannot both be true but may both
be false
Sub-contraries: the two particular forms may not both be false but
may both be true.
Subalterns: if the universal is true, the particular on the same side is
also.
Converses: in i and e, the order of the predicates is irrelevant, i.e.,
SiP <=> PiS and SeP <=> PeS.
Contrapositives: If we allow the complement of a class, C' of  C,
then, in a and o, we can equivalently change the order if we
complement each class, i.e., SaP <=> P'aS' and SoP <=> P'oS'.
Obverses: a sentence is equivalent to the sentence horizontally
opposite it with the complement of P: SaP <=> SoP' and so on.

All this logic was developed in a practical context, talking about
ordinary things.  When it became a matter for theoreticians,
however, they found that they could not maintain all of these claims
when they allowed empty classes for S (or, with complementation,
universal ones either).  The discovery of this problem (we're in the
fourth time around for it) tends to lead either to the abandonment of
logic or the quest for a patch.  Although there are 16 ways of
interpreting the square in terms of which sentences require that there
are Ss (or Ps for that matter), only a few have been seriously
considered and two regularly appear as the norms.  One of these is to
take the particulars as having existential import and deny that to the
universals.  This saves the secondary relations and keeps only
contradictories of the basic ones.  The other takes the affirmatives to
have existential import and the negatives not.  This keeps the basic
relations but loses the secondary ones, except conversion.  On
technical grounds, the second form is preferable, not just because it
preserves the basics (that distinction is largely conventional), but
because it is functionally complete: all the various ways of
interpreting the four basic expressions are definable within this
system, using either the given forms or their obverses.  For example,
an importless universal affirmative is just SeP', just as an importing
universal negative is SaP'.

In the current round of dealing with these notions, there is also a
second -- extraneous -- reason for preferring the second pattern for
the four-quantifier system.  Since the last century a second system of
quantifiers has come to be dominant in logic.  This system takes only
the affirmative side of the square and takes S to be always the same,
namely the universal class (of the given domain of discourse).  Any
reference to more specific classes is then reconstructed in a complex
predicate: the subclass being mentioned in the antecedent of a
predicate conditional in the case of universals.  The four-quantifier
system underlying this is the old basic system, in which all the
corners have existential import (notice that SoP does, for example,
since when it enters as the result of negating the universal, it
immediately obverts to SiP').  This seems quite reasonable, since,
whatever may be the case with specific classes discussed, it is hard
to imagine the whole universe of discourse is empty -- talking about
literally nothing is very hard to do.  The effect, however, is to make
universal claims about the specific classes non-importing, because
the class is tied to each individual in the universe only conditionally
("if a is an S") and the material conditional is true if its antecedent is
false.  Thus, if we represent "All S are P" in this form we get ("Ax"
short for "(AxUx)x") Ax:Sx => Px, true whenever "Sa" is true of no
thing a, in which case Ex:Sx & Px -- the corresponding form for
"Some S is P" is false. Thus, the system with universal lacking
existential import need not be used in the four-quantifier system, for
it can be represented in the other system, which, in its turn, can be
defined within the recommended four-quantifier system.

Lojban (finally to the point), as a logical language in the sense of
being based syntactically upon the language of logic, has one
universal quantifier and two clear idioms for using it: the restricted
form, matching the older, four-quantifier,  system, and the
unrestricted form matching the newer, two-quantifier system.  The
first of these has the pattern _ro da poi broda cu brode_  and its
variants.  The second has _ro da cu ganai broda gi brode_ and its
variants.  In addition, Lojban has two other forms that correspond in
some way to (logical) English "All S are P,"  _ro lo broda cu brode_
and _ro broda cu brode_.  Presumably, each of these is equivalent to
one or the other of the full bound-variable forms, but there is
controversy about which one each attaches to (and about how each
differs from the basic form, if at all).  Presumably the basic form in
each case decides the issue of the emptiness of the class of brodas.
In the restricted form, the class of brodas, as the class which is the
range of values of _ro da_, is nonempty.  In the unrestricted form,
only the universe of discourse is the range of values for _ro da_ and
so assumed nonempty, the class of brodas, mentioned only
conditionally, may be empty.  In any case, both possibilities are
covered at least once in Lojban and at least one is covered twice (but
it is not clear which nor how nor what difference the other forms
make).

Lojban is slightly defective in representing the four-quantifier
system, for at present no one admits to remembering what form was
assigned to the SoP position (lower right).  This is a correctable and
minor problem, since we can easily assign some cmavo to the job
(xorxes kindly reminds us that the nonce form I was using, _nairo_,
won't work because _nai_ is a suffix) and the main use of this form
is only to carry denial of SaP, so even _na roda poi broda cu brode_
covers most uses.

Lojban does not provide any means of dealing with one theoretical
position in Logic, the suggestion that the range of quantifiers might
be empty.  This strange position seems to have arisen out of a
misreading of the particular quantifier as being about existence
rather than about being a topic of discussion, about the "real world"
rather than the universe of discourse.  Once suggested, however, it
has been pursued in a variety of ways, most interestingly from the
point of this discussion by a structure which would make all
universally quantified sentences true and all particularly quantified
sentences (and all sentences  involving only nonquantified terms)
false in the empty universe.  That is, it would introduce the
possibility that an affirmative universal quantifier might lack
existential import.  The result of this is that, in this system, universal
instantiation is not valid (the move from "Everything is F" to "a is
F") nor is the longer jump from universal to particular (the basic
form of existential import).  While, as noted, there is little practical
value to this system as it stands, we can reconstruct the essentials of
it in ordinary logic and in Lojban by representing the desired bare "Q
S is P" by the modern form of "Q existent S is P," _ ro da cu ganai ge
zaste gi broda gi brode_ for the universal.  We could also, of course,
achieve the effect by working from an unrestricted form derived
from the restricted quantifier _no_ (and the usual _su'o_).  IF we can
think of a reason to want to.

i,n
(Ax: ((Ay) receives-bill(x, y) -> pays [by 15th of the month] (x, y)))
deserves(x, 15% discount on all publications)
pc:
Presumably
Ax: (Ay) (receives-bill(x, y) -> pays [by 15th of the month] (x, y)) ->
deserves(x, 15% discount on all publications)
That looks to be about what the original says and does allow those who
receive no bill to get the discount (I'm not sure that is really what is
wanted, since the bargains usually go to the good old customers).
pc>|83