Step C:
Two-formulae sentence makers
The two-formulae
sentence makers start with some adequate selection of the
“propositional connectives” (the Loglans take AND, OR, IFF and
REGARDLESS, though the last needs some extra work like the argument
reordering for predicates). Added to these are similar connectives
that go outside truth value logic to causation in various senses and
various sorts of modalities. Like subjunctive conditionals
(hypothetical, contrary-to-fact, etc.) as well as alternate logics
like strict entailment or analytic entailment or relevant entailment
(and relevant or analytic disjunctions as well) and so on through the
plethora of logics. But, for the most part, these additions do not
make grammatical differences and so do not need to be discussed
separately here, even though the Loglans do accommodate some of them.
(There is a similar plethora of logics for one-formula sentence
makers and the Loglans have some of them as well, but again, they are
grammatically of a piece with the standard items.)
Historically there
are two ways that these sentence makers (conjunctions) are
represented. The dominant form is infix – or Principia –
notation, where the mark of the conjunction goes between the two
formulae and the a pair of parentheses enclose the whole. The
alternate form is prefix – or Polish – notation, where the mark
goes before the pair and no further parentheses are needed. (There,
admittedly, a third possibility, postfix or reverse Polish notation,
where the mark comes after the pair. This was used on some
calculators back in the day, but never had much play in Logic). From
the point of view of an attempt to eliminate detritus, prefix is
obviously the most desirable version. But as a feature in a spoken
language, it seemed to put a strain on memory and and analysis. It
seems to be harder to grasp CCpKqrKCpqCpr than even the fully
parenthesized ((p→(q&r))→((p→q)&(p→r))). And, in
FOPL as used, numerous abbreviations were possible, dropping
parentheses under a variety of rules, including various additions to
the the markers to show relative depth and the like. Prefix notation
does not offer much in the way of abbreviations, except marking when
a string the same connective occurs and this rather obscures
structure than reveals it: C3pqpp is even more opaque that CCCpqpp.
The Loglans use
both forms and, indeed, mix them in a single sentence. Obviously,
this requires some care and, especially devices for showing
boundaries of component sentences: Kpq&r is just ambiguous as it
stands, requiring parentheses somewhere or a convention that tells
where they go: (Kpq & r) or Kp(q&r). But such explicit
parentheses or conventions or other devices are needed already for
the infix forms in any case. As noted earlier, right parentheses are
generally detritus – except in various situations where they are
not. Right parentheses are needed more often, but they, too, can be
dropped in many cases (and always the outermost ones if they begin
the sentence). The rest of the infix cases depend upon conventions
involving order of grouping (left grouping of similar conjunctions
does not need parentheses – this and the following are not
necessarily the Loglanic conventions, but familiar types) or type of
conjunction (AND and OR don't need parentheses as components of IF).
The Loglans also have depth markers, so that a conjunction marked n+1
is of a component of a sentence with a conjunction marked n. And
there are convention about whether the prefix or the infix marker
dominates in a mixed sentence.
There is one more
marker that is needed in the Loglans. In prefix notation in FOPL,
the boundary between the two connected sentences does not need to be
marked, since the new sentence always begins in a distinctive way: a
new conjunction or a one-formula formula maker or a predicate, any of
which close off the previous sentence, which was down to a string of
terms, into which these new markers do not fit. But in the Loglans,
a new sentence can begin with a term or a quantifier, which now
counts as a term, and so can appear to continue the string of terms
of the previous sentence. One could, of course, require closing out
all the terms and the previous entences to start afresh, but it is
clearly more efficient to have, as in the case of the separation
between subject term and predicate, a single marker to accomplish
this necessity. As a plus, the separator can carry negations, which
means that the initial conjunction can be simple and yet all of the
logical relations be expressed.
With all these
devices, it seems likely that any formula of FOPL can get an
reasonably efficient unambiguous Loglanic formulation, though, short
of a fully parenthesized one, I am not sure this has ever been proven
(or questioned, even). What is less certain is whether a given
formulation is in fact unambiguous and, even if it is, that it is an
unambiguous representation of the formula intended. As will be
discussed later, the test for anamphiboly is not directly tied to the
structure of FOPL and the presumed indirect connections have not been
tested (or, for the most part, stated). For now, however, the general
expectation is enough to continue the claim that the Loglans are
spoken FOPL.
But conjunctions
introduce several new kinds of repetitive redundancies. And removing
this detritus introduces new kinds of expressions into the Loglans,
which, in turn, suggest new kinds of expressions in FOPL, expressions
which may have been there but were not discussed earlier. Some of
these cases are just matters of convenience (more efficient usage, a
branch of speakability), others are genuine new notions. Similarly,
so merely expand on already given categories, others change the
boundaries of familiar structures.
To take a simple
case, “Sam is tall and Sam drinks beer” (symbolically (Ts &
Bs)); do we really have – in a human language – have to (or want
to) repeat the “Sam” Just about every L1 experience says not.
The Loglans could, of course, use a pronoun here, but that is hardly
a savings. So we want to collapse the two sentences into the single
subject and a complex predicate. Now, in the logical tool kit there
is a device for doing just this, using a predicate making operator on
a formula and a variable. This would result in \x(Tx &Bx) for
the predicate and the desired sentence would be the \x(Tx & Bx)s,
not an improvement. But we have some experience with which suggests
immediately that we 1) move the subject to the from an replace the
operator, 2) assume the bound variable inside is the subject and so
drop it as covered in front, and 3) drop the superfluous right
parenthesis. This gives s(T&B, or even sKT,B. We do need the
left marker still, since B might be a sentence in its own right under
some circumstances. It also turns out, that if the & here is a
different word, peculiar to joining predicates, the left parenthesis
is not needed (except in more complex cases) , so we can get down to
sT+B. Curiously, this sort of change is not needed with K, since what
follows the K up to the separator shows what sort of expression is
involved. This factor will recur in what follows.
We can complicate
this example slightly: “Sam is tall and Sam is going to San
Francisco”: (Ts &Gsf). The first step in the collapse is\x(Tx
& Gxf)s. But now, we need to proceed with some care, since the
simple sT+Gf is unclear: f might be an argument to both predicates,
especially if T is (as is usual in the Loglans) a predicate of more
than one place with some later ones just not mentioned. There are
two simple possibilities: either mark the end of the compound
predicate to show that the following term goes with both or mark the
term as being connected with just the last predicate (similar to the
connection within terms). The general dislike of RHE markers favors
the second approach, sT+G-f, but, in fact, as cases become more
complicated, with some terms going with only one predicate and some
with both (and with more predicates involved), both systems have to
be used, so sT+Gf is also correct for this case (the final
parenthesis, after the f, not being needed).
All of this amounts
to a change like that seen earlier with quantifiers, a formula maker
has become a more inner grammatical type, a predicate maker in this
case. At least, unlike the case of quantifiers, the relative scope
of the collapsed sentence is not a problem, always being a component
of what larger sentence it lies immediately within. When the
collapse is extended, the abstracted sentence itself more than one
level deep, there may be internal problems of relative depth, but
there are surely enough mechanisms in place for the fully sentential
forms that fairly straightforward modifications can be made for these
cases.
This pattern calls
attention to another. A logician confronted with “This is a tiny
galaxy” would likely transcribe it as “This is tiny and this is a
galaxy”, KTt,Gt, which a Loglanist would immediately want to turn
back into tKT,G. But that Loglanist would also recognize that this
is just not right, even the tiniest galaxy is not tiny (or even
small). So, how do we deal with these? Logic has a series of
suggestions. The first is to simply say that “tiny galaxy” is a
separate predicate, related to smallness and galaxies, if at all,
only semantically and not formally. So a tiny-galaxy is indeed a
galaxy and smaller than most other galaxies, but this is all
additional information in the dictionary, not available
grammatically, as it appears to be in the English. That is, the
correct transcription is tW. This seems pretty unsatisfactory, even
aside from the necessity of constantly creating new predicates which
are related to existing one in similar ways. The second approach
(and Loglan proper did this at one time) is to say that a number of
adjectives (call them) are in fact two-place with the second place
for some reference class, so “tiny” is actually “tiny for a
...” with the argument “a galaxy” or “galaxies” or some
such added somehow (and just how is open to several suggestions) but
presumably as a term (*G in the Loglan, say). So, we end up with
tKGT-*G. This is clearly better, but the repeated G looks like
redundancy. To be sure, we do occasionally want to use predicates of
this sort non-redundantly: “He is tiny – for a walrus”, say
(meanly), hKHT-*W. But, when the reference class is given directly,
this seems unnecessary (and so to be eliminated for speakability
purposes). So, the third approach is to produce a predicate maker
which, in this case, asserts one predicate of the arguments and
relativizes the other to that first and then assert that whole of the
arguments again. While this case is typical, fine analyses have
found other cases where two or more predicates interact to create
something new, though related in regular ways to the underlying basic
predicates (adverbs, for example, like “very” or “rapidly”).
While the Loglans have developed experimentally a number of markers
for different sorts of such situations, the general approach has been
to use simple concatenation (as in English), so back to tTG (the
reference class comes last). Since both predicates may well have
other relevant arguments than t and may be complex in the way
discussed in the previous paragraph, some markers of grouping and
subordination may be needed, but there seem to be enough of those,
either in the forms used for sentential cases or in slightly modified
versions, to guarantee that an unambiguous expression can be found
for these cases. In addition, one of the concatenated expressions
might itself be a concatenation, not a buried sentential conjunction.
Sorting out the half-dozen or so readings of “pretty little girls
school” (tested later on such thing as “pretty little girls
school teachers union regulations compliance monitors”) led to
another system of prefix and infix and closure markers. parallel to
those for collapsed sentential connectives – and some devices for
resolving indeterminate scopes.
The opposite
situation also often occurs: same predicate but different arguments
“Sam is going to San Francisco and Bob is going to San Francisco”.
Again, an anaphoric solution is possible, but offers no advantages
over the original. So, as expected, the Loglans create a compound
term here – not corresponding to anything at all common in FOPL and
its kin. So, we get something like (s&b)Gf or, again with less
detritus, Ks,bGf; the occurrence of only a term between conjunction
and separator shows that this is a term maker. The infix system
needs a different form of the conjunction again (neither sentential
nor predicate), s^b,Gf, more or less. Once you start on this course,
of course, it is hard to stop. So “Sam is going to San Francisco
and Bob is going to Los Angeles” is Ksf, blG (non-first arguments
could always move in front of the predicate for rhetorical reasons
and so this poses no new issues) or sf^blG, with parentheses as
needed in each case. These moves can be iterated to, say, Ksf.bDlvG:
“Sam is going to San Francisco and Bob to either Los Angeles or
Las Vegas.” The subordination of the components, though moved from
the sentential to the nominal level remains clear. But, in a case
like KsbGDfl “Sam and Bob are going to San Francisco or Los
Angeles”, some doubt remains: are both of them going to one of the
places or is each of them going to one, perhaps a different one:
going back to the sentential level, DKsGf,bGf,KsGl,bGl or
KDsGf,sGl,KDbGf,bGl. The usual possibilities are available: we
might reorder the terms so that the topmost conjunction comes first
and so on, or we might mark each conjunction for relative depth.
This whole approach can even be extended to cases which are not
exactly parallel: “John is going through Chicago or by auto”
jGD4c5a. As noted earlier, the prefix notation is generally simpler
here, since the same form can be used for sentences and most
collapses (and markers added for nonsentential conjunction); the
infix forms require new forms (typically related) for each sort of
case: terms, predicates, term strings, and even subtypes within
these.
When we say that
Sam and Bob are going to San Francisco, there is no obvious
suggestion that they are going together (whatever that means: on the
same plane, in adjoining seats, for the same meeting, etc.), just
that one is and the other one is, too. But sometimes it is
significant that they are going together and that should be marked.
The straightforward way of doing this, a term-maker (of extendable
number of terms, since the group need not be just two) raises some
problems. In standard FOPL, terms refer to individuals, though that
is not very precisely defined. These new terms clearly refer to sets
or, at least, to more than one individual simultaneously (a little
excursion into Logic gets these two to amount to the same thing
eventually). The fact that the collapsed sentential forms above also
seemed to do so can be dismissed as being merely an appearance, not
the ultimate situation. To be sure, the present new situation can,
with some degree of plausibility, be reduced to the sentential case
by a variety of devices: as a collapse of “Sam is going to San
Francisco and Bob is going with him”, the latter predicate probably
concatenating with the former, or , more simply, as an preposition
“with Bob” attached to the main predicate and its argument raised
somehow (but quite regularly). Neither of these feels quite right
and so the term maker is used, iterated for more than two involved
terms. These terms can, obviously, interact with other types, from
above, so markers for relative scope are needed throughout.
There turns out to
be a similar situation with predicates as with terms, one thing with
two or more different components. So, along with blue and black
balls that are some blue and some black, there are blue and black
balls that are each partially blue and partially black. This seems,
possibly because it uses “and” in English) to be a special case
of combining predicates, different from the modifying sort and the
sentential collapse, and so it also receives its own markers (related
to those for set building above, perhaps). And, of course, devices
for marking relative scope.
And scope is the
last issue to deal with, the scope of those prenex 1-formula markers
moved inward early on. The Loglans tend to be very careful with
negation, keeping it clearly over compond sentences by attaching it
to connectives and making appropriate changes in quantifiers and
modals in the move. The situation with quantifiers and modals is
less clear. A prenex quantifier tends to be moved to the first
occurrence of its variables, which may be deep in some compound
sentence. Though there is a rule about heeding changes brought about
by negations, passage through a negation scope is not always obvious.
It may be more obscure if the quantifier is caught in a collapse and
is buried in a term, not even a sentence. So, while the general story
is that the scope of a quantifier is the shortest complete sentence
that contains all the occurrences of its variable, it may not be easy
to see what that is. And reconstructing the sentence may only work
up to equivalence, not the real original (not that that is a bad
thing). For the restricted quantifiers, which have no variable to
keep track of, the limits are the last pronoun that picks up that
quantifier expression – and there may be several such in the course
of a complex sentence. For the modals, there seem not to be strict
rules but rather loose habits: a tense marked predicate refer to
events at that time, subsequent ones (unmarked) refer to that same
time or ones later as the eventss flow naturally. Subsequent marked
ones place their event according to the mark relative to where the
time was when the predicate came along. Except, of course, there are
markers for radical shifts – to Now, for example, or some specified
event. The case for non-tense modals is even less clear: one tendency
is to take each as referring to the smallest possible sentence, the
other is to take them as lasting until and countering modal comes
along (and “in fact” to the ongoing “supposing”, say.)
In Summary,
The Loglans can be
said to be spoken FOPL (or its current equivalent) in the sense that
every sentence of such a language can be viewed as derived from a
formula of FOPL by a series of transformations, which preserve
meaning and structure, while reducing repetition and irrelevant
items. I have sketched the major types of such moves above, skipping
details, which are both very detailed sometimes and also have changed
over the history of the Loglans and in the different separate
languages. The crucial point is that these transformations are all
reversible, that the original formula can, in principle, be
recovered. A related feature is that the basic structure of that
underlying formula is close to the surface, easy to see, since the
transformation do not run deep.
Interestingly, the
books about the Loglans (Loglan 1 and The Complete Lojban Language,
preeminently) say little about all of this, but are focused more upon
the relations of the language described to familiar languages
(English first, of course). One would not really learn the grammar
of FOPL from any of these books and so lines like “this structure
in FOPL gets transformed to this structure in Loglan” do not play
much of a role, either as instruction or explanation. We do learn
that basic sentences consist of a predicate and a string of terms in
order, without any special marking for the roles of the terms and
that changing the order of some items is not to be done unless
caution is used (with some English cases of what lack of caution
could do). And that compound sentences come with a choice of
representations, which will carry over to sentences of similar
meaning which have compound predicates or compound terms. And we
learn that certain sorts of delimitors can be dropped and others not
in various situations, although this is based on problems about what
comes next in a string of words, not about the end of structure as
such. So, since the original transformation is not much discussed,
the reversal plays no role; it is enough that the sentence is
grammatical in this language, without considering whether it really
represents FOPL. Originally, this is not surprising, since the
scientific foundations for this sort of description only appeared at
the same time as Loglan began (1955) and the Loglans lost their
contact with academic linguistics (they never had much with field
linguistics) in the early 1960s, when these theories began to make
some way. On the other hand, the epigones of the Loglans were
largely computer scientists, and so theories of computer languages,
which are more static – not to say linear – dominate most
theoretical discussions of the grammar of the Loglans. This theory
has been directed mainly at producing parsers to derive a grammatical
description linearly (YACC and PEG seem to be the current models).
But surprisingly,
had the Loglans kept in contact with Linguistics outside the computer
field, in Anthropology and Philosophy and just pure Linguistics, it
would have found that it was in the forefront of the field.
According to not a few schools of Linguistics, every sentence of
every language is derived from a formula of some worthy successor of
FOPL, by some appropriate form of the moves outlined above. The
theoretical base is not, of course, strictly FOPL++, but an
abstraction with essentially the same structure. And the moves will
be different for each language, but basically of the same sort:
shifting linear order, collapsing commonalities, eliminating detritus
and so on. The major difference for natural languages, aside from
generally a much larger set of rules, obligatory and optional, is
that they are not required to be reversible. That is, a single linear
string of words can be derived equally correctly from very different
formulae. So, again I come to the point that the Loglans' interest
lies entirely in its monoparsing.
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