Step B: One-formula
sentence makers
Sentence makers
that require one formula (and perhaps something else) fall mainly
into three groups: Negation, Modals, and Quantifiers (which also
require a variable and possibly even another subordinate formula).
In the standard presentations of FOPL, these go at the beginning of
the sentence and their order is significant, since each attaches to
the sentence formed by the sentence makers to its right back to the
unmarked sentence. But beyond the structural significance of the
order, there are clear semantic differences between, say “it is was
the case that someone [then] was a witch” and “someone [now] was
a witch”, that there used to be witches and that there are still
former witches around. So moving one past the other is generally not
allowed. On the other hand, many modals and quantifiers come in
pairs, strong and weak, such that passing negation through them
changes one to the other: ~1~ = 2, so ~1 = 2~ and conversely. Of
course, in any case, the maker governs the whole sentence that
follows, negating it, casting it into the relevant alternate reality,
or binding all its free occurrences of the indicated variable.
But in adding these
items to a Loglan sentence, we discover something more about what is
bound up in the expressions “speakable”. We began by giving
voice to the expressions of FOPL. Then we pruned the mass of
punctuations to just those actually required to keep the structure
fully marked, Now, something new seems to have been added, which
seems at the moment to be loosely familiarity. That is, the changes
here from formula to Loglan neither gives voice to new symbols nor
eliminates detritus, but merely puts things in positions familiar from
the L1s of likely learners of the language. In light of this, we can
look back at the shift of the first argument from after to before the
predicate and wonder. Was it just to make for a more efficient term
formation or was it also to bring the sentences into something very
like the familiar Subject-Verb-Object order of the L1s of many likely
students of the new language?
What happens with
Negation and Modals is that they are regularly shifted from the front
of the sentence to a place immediately before the predicate. The
original position is always possible but is, in fact, rarely used,
even with compound sentences. Quantifiers are also shifted inward,
from before the sentence (prenex), to the place in the body of the
sentence where the bound variable first occurs. To be sure, this
move is done carefully, in that, for negation and modals, the
original order is preserved – though negation tends to move left
when the standard dualities allow (but this is stylistic, without
logical significance). Similarly, the order of quantifiers of
different sorts is preserved, argument places being rearranged to
preserve order and further rearrangements forbidden if that order
would be disturbed. So, assuming “x loves y” is xLy, “Everybody
has someone who loves them” is basically AxSy yLx, which becomes
AxL[1,2]y (Loglan has no free variables; every variable is assumed
bound particularly unless otherwise bound explicitly). The movement
of the quantifier over quantifiers is also treated carefully, though
there is some controversy (resolved in different ways every few
years) about exactly how that works. The basic positions are that
negation, while represented just before the predicate, is to be
understood as as far left in the sentence as possible, and that
negation is to be taken as being where it appears to be. In the
first case, quantifiers (and modals) may have to through the logical
place of the negation to get where they belong and so are transformed
in the usual way. In the second view, only makers that came before
the original negation need changing. Whichever way is current, the
proper original form remains (subject to the position finally
assigned to the negation), though it will be different in the two
cases. Ax~Fy, on the first view, might be from ~AxSyFxy or Sx~SyFxy
or SxAy~Fxy, which are all equivalent. On the second view, it would
be from Sx~SyFxy, since Sx, but not Sy had to pass through the
negation to get to its place. (We will later see cases where the
negation comes at the end of a sentence and the matter is slightly
more complicated, but still resolvable). In all these cases, then,
the logical form is preserved – up to equivalence, anyhow. And, of
course, the prenex version remains available (at slightly extra
cost), just as the L1 probably contains the equivalent of “it is
not the case that” and “it is possible that” and even
“everything is such that”
The case is less
clear with some modals that do not comes in pairs, like Past and
Future. In standard Tense Logic, each of these is one of a pair of
the usual sort: “somewhen in a past” and “somewhen in a
future”, roughly speaking, but paired with “everywhen in pasts”
and “everywhen in futures”. Thus, the usual negation movement is
validated. But Loglan tenses are not quite like that. Nor are they
like natural language tenses, built on a system of points (present,
past, future and retrofuture) and vectors (before, now, and after).
Rather they are (mea culpa in here somewhere) an uneasy compromise
between the two: vectors to begin with, but once the vector is
traced, its head become a point from which a further vector can
extend. As a result, PFa sometimes says no more than that Fa was
once true, but at other times it says that Fa was true at a
particular, but unspecified, past time. And negation reflects this
in that ~PFa is sometime unresolvable and sometimes means P~Fa (The
easy analogy is when we say “There is a man in the house. He...”,
so here “There was a time when … Then ...”). The best solution
seems to be not to move tenses relative to negation, but the rules
for this are neither so well spelled out nor so carefully followed.
There are other modals with similar problem, “probably” and
“certainly” for example, but for different reasons. Still, with
care, the original structure is retained, if not quite transparently.
The case of
quantifiers, and especially restricted quantifiers, is a more
profound change. Not only are the quantifiers moved inward from
their prenex position, but they change their grammatical status,
Quantifiers are no longer a separate sort of thing – a 1-formula
sentence maker – but become simply terms. Syntactically, there is
little difference between AxFb and @G,Fb, and there is none in the
case of the resolution of AG,Fb, from [AxGx]Fxb, where the values of
the quantified variable are restricted to the non-empty class of Gs
(the comma mark a separator between the predicate in the term and the
one in the sentence, to prevent them being taken as a compound – on
which more later). Quantifiers thus get involved in the place
shifting predicate changes, where care has to be taken to prevent
changing the relative order of different quantifiers, although terms
generally can move about freely. As with negation and modals, these
problems could have been avoided by leaving everything prenex, though
arguably this would make sentences of any appreciable length harder
to understand – and it is not a common pattern in natural languages
(so maybe not something hard-wired in our understanding?).
Moving
the quantifiers inward to the first place the bound variable occurs
also means a loss of direct information about the scope of that
quantifier (this is true for modals and negation as well, though the
negation tends to get dealt with in various ways, using DeMorgan and
the like). If the variable bound by a quantifier occurs in some
place other than the first one, the connection has to be made. In
the case of simple quantifiers, this is done by repeating the bound
variable, so AxLxx becomes AxLx. But with the restricted
quantifiers, the ordinary anaphoric pronoun resources must be used,
the variable having been swallowed [AxGx]Fxx becomes AG,F[it], for
some pronoun [it]. The Loglans have a plethora of pronoun systems,
including assignable ones, and ones that can be used on the fly,
depending on such factors as the initial letter of the predicate in a
term or the structural position of the original term in its home
sentence and so on. Despite this, it is not clear that every term
can be represented unambiguously by a pronoun in every position, and
certainly not clear that this can be done in a way that is easily
interpretable by a hearer. Keeping the variables somehow would have
eased this problem, which, admittedly, looms larger in theory than in
practice. In any case, the scope of a quantifier is now determined
to be the shortest sentence which contains the quantifier and all its
anaphora (variables or pronouns). Aside from taking some care about
what variables to use, this gives a practical solution, even if the
sentence represented is only an equivalent to one started with.
The
issue of repetitions arises for regular terms as well, of course, and
does not have a variable solution although a variable has been hidden
in going from FOPL to Loglan. With regular terms there is the option
of simply repeating the term rather than using a pronoun, and this
can be used when clarity advises it. Repeating a quantifier tends to
make for confusion: is this just a repetition or is in a new
quantifier with the same range? The convention is that “repeated “
quantifiers are actually new ones with the same range. So, AG,FAG is
[AxGx][AyGy]Fxy, a very different claim from [AxGx]Fxx (“Everybody
loves everybody, versus “Everybody loves himself”). (By the way,
there is a version of FOPL in which term makers are in fact treated
as quantifiers. In such a system – or even in the present one with
minor changes, if the variables were not suppressed, a large part of
the complications of the Loglans' various pronoun systems could be
relieved by using the variables. This added “detritus” would pay
other benefits as well, eliminating a major need for place-shifting
and for the separation between predicates – assuming we could also
eliminate the shift from VSO to SVO order. Of course, place shifting
has other virtues, like providing easy ways to match familiar
concepts with very general predicates, as “destination” is hidden
in “go” as the second place, to be shifted to first for
independent use. The need for predicate separation markers – and
term enders, for that matter – does not seem to have any separate
use and adds a variety of complications.)
No comments:
Post a Comment