Saturday, June 6, 2015

The Role of Errors in the History of Loglans 1C

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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