logic Language Draft 2.1

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To apply all of the information about AFOPL to Lojban, we need to notice one more point. We have already a couple of times slipped past an uncertainty about what the syntax of AFOPL is. We have mentioned what is going on and then said how it is represented when written out and each of these is at one point or another called the grammar of logic. We need to carry this over to our discussion of Lojban as well, now in a double way. We have what is going on, how that would be represented in AFOPL, and how the AFOPL is represented in Lojban. Once we do this, however, Lojban's claim to be a logical language in the present sense may seem to be trivialized, for not a few people hold that all languages are based on underlying AFOPL structures, which they represent in various ways. Lojban's claim to a special status comes, in that case, to a claim that its representations are particularly close to AFOPL or that its derivations (or deviations) are particularly clearly marked. At best, the claim comes down to one that the structure of the representations of many Lojban sentences is exactly that of the corresponding AFOPL sentences and, where not, the AFOPL forms are automatically recoverable when the notion is applicable. We will be commenting as we go on the extent to which such a claim is justified.

Clearly it works pretty well for simplest sentences and the corresponding atomic wffs. A predicate with all its arguments in place (as they always are in AFOPL) matches a selbri with all its sumti in place, up to the question whether the first argument comes before or after the predicate -- and we have seen that some versions of AFOPL have it come before (SVO rather than the VSO usual for logic). Deviations from this basic pattern, however, immediately introduce questions.

If we drop some terms -- as we usually do, the relation with the logical form requires some device, which ought to be very obvious, on our assumption that Lojban makes obvious any deviations from AFOPL. But what we get is simply a nullity, an absence -- which may not even be apparent if it comes in the last place(s) of a selbri with an unfamiliar place-structure. And what exactly is the change? Projected back onto AFOPL, there are several equally unpalatable possibilities.

  • One is simply to leave out some arguments required by a predicate. But that makes the string of symbols ill-formed, not a wff. This cannot be the underlying logic of a correct sentence.
  • The second possibility is that the predicate has been changed to a different but related one, which lacks just the arguments not stated. But AFOPL does not have -- at this elementary level at least -- any way of doing this, so it cannot be represented in the AFOPL form. And even if it could, it would clearly be complex and so require more -- not less -- in the Lojban analog. Further, if the omitted sumti is not the last one, the following one has to be marked for its place, which is one that assumes that the omitted place is still a proper place of the selbri.
  • Or we may insist that the places really are filled, but by items not presented -- for rhetorical reasons. The question is then what these items might be:
    • They cannot be bound variables (more on these directly), since these are major logical features, which AFOPL always displays.
    • They cannot be things picked at random, since such things might well falsify what was intended to be a true claim, or verify a false one (the bound forms might also have this effect).
    • So, they must be things picked in each case to give the right result -- an unstated {zo'e}, in fact. Notice that {zo'e} itself -- with the understanding that each occurrence has a different referent -- does not present the same problem as a gap. It does present the problem of how to represent this character in AFOPL. The nearest that is immediately available is to fill in unspecified places of an argument with terms not otherwise used (the next ones in alphabetic order, say). But in AFOPL this has the effect of using random things and so may affect the truth-value. At this point, we have to appeal beyond logic to pragmatics, which deals with how an expression is used in a context, beyond what its structure and meaning are absolutely. The context picks what things could be referred to by the dummy terms, even though there is nothing in grammar nor meaning (and so in AFOPL) to guide this choice.

Notice that with all the {zo'e}s in place, the problem disappears for AFOPL syntax, whatever remains for pure semantics, before pragmatic considerations are introduced. But the elision of {zo'e} is an unmarked deviation in Lojban from AFOPL and so an early -- though minor -- point against the basic claim.

Shifting the surface order of the arguments presents less of a problem, so long as the underlying order is preserved in the marks, as it is by FA. The changes are marked and the original automatically recoverable. Notice that the same does not hold for conversion (SE), which involves a different -- and more advanced -- logical operation, not merely a surface change. Of course, moving more of the later sumti to before the selbri -- from SVO to SOVO or even SOV -- requires no comment at all, since ultimately the surface position of the predicate with respect to its arguments is not significant. For Lojban, however, moving the first argument back -- from SVO to VSO -- does require a mark, since it is otherwise indistinguishable from VO, which has a different function.

In a given presentation of AFOPL, one or the other of Principia or Polish notation will be used for the connectives, and, if the former, one set of conventions for dealing with grouping: full parentheses, conventions about order or strength or when to reverse, or separate marks to indicate relative depth or some combination. Lojban has pieces of all the options: basically Principia but with Polish also permitted, left grouping with various ways to mark right-grouping segments, and devices to show depth, and full parentheses as well. The potential interplay of these systems makes for a fairly efficient way of stating complex compounds, although decoding them from speech -- or even from reading -- would seldom be easy.

As noted earlier, Lojban takes as basic connectives NOT, AND, OR, IFF and the two "repeat one, ignore the other" forms (TTFF and TFTF). The last of these (TFTF) is actually presented as derived from the previous (TTFF) by conversion, but, since we won't talk about conversion for a while, we will take them both as primitive (the process cannot be generalized for the connectives, since all the other basic ones are symmetric). The other nine two-place connective (aside from Tautology and Contradiction) are derived from these by negating one or both of the components. The Lojban forms for these connectives are quite explicit, placing a negative word adjacent to the negated component in each case. The system is not optimally efficient in the sense that it generates some redundant forms (it doesn't matter which component of IFF is negated, and negating both gets back the original, similarly negating the ignored component of the last pair of connectives gives the original still, so there are six unneeded forms). On the other hand, none of these definitions requires a component to be repeated, which is not the case with most other definitions.

  • TTTF (OR) : P ija Q , ga P gi Q
  • TTFT : P ijanai Q, ga P ginai Q
  • TTFF (1 only): P iju Q, gu P gi Q
  • TFTT: P inaja Q, ganai P gi Q
  • TFTF (2 only): P iseju Q, segu P gi Q
  • TFFT (IFF): P ijo Q, go P gi Q
  • TFFF (AND): P ije Q, ge P gi Q
  • FTTT P inajanai Q, ganai P ginai Q
  • FTTF P ijonai Q, gonai P gi Q
  • FTFT P isejunai Q, segu P ginai Q
  • FTFF P ijenai Q, ge P ginai Q
  • FFTT P inaju Q, gunai P gi Q
  • FFTF P inaje Q, genai P gi Q
  • FFFT P inajenai Q, genai P ginai Q

(Notice that Lojban needs a divider {gi} in Polish notation because there is no way to tell when one concatenated sentence ends and the next begins. Well, strictly there are several, but they are all longer than {gi} and less familiar.)

But, before we deal with two-place connectives, we need to say a word about NOT. In AFOPL, regardless of notation (well, almost), this is marked by a sign placed directly in front of a wff. For complex wffs in Principia notation, the outermost parentheses on the wff, which are dropped in most versions, need to be in place to negate the whole (or the negation mark needs to be given highest priority and all others dropped by one, if that is the way things are being done). This is the same regardless of the structure of the wff.

In Lojban, in simple sentences, the negation ({na}) over the whole sentence is placed directly before the selbri, not at the physical front of the sentence. This sign can in fact be placed any number of places in a simple sentence, but must always be presented as {naku} in other locations as derivations from basic (although the {ku} is needed even at the very beginning of a sentence, in what looks at first like the basic place). Unfortunately, sentences which are not (logically) simple have the apparent grammar of simple sentences and so the placement of {na} in the middle can mislead a person about the logic of what is going on. We will return to this after we have discussed some of the complexities involved.

In addition, the connectives may introduce further negations. In Polish notation and with following negations, the {nai} always comes immediately before the sentence negated and so interchanges with a leading {naku} and can be dealt with accordingly. With a leading {na}, however, the negation comes at the end of the negated sentence. This is still equivalent to a leading {naku} (not a trailing {naku} as it might appear) but the rearranging is often harder to do on the fly.

Negating compound sentences is also often hard to do on the fly. One can negate a whole complex in Polish notation (or beginning that way) by putting {naku} before the first connective. But compound sentences in Principia notation are not so easily dealt with, since explicit parentheses are rarely used. Generally, people seem to do a mental De Morgan: changing A to E and conversely while changing all signs, changing even numbered negations around O to odd and conversely (0 or 2 to 1, 1 to 0), and changing the sign on the affected component with U (compare the upper half of the chart above with the lower in reverse order). But, failing that, parentheses seem to be the best way to go: {P ijenai tu'e Q ija R tu'u}, since whole sentences are involved, or, since this is after a connective, {P ijenai ke Q ija R ke'e} or -- always safe -- the mixed notation {P ijenai ga Q gi R} (the right parentheses {tu'u} and {ke'e} are usually elidable). The straightforward {P ijenai Q ijabo R} ought to mean the same thing but is much too easily read as P & (nQ v R) rather than P & n(Q v R) ("n" for NOT, wiki lacking a tilde, too).

The previous paragraph shows the main features of longer compound sentences in Lojban. As the Lojban name, "afterthought connectives" suggests, the unmarked forms group to the left; the main connective is the rightmost. Right groupings can be introduced by parentheses ({tu'e ... tu'u}around whole sentences and, after connectives only, {ke...ke'e}), by forethought connectives (Polish notation) or by suffixing {bo} to raise the precedence of the connective in the right grouping. As the complexity of a compound grows, mixtures of these may be called into play, although the best plan for a really complex case is to think it through beforehand and then put it into forethought mode from the start. Trying to construct a complex case by additions is unlikely to succeed exactly the first time. The exception to this rule is the construction of normal forms, the disjunction of the lines on which the whole is true, each represented by a conjunction of the components plain or negated as they occur true or false on the line. But this involves many repetitions of the same sentence (as may an accurate construction of a shorter sort).

LoCCan note 1

Given the centrality of the truth-functional connectives in logic, one might expect them to be prominent in a logical language and thus to receive simple forms. The forethought connectives do come close to this, being marked only to distinguish them from afterthought. But the afterthought connectives -- the usual ones for logic -- are fairly complex, containing both a sentence marker (i) and distinguishing marker (j) as well as the basic connective forms. In logic generally the connectives are only sentential, so the i and probably the j would be unnecessary. But in Lojban connectives can come in various places within a sentence, not merely at the sentence boundary, even though what is represented is always ultimately a connection between sentences. The variety of distinguishing marks placed on the bare connectives is needed to show which place in the sentence is being doubled, since the parsing program cannot work that out just from what follows (though it can for forethought connectives, since the characteristic chunk is just what lies between the initial part of the connective and {gi}).

What can happen is that two sentences joined by a connective may share some parts, anything from differing in only one piece to being the same in only one piece. And language, seeking efficiency in a way that logic does not require (or even allow if it might impinge on clarity), compresses all this as much as possible, saying the common bits only once and attaching the connective to the differences. Since (so the argument goes) in real conversation, joined sentences typically have something in common, such collapsing is more common than fully expressed connections and, since the common items are most often terms, the simplest connective forms are used to join single terms: {la djan klama le zarci ije la maris klama le zarci} collapses to {la djan e la maris klama le zarci}. (Note that the collapsed form means exactly the same thing as the full form, with no hint that they went together or even at the same time, or -- thinking of the elided places -- from the same place or along the same route or in the same vehicle. This last creates problems for logic again, since the parts are not really the same -- from the logical point of view -- and so should not be collapsed.)

There may be more than one such duplication or a single one may involve several duplications:

{la djan e la djordj prami la meris a la delores} or {la djordj e la klaid o la xerold prami la meris}

Given Lojban's regular left grouping, the first of these says that either both John and George love Mary or they both love Delores. The alternative, that they both love one or the other of Delores and Mary (or perhaps both), can be dealt with in any of the sentential ways except {tu'e}: {bo} attached to {a}, or replacing the {a} with {gi} and preceding {la meris} with {ga}. Further, the terms can be rearranged to get the wanted left to right order: {la meris a la delores se prami la djan e la djordj}. The second example above means that either all three of George, Clyde and Harold love Mary or Harold and at least one of the others does not (at most one of George and Clyde does). The other reading, that George definitely does love Mary and that Clyde and Harold either both do or both do not, is again obtained either with parentheses ({ke) right after the {e}), or with {obo} or by polonization ("Polishing" tends to get misread).

Of course, we sometimes want the various sites of a compound to covary, to collapse, for example, {la djan prami la meris ije la djordj prami la delores} around a single instance of {prami} while still keeping John aligned with Mary and George with Delores. As things stand, the only device available is a "do not distribute" marker, not strictly a logical connective for all that it works similarly. So we get {la djan fa'u la djordj prami la meris fa'u la delores}. This happens because the two pieces to be coordinated are separated. If they occur together (as they can usually be made to do), they can be converted into term sets and joined directly by {e}: (assuming the second argument is moved to before the predicate) {la djan ce'e la meris pe'e e la djordj ce'e la delores prami). And this pattern, which takes a string of terms as a single term (in effect) can be used anywhere in the sentence.

Another common situation is that one thing is mentioned in several connected sentences, doing/being different things in each. In this case, Lojban divides a sentence in a way that does not correspond to logic, although it is fixed in the Lojban grammar: it separates what is before the predicate from the predicate and all that follows, the "bridi tail." Bridi tails can be connected after a single occurrence of the repeated preceding item(s), using connectives marked by {gi�}, which forms combinations like {j}. So {la djan klama le zarci ija la djan stali le zdani} becomes {la djan klama le zarci gi'a stali le zdani}. Only the selbri is essential here, so simply changing these would use this same technique. However, if there are terms in common which follow the connected bridi tails, the last member of the connected forms has to be closed off with {vau}, to attach the common elements to all, not just the last in the string.

In all of these, the Polish forms are still the same. There are some further kinds of compound sentences in which the Polish forms also need some further modification: {gu'} instead of just {g}. The main such case is of connections within tanru. Thus, {ge la djan citno nanmu gi la djan prije nanmu} collapses to {la djan gu'e citno gi prije nanmu}. The corresponding afterthought form is also distinctive: {la djan citno je prije nanmu}. In both cases the different form is to exclude confusion (probably not just in the grammar) with connected bridi tails that consist only of the selbri.

There are a number of other situations in which this collapse can be done. Most of these require the {j} form of afterthought connectives and either cannot be connected in forethought mode or require {gu'}. There are in Lojban a number of connectives that do not derive from the logical ones. These also occur in forethought and afterthought mode and work pretty much like the logical ones. They do not, however, need different forms for different places (except an {i-} prefix when joining sentences). The afterthought forms cannot be used to join bridi tails, and the forethought forms (which prefix {gi} to the basic forms) cannot be used where logical forethought forms cannot be used or where they have to be {gu'}.

Logic has nothing that directly corresponds to this kind of collapsing, of course. Within the logician's tool kit, there are devices for achieving the same result, usually -- as we will see -- much more ponderously. The various kinds of collapses are also achieved in rather different ways, and the different cases do not correspond to the differences that turn up in Lojban. The present system does work rather well however -- once you get used to it -- and it seems unlikely that another system, based more firmly on logic would be an improvement. Still the fact that non-logical connectives can cover almost all cases without change makes the need for three versions of the afterthought logical connectives and the restrictions on forethought ones a tempting field for further investigation.

LoCCan Note 2

Logic Language Draft 1.1

Logic Language Draft 3.1

There may be more than one such duplication or a single one may involve several duplications: (...) The alternative, that they both love one or the other of Delores and Mary (or perhaps both), can be dealt with in any of the sentential ways: {tu'e} before {la meris}, {bo} attached to {a}, or replacing the {a} with {gi} and preceding {la meris} with {ga}.

  • {tu'e} is not an alternative here, it only serves to bracket full texts (sentences and paragraphs, not terms).
    • True. Another reason for reforming the connectives if even parentheses do not work reliably (The rule is nowhere explicit except in the formal grammar, so far as I can tell.) Corrected.
  • My system messed up your dashes, apostrophes and quotes. Sorry about that. --xorxes
    • Everything looks to be back in place -- the screwed up tables seem to be just a feature of wiki.

  • in "otherwise indistinguishable VO", insert "from"
  • in "by negation one or both", negation > negating
  • in the table of connectives, in the first three lines there is a comma to separate the afterthought form from the forethought form; this comma is missing in the rest of the table
  • in "Well, stricly there is, but they are longer", "is" and "they are" do not agree in number; I'll leave it up to you how best to fix this
  • "too easily read as P & (Q v R) rather than P & (Q v R)" umm... I don't see a difference
    • Like the table above, this was the victim of wiki's curiously defective symbol set: no repeated spaces, no tilde. This is not the ideal format for presenting this stuff, but hopefully it will be saved by the speed of useful correction it makes possible.
  • in "introduced in by parentheses", delete "in"
  • in "suffixing {bo}to", insert a space
  • in "common that fully", that > than
  • in "with not hint", not > no
  • the sentence starting "Of course, we sometimes want" confuses me
    • Hope this is better
      • Much better
  • in "converted into terms sets and joined by directly by", terms > term, delete 1st "by"
  • in "assuming second", insert "the"
  • the sentence starting "Bridi tails can be connected" appears to be garbled
    • Better?
      • Yes, better

Thanks again.

  • My pleasure. I don't yet know enough Lojban to contribute in other ways, so it makes me feel good to be able to at least help with this kind of proofreading. mi'e ScottW