Thursday, June 4, 2015

The Role of Errors in the History of Loglans 1B

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.)  

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