logic Language Draft 3.1

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Turning from the molecular to the subatomic, a wff of AFOPL is composed of a predicate and some fixed number of attached terms. Predicates differ (t this level at least) only in the number of terms they take. There are, however, several kinds of terms. So far, we have been talking only about simple terms, which directly name individuals in the universe: the analogs of names and their equivalents (generally pronouns). These are generally represented by lower case letters. However, it is customary to set aside some of these letters for a second type of term, variables, typically using letters from the end of the alphabet: "x, y, z, w" and so on. In addition, AFOPL may contain terms based on function symbols (typically letters from "f" on) and their following terms, and descriptions, which are built on other wffs.

Simple terms, names, which can also be counted as 0-place function symbols, require no further comment now. Terms based on function symbols are defined much like atomic wffs: a function symbol (analog of a predicate) "followed by" its associated terms. These following terms are officially to be recognized (as with predicates) by counting out the number of terms giving in the superscript of the function symbol. In practice, however, they are indicated in some other way, usually enclosure in parentheses or angles. Descriptions require a discussion of variables, so we will take them up after this section.

Variables are, in the syntax of atomic wffs, no different from other simple terms; their distinctive features are 1) semantic and 2) how they are used to build up other complex sentences and descriptions. A wff containing a variable is called a propositional function. The semantics of a variable is that a propositional function, though a perfectly good wff, is not like a sentence. It is neither true nor false in a given interpretation, since the interpretation as such does not assign an object to any variable. Rather, like a function generally, it returns a truth value for each object in the domain that is taken as the value of the variable. The taskof running through these objects is carried out by an assignment attached to an interpretation. It used the same domain as the interpretation and, within that domain assigns variables just as the interpretation assigns simple terms. When variables are involved, truth in an interpretation usually involves all of the possible assignments in the domain of that interpretation, all of the ways of hooking up objects to variables. Exactly how this is done will depend upon how the variables are used in the sentence at hand. A wff involving a variable is true in an interpretation + assignment, but not in an interpretation alone.

That is if the variable in question is free, not in the scope of a variable binding operator. A variable binding operator always comes with a variable attached and attaches to a wff, its scope. An occurrence of a variable is bound if it is attached to a variable binding operator or is in the scope of one. It is bound by the nearest variable binding operator which is followed immediately by the same variable. Any variable in a wff that is not bound is free.

For now, variable-binding operators are of two kinds, wff-makers and terms-makers. Each of these takes a variable and a wff and combines them to give a wff or a term, respectively. The syntax of both is about the same: the operator followed by the variable, followed by the wff (with its outermost parentheses in place if any). This unit then functions, in the one case like an atomic wff, in the other like a term, indeed, a function symbol followed by 0 or more terms, depending upon how many free variables the whole contains.

While there are a number of different types of operators of each sort, we will now be concerned with only quantifiers among the wff-making and descriptors among the term making (most other kinds are, in logic, reducible to these, by definitions, but not always in syntax). For now we shall confine our attention even within these groups to the traditional quantifiers and the range of definite, definite and "thatty" descriptions. There is room enough with these to fill our time.

The traditional quantifiers -- "all," "some," "no," and "less than all" (with various variants) -- actually began their careers in logic as something more like term-makers: "some man" was grammatical parallel to "that man" or "Socrates" as were "no man" and "every man" (a version of "all"). Indeed, they still have that character in English (and Lojban). Indeed, they did not originally involve variables at all, as the examples show. However, in the 19th century development of logic, as more complex sentences were subjected to logical analysis, a more flexible language was needed and variables came in to provide some of that, for the old system could only use the "some man" format once and subsequent references required pronouns or repetition, both of which were notoriously imprecise. A repeated "some man" might mean another man as easily as the same one, and "he" might refer back to "some dog" as easily as to "some man." And the developers were all mathematicians, used to the variables of algebra, which covered the same sorts of cases quite successfully.

You might expect that the transition would take the form of having sentences involving the quantifier terms be replaced by quantifier term + appositive variable and the use the variable ever after: "some man, x, ... x ...". But these expressions never occurred in the 19th century writings (or only in obscure and totally uninfluential ones). Rather the move was directly to a wff-making quantifier (in effect, there were no symbolic logical languages until late in the 19th century) with variable having universal range rather than each restricted to some class. Class restrictions were then built into the sentences quantified over. Only late in the 20th century did something closer to the old version appear: wff-maker quantifiers but with their variable restricted to particular classes.

The exact meaning of some of the quantifiers has also gone through a range of changes. In the earliest version, the four basic quantifiers formed an implicational square: "all" (A) and "no" (E) across the top, "some" ("more than none," I) and "less than all" ("some not," O) respectively beneath them. In each case, the upper, universal, implied the corresponding lower, particular, form (subalterns). The diagonally opposite forms (A-O, E-I) were negations of one another (contradictories). Either of the top two implied the denial of the other (contraries), while the denial of one particular implied the other (subcontraries). (These can all be derived from subalternation and contradiction alone, and all four forms derived from any two along an edge, by negation contradiction, and the rule for that edge.) Since the I form, "some S" pretty clearly says that there are Ss, the A form must do so as well, given subalternation. Consequently, the O form will be true even if there are no Ss, as will the E also -- given contradiction. In addition, E and I convert, that is, exchange the restrictive predicate of the quantifier with the descriptive one of the sentence: "some S is P" is equivalent to "some P is S," and similarly for "no." In addition, the universal partially convert, changing the order but shifting to the corresponding particular: "all s is P' entails "some P is S."

In the Middle Ages, however, the possibility of an empty class was discounted and the E and O were also taken to require that the mentioned class was not empty. The diagram remained the same, so long as the assumption that there was no empty class (or universal one, for that matter) held. In addition to the inference in the classical form, the medieval also allowed obversion (each form was equivalent to the one horizontally opposite with a negated predicate: O as "some S is not-P" comes from this period, and E is "all S is not-P," properly understood). And, once negative terms were allowed, contraposition was also recognized: A was equivalent to "All not-P is not-S" and so on.

The recognition that classes could be empty -- or that such classes might play a role in interesting arguments -- began in the 17th century and led to a reexamination of standard arguments, resulting in the new system of the 19th century. In this system, both universal were without existential import, that is, were true when the subject class was empty, while both particulars had existential import and so were false in those cases. As a result, all of the inferences on the square, except contradiction, were invalidated. Conversion (but not partial conversion), obversion and contraposition continued to hold.

The founders of symbolic logic in the late 19th century reproduced this system in their new format. They reduced the basic quantifiers to two, corresponding more or less to A and I, and separated the restriction on the quantifier from the quantifier, placing it in the body of the following sentence, conjunctively for I, conditionally for A: "some S is P" becomes "for some x, x is S and x is P," and "all S is P becomes "for all x, if x is S then x is P." The other quantifiers are defined by obversion, and conversion and contraposition follow from corresponding rules about "and" and "if." And, of course, new forms appeared that could not be formulated in the older systems. But one distinctive feature of the original system remained: the A used is the existentially importing A with respect to the domain covered by "x". The lack of existential import for the representation of "all S is P" comes from putting the subject term "S" in a condition, not from the quantifier itself. Only late in the 20th century did logicians consider the possibility of an empty domain and examine several systems that would fit into that (all "some" statements are false, of course, and universal ones are either 1) all true or 2) all false or 3) variable, depending on the subsequent structure: conditionals and equivalents true, conjunctions, disjunctions, atomics, and existentials false.

The original system is clearly the basic one, since all of the others can be defined with in it by obversion. An E with existential import is just and A with a negated predicate, and existential-free A is similarly derived from E. Within the modern system of unrestricted quantifiers reconstructing the various forms requires various additions. The modern system of restricted quantifiers presumably follows the classic pattern, with the 19th century pattern introduced by explicit additions. It would be possible also to take the 19th century pattern as basic and introduce the classic one by definition, but this would only make sense if the unrestricted quantifiers were already of the empty-universe sort.

The quantifiers of modern logic are written -- like most logical constants -- in an astounding variety of ways: "All x" "x" enclosed in parentheses or following inverted "A" or inverted "V" or simple capital pi, to list the most common; "some x" as "Vx" or inverted "E" followed by "x" or capital sigma followed by "x". But the syntax is the same in all cases, although some of the formats require that the quantifier phrase (quantifier + variable) be inclosed in parentheses. The restricted quantifiers are typically the quantifier of choice followed by the restrictive propositional function, the whole enclosed in brackets (here represented by parentheses becasue of wiki screw-ups): "(Vx Fx) Gx", "Some F is G." In any case, a universally quantified sentence "(x)Fx" is true in an interpretation just in case "Fx" is true on every assignment based on that interpretation, and "Vx Fx" just in case it is true on some assignment. The corresponding restricted case, "(Vx Fx) Gx," look at assignments that give to "x" some member of the set assigned by the interpretation to "F," and is true if at least one such assignment makes "Gx" true, false otherwise, including cases when the "F" set is empty ( this latter restriction holds for universals as well).

One of the descriptors of modern logic, the indefinite descriptor -- not the most common one -- takes up the classical particular affirmative quantifier "some S" (perhaps better "a (certain) S"). Those who use this descriptor typically write it with e (epsilon): "G exFx", "Some F is G." It differs from the usual particular quantifier (or either sort) in that repetitions refer always to the same object, like the repetitions of the variable bound by a quantifier. In addition, an interpretation fixes the object referred to by this construction; it does not change with different assignments, as it would for the quantifier. The referent is, if possible a member of the set defined by the propositional function that is the scope of the descriptor, the Fs in the example. If the set is not empty, of course. If the set is empty, there are a variety of possibilities, most of which have been explored:

  • every wff containing the description is false
  • the truth value is that of the wff obtained by taking the smallest component that contains all the occurrences of the descriptor, replacing those occurrences by a new variable, and then binding that whole component with a particular quantifier
  • every atomic wff containing the description is false, molecular sentences working out as may be
  • the description always refers to something, in this case an arbitrarily selected object, with truth worked out as may be
    • the arbitrary object is the same for all empty sets (the "null entity")
    • the arbitrary object is selected for each set separately (the "null F")
    • the arbitrary object is selected for each "failed" description (that is, even though two descriptions pick out the same set -- even necessarily, they may have different null entities.

These various possibilities give slightly different results -- well, the first major choice reduces the class of tautologies (if "exFx" is a real term). That case aside, the choice among the various readings is hard to make on theoretical -- or even practical grounds. In general, though, the view that a term always denotes something inclines the choice toward the fourth major choice, and -- for practical reasons -- toward the first subchoice: the same null entity for all.

Part of this preference comes from the problems that arose with the most common descriptor, the definite descriptor, written i or some stylized or inverted version of iota. Russell introduced this by a contextual definition, making it not a genuine term but an "improper symbol," an abbreviation for a longer expression in which no one part corresponded directly to the symbol, but the whole expression corresponded to the expression with the symbol in it. In this case the definition of "GixFx" is "for some x, both for all y (Fy <=> y=x) and Gx." This notion requires a complicated system of marks to indicate the scope of the description, that is, how to expand complex sentences containing such descriptions: is "~GixFx", to take the simplest example, saying that the unique F lacks G (which implies that there is a unique F) or just the denial that the unique F has G (which does not)? That taken care of, however, the truth of any wff containing this term can be worked out, regardless of whether there is exactly one F or not (compare the second solution for indefinite descriptions).

Before Russell introduced this expression, Frege had already introduced a similar notion -- with a different notation. But Frege's definite description was a proper symbol, a real term needing a referent assigned through the interpretation and, consequently, having a referent even when the set of Fs was not a singleton, was either empty or had more than one member. Frege used the null entity solution, the same entity for all failed definite descriptions. Since his time, all of the variations have been explored, including those that give a different null entity for an empty set from that given for a multitude. As with the indefinite quantifier, the Fregean solutions have tended toward the null entity solution Frege originally proposed.

Once you have a bunch of descriptors which purport to refer to objects of a certain sort, but that refer to something even if there are no objects of the sort mentioned in the description, it is, I suppose, inevitable that someone would suggest a descriptor in which the reference was fixed first and what propositional function was used was largely irrelevant. Of course, within logic, the simple terms function pretty much like that, referring without even any mention of a property that applies -- like proper names, in fact. Well, actually, some earlier logicians had held that proper names were always descriptions of some sort, that "Socrates" was a disguised way of saying something like "the irritating Athenian philosopher who taught Plato," for example. In denying this, the logicians of the sort presently being considered went on to note that a number of things that appeared to be descriptions also were not so in the strict sense (the one above for Socrates, for example would pick out Socrates even if we found out that Plato had several irritating Athenian philosophy teachers). The present descriptor arises to cover these cases: "G dxFx" (usually with a delta) means that G holds of whatever "dxFx" refers to, which may or may not be an F (though, for pragmatic reasons -- which play no real role in logic -- it should be an F, if one is available in context, or be somehow related to Fs).

Lojban takes a diverse sample from this historical accumulation. It has classical term quantifiers, both domain-wide and restricted variable quantifiers, and both the epsilon and delta descriptors. What it lacks -- at the current level at least -- are propositional functions: Lojban variables are never free. It also presents unique understandings of some of the items that it has, and raises some unique problems, mainly from the way that quantifiers are introduced into sentences.

Of course, most logical systems, when used, contain not free variables, since a propositional function makes no assertion and thus raises no truth issues -- what logic is ultimately about, after all. But I know of only one reasonably standard system -- Peirce's two-dimensional graphs -- in which unbound variables cannot occur syntactically. That system is also the only one I know that does not use explicit quantifiers for all the sorts of quantification it allows. But Lojban's variables are bound by an unexpressed particular quantifier unless bound by some explicit quantifier. This semantic point aside, they function like other pronoun terms.

In AFOPL each quantifiers is introduced at the beginning of the wff that is its scope. A quantifier binds occurrences of its variable only in that scope (and there only if they are not already bound by a close quantifier -- but that sort of doubling up is bad form). In Lojban, however -- like Peirce again -- quantifiers (explicit and, obviously, implicit) are usually introduced where the variable would first occur in the function, inside a sentence. To be sure, the rules say that quantifiers within a sentence are to be understood as occurring at the beginning of the sentence (well, right after any negation from just before the selbri) and that, when there is more than one quantifier in a sentence they are to be understood as being in front in the same left to right order as in the sentence. Further, a repetition of the same variable in a sentence is taken to be bound by the same quantifier as the first occurrence -- unless explicitly bound by another (but this again is bad form, if the two quantifications are meant to be independent).

The problem here is that the Lojban notion of a sentence -- unlike the AFOPL notion of a wff -- is at least vague. For the present question -- what is the scope of a given quantifier -- cases can be made for it being 1) the shortest full sentence which contains the first quantified variable 2) until the next unextended {i} (that is, not an ijek), or 3) until the last occurrence of the variable involved short of some "reset" expression ({da'o} and {ni'o} at least -- perhaps others) or 4) "long enough." The last is hopeless, of course, but the others present problems as well. The official line (discussed in a different place from quantifiers generally, 16.10, p.404) is 3, which poses the practical problem that the full impact of the quantifier is not known when it is introduced or at any time until reset. This is particularly problematic in cases -- also covered by 2 -- where some occurrences of the variable are superordinate to the original one or where negations are introduced onto the sentence within which the original quantifier lies. There does not seem to be a clear answer to how to say explicitly what {da broda ije da brode ija da brodi} says, since some occurrences of the variable are logically prior to the initial occurrence. Still, in this case, the explicit form seems to be {da zo'u tu'e da broda ije da brode ija da brodi} (and I am not perfectly sure the {zo'u} is needed). But the same cannot be said for {da broda ije da brode inaja da brodi} or even {da broda inaje da brode}, where the original quantification is now found to be in a negative context relative to its whole scope. This may just be the problem of postposed negations again and the answer is simply to bite the bullet and perform the changes in pulling it out. This makes sense for the first case (we are used to "If a boy comes to the dance, all the girls will dance with him" being a version of "All the girls will dance with any boy who comes to the dance" and to universals going with conditionals), {roda zo'u tu'e da broda ije da brode inaja da brodi}. But in the second case, the intention seems pretty clearly (influenced perhaps by the affinity of particulars with conjunctions and disjunctions) to have been to pull out the form unaltered: {da zo'u tu'e da broda inaje da brode}. The difference is, of course, what sentence the quantifier is at the front of originally (the negation is pretty clear, being outside the simple sentences already). We may -- as we have in fact done here -- use pragmatic considerations ("What is a person likely to be saying?") to decide in individual cases, but that is not a very firm basis for a logical language in any extended sense. And however this issue is decided, some apparently natural expressions are going to end up saying the wrong thing ("Gotcha"). In addition, quantification scope is inherently rightward grouping, thus running counter to Lojban's regular left grouping: in {tu broda ije da brode ija tu brodi da} one rule requires the whole to become a (particularly quantified?) disjunction with a conjunctive first part, another requires it to be a conjunction with a particularly quantified disjunction as second part. Presumably, the first is the official solution, but not necessarily the logical one.

All of these examples used implicit quantifiers to begin with, but the problems are the same with explicit ones, whether domain-wide or restricted, or in term form -- though the last uses anaphoric pronouns to pick up the same referent in other places. Logic supplies one solution -- which is not the one in Lojban usage as far as I can see. What I have called Lojban's representation of a term quantifier and what I have called the representation of an epsilon description are (virtually?) the same {su'o/lo/su'o lo broda}. But in several versions of epsilon descriptions that description is made referential outside the context in which it occurs: it is the selected member of the indicated set, even when it is referred to within a negative context. It is in effect, then, a particular quantifier fronted all the way to the beginning of the paragraph, right after the last previous reset at least. Taking this into Lojban would solve one set of problems (making a corresponding version for universal quantifiers {ro (lo) broda} would be harder to justify within logic, even thought the class of broda is determined outside context), those mentioned just above. But it would, of course, create a whole other set dealing with moving negations or even what an unmoved negation really meant. For now we leave the uncertainty as yet another mark of how far Lojban deviates from AFOPL.

Seeing that much of the work of quantifiers in one form can be done with those in another, the question arises of why the diversity, which seems greater than that of a natural language. Part of the answer seems to be a balance between efficiency and clarity: term quantifiers are short but occasionally imprecise, variable quantifiers open the way for precision, but -- when precise anyhow -- are prolix: compare {da zo'u ge da broda gi da brode} with {lo broda cu brode} which arguably says the same thing. Another, perhaps related, factor is that, even without the full apparatus of variable quantifiers and sentential connectives, some things are just easier to say with variables than with terms and anaphora: variables require no overt assigning with {goi} nor are they subject to the uncertainties of the various Lojban anaphora systems.

As for having both domain-wide and restricted quantifiers, the need seems to be more rhetorical than logical (questions of epsilon descriptors aside). Restricted quantifiers retain the natural subject-predicate, topic-comment, character-action dichotomy which is lost in the domain-wide forms: {da zo'u ge da broda gi da brode} could as easily be a version of {lo brode cu broda}, the focus is lost and yet is what makes the tale. In extreme cases, you get a whole story being nothing more than a claim that there is something with all these properties, rather than isolating a protagonist and the events. Not, perhaps, logically significant, but important for human communication.

Finally, there is the question of where Lojban quantifiers lie in scheme of quantifiers possible for the classical system, which ones have existential import, which do not. For the domain-wide quantifiers, this is a basically uninteresting question. Talking in this world presupposes that there are object in the world (else there could be no communication, indeed not sentences). And when there are objects in the domain, the importing and non-importing quantifiers behave exactly the same. If we really want to talk about the empty universe (the one God mysteriously did not let continue), the change is surely dramatic enough to call for some special attention by marking the quantifiers (though just exactly how they are to behave is open, as noted, to a variety of rules). It would seem then that assuming that the domain-wide quantifiers of Lojban are the traditional ones -- affirmatives importing, negatives not -- is the move that takes least explaining.

The issue with restricted quantifiers is rather different. The tradition again has the pattern just noted, but we do want to speak about sets we know to be empty and, more often, about sets that may -- for all we know -- be empty, but about whose members we can still say interesting things. Lojban has come down more or less clearly on the side of logical tradition here: the implicit "internal quantifier" (getting rid of which is another LoCCan note) is {su'o} and the subalternation inference is explicitly endorsed: {ro...} => {su'o...}. Further, of course, if the domain wide quantifier are traditional, it is hard to make the restricted otherwise, since the usages fade into one another: {da broda ije da brode}, {da poi broda cu brode}, {su'o broda cu brode}. As noted earlier, it is relatively easy to reproduce the opposite type of quantifier from the classical by obversion and even that is unnecessary, since the quantifiers, being numbers, can be marked "+" and "-" with the import interpretation. Getting the classical from the non-importing is more difficult, as is reproducing the effect of the importing restricted from the domain-wide: {ro da zo'u ganai da broda gi da brode} needs {da broda} conjoined on to get the effect of {ro broda cu brode} or even {ro da poi broda cu brode}.

At the quantifier level, Lojban allows any sort of numerical expression to occur as well as {ro} and {no}, {su'o} and {me'i}. In logic, some such expressions can also occur: "exactly n," "more than n," "less than n," "at least n," "at most n" and "between n and m" for natural numbers n and m. In logic, all of these are improper symbols, defined in terms of regular quantifiers and identity, in ways so that no proper part of the definiens corresponds to the definiendum directly. They function in the two cases exactly the same, presumably, when used as external quantifiers. Lojban also allows several other types of quantifiers that do not belong to basic logic: "few," "many," "most," and "enough," and a few others. Some of these may have analogs in non-standard logic, others involve factors beyond the scope of logic altogether ("enough" is purely contextual -- pragmatic).

Lojban also has variables and so quantifiers (and perhaps descriptors) for predicates, brivla, which AFOPL, being First Order, does not have. Moving up to Second Order destroys what virtues First Order Predicate Logic has; Second Order Logic, since it contains Arithmetic, is incomplete as well as undecidable -- and, if care is not exercised -- inconsistent as well. So even in standard logic there may be -- indeed are -- several essentially different Second Order Logics. But moving to Second Order, however undesirable it may be in logic, is essential of a human language that is to be used in communication. We shall return to this point directly, starting with to what extent even bare First Order Logic in practice uses Second Order features.

When talking about descriptors from the logical point of view, the main question is always "What happens when the defining function is false on all assignments?" JCB's descriptor, in Lojban {le}, is one answer to this (earlier, but with a different motivation, than the delta descriptors of logic). {le broda} is a proper term and so always refers. Pragmatic restrictions are in effect here, so that, when there are brodas in the context, the description preferentially refers to some of them. In the event there are no relevant brodas -- and in special cases even when there are -- the referents are things plausibly connected in the context with brodas. This does not say much but prevents just arbitrarily, outside the conversational contract, calling any old thing a broda (Humpty-Dumpty is restricted by the need -- which he probably never felt -- to communicate).

The case with {lo} is, of course, quite different, since {lo} is veridical, requires that its referent actually satisfy its function. Given the close connection with particular quantification, the obvious move to make here is to call any sentence containing {lo} false, with all the usual questions about what "sentence" means here: simple, whole complex in which occurrences of the {lo} phrase or its anaphora occur, whole paragraph in which this happens, and so on. In other words, we can treat {lo broda} as a quantifier. Taking {lo broda} seriously as a term leans heavily toward the last choice, the preposed existential claim over the whole context, making it all false. But that has also the look of presuppositional failure, so rather than denying the whole lot, it might be more correct to {na'i} it.

The official Lojban line, however, is neither of these, but to take a {lo} description that fails in the present universe of discourse as a sign that the discussion has shifted to another universe of discourse, where the description holds, an implicit {da'i da broda} or however that shift is to be marked. This move is not available in logic, of course, and is objectionable in language from a pragmatic point of view. The shift is made unilaterally and most likely unconsciously, which goes against the basic contract. Further, the shift renders illegitimate the natural response of another member of the contract, {no da broda}, since, in the new universe, this is false. On the whole, the {na'i} solution is both the most revealing and the least denigrating of the possibilities -- it keeps the whole {lo broda}-containing discussion from being true, without calling it false.

Notice that the other question that disturbs logicians about descriptions does not arise here. Lojban descriptions are inherently plural and only actually singular. They are about sets of individuals, not individuals, even though the set may be a singleton. This means that some of the features of logical terms do not apply directly to them as the do to names and variables (though even names may have plural referents in Lojban). Projecting back into logic from Lojban, then, makes all the description technically improper symbols, statements containing them to be expanded in such a way that no part corresponds directly to the description. In fact, each description selects a set of objects and them makes a claim about all of the members of that set (which usually is a subset of the larger set given by the function). In this context, further quantification on a description makes sense. This does leave the question of whether {re lo broda}, {re broda} and {re da poi broda} really mean the same thing, even if they usually play the same role in the same claims.

It is in this context that requantifying a bound variable seems to make sense. If {da broda} selects a set of objects, as {lo broda}, taken as a descriptor does, then to use {re da} in the scope of that original {da} is just to take two of the things selected by the original quantifier out of the brodas. On the other hand, if there is to be in Lojban any connection with concrete individuals, variables have to remain singular, as they are the only things left for that purpose. Since I think that this connection is vital, I would recommend against requantification. The second {da} is in fact a different variable, carelessly misspelled. The effect of requantification requires a description {lo}.

Which brings us back to an earlier question: are {lo broda cu brode}, {su'o lo broda cu brode}, {su'o broda cu brode} {da poi broda cu brode} and {da broda ije da brode} all the same? Is there a grammatical clue to a distinction between a quantifier and a descriptor here? Are there contexts in which using one rather than the other makes a difference -- especially in truth value, but in pragmatic ways as well? The previous discussion shows that the answer to the last one is "Yes." So, the second question -- and thus the first -- is significant. CLL and tradition seem to come down rather heavily on the side of "Yes" for the first question and so "No" for the second. Is there a way to introduce a break in this pattern? At the moment, there does not seem to be. The matter is not terribly pressing (it can go into LoCCan), since the hard cases are rare; none seem to have occurred so far -- or at least to have been notice.

Logic Language Draft 1.1

Logic Language Draft 2.1


But the same cannot be said for {da broda ije da brode inaja da brodi} or even {da broda inaje da brode}, where the original quantification is now found to be in a negative context relative to its whole scope. This may just be the problem of postposed negations again and the answer is simply to bite the bullet and perform the changes in pulling it out.

  • Can you be more explicit? {da broda inaje da brode} is {genai da broda gi da brode}. Are you saying that this is also equivalent to {naku ga da broda ginai da brode} = {naku su'oda zo'u ga da broda ginai de brode} = {roda naku zo'u ga da broda ginai de brode} = {roda zo'u genai da broda gi de brode}, instead of being equivalent to {su'oda zo'u genai da broda gi da brode}?
    • I don't know about the also but yes, a good case can be made for that being exactly what the original means. Now, it may not be the Lojban case, but the rules do not clearly exclude it, and some seem to imply it.
      • But {da brode ijenai da broda} would in any case be {su'oda zo'u ge da brode ginai da broda}, right?
        • As things are going at the moment, yes.

This makes sense for the first case (we are used to "If a boy comes to the dance, all the girls will dance with him" being a version of "All the girls will dance with any boy who comes to the dance" and to universals going with conditionals), {roda zo'u tu'e da broda ije da brode inaja da brodi}. But in the second case, the intention seems pretty clearly (influenced perhaps by the affinity of particulars with conjunctions and disjunctions) to have been to pull out the form unaltered: {da zo'u tu'e da broda inaje da brode}. The difference is, of course, what sentence the quantifier is at the front of originally (the negation is pretty clear, being outside the simple sentences already).

  • But how could the quantifier possibly be not at the front of the connective, given that both connectands contain the variable?
    • The question is not whether the quantifier is at the front of the connective; of course it is. The question is how much the quantity of the quantifier is affected by the environment in which it initially appears at the surface level.
    • And it seems that whatever answer we give is going to be "wrong"; in some cases, that is, counter-intuitive.

We may -- as we have in fact done here -- use pragmatic considerations ("What is a person likely to be saying?") to decide in individual cases, but that is not a very firm basis for a logical language in any extended sense. And however this issue is decided, some apparently natural expressions are going to end up saying the wrong thing ("Gotcha").

  • An even more clear case is {da ganai broda gi brode}. Requiring that to mean {roda zo'u ganai da broda gi da brode} would be perverse.
    • Well, "perverse"; is rather strong. IF it means the same as {ganai da broda gi da brode}, then it seems pretty clearly to have the "perverse" interpretation (as does "if something is a broda then it is a brode"). And, of course, if it does not mean the same thing, then all that stuff about compacting compound sentences is out the window -- hanging by its nails, if not crashing to the ground below.

In addition, quantification scope is inherently rightward grouping, thus running counter to Lojban's regular left grouping: in {tu broda ije da brode ija tu brodi da} one rule requires the whole to become a (particularly quantified?) disjunction with a conjunctive first part, another requires it to be a conjunction with a particularly quantified disjunction as second part. Presumably, the first is the official solution, but not necessarily the logical one.

  • I see three alternatives:
  1. {su'oda zo'u ga ge tu broda gi da brode gi tu brodi da}. Since {da} appears within both members of the disjunction, its quantifier must be in front of it. (I suspect this one is the official solution.)
    • Yes, I am reasonably sure this is the official line. We can probably come up with some good story about how the {su'o da} gets over the whole conjunction when it appears in only one conjunct. But notice that the two stories are opposed to one another. Why not have it go over a compound in which it does not occur at all?
      • Good point. The more I think about it, the less I like having the implicit quantifier leave its home bridi.
        • But that means either that there can be no short-form quantified complexes or the complexes are not equivalent to their unpacked versiosn. Neither of these is desirable -- even feasible, I think.
          • {da ganai broda gi brode} would not be equivalent to {ganai da broda gi da brode}, that's right, but it could still be unpacked with a prenex: da zo'u ganai da broda gi da brode}.
    • Notice also that this problem arises even without negtions of any interesting sort. In fact, there are in logic a number of theorems about moving quantifiers over connectives. Very few of them are equivalences. But they all assume that we know where the quantifier is in the first place, which is just what we do not know here.
      • It now seems to me that the only sane alternative is to force it to its closest prenex. That means interpretation 2 below in this case.
  1. {ga ge tu broda gi su'oda brode gi tu brodi su'ode}, i.e. two different quantifications, assuming that the quantifier of da can't jump so far outside of where the variable first appears.
    • We could get away with this only if "sentence" in the definition of "scope" was limited by anything that contained {i} , which would ruin countless other cases.
      • Which cases would be ruined? This is my favourite interpretation at the moment.
        • Any -- every -- case where we wanted the variable bound in two different components -- which is most of them, I suppose. We could, I guess, insist that only fully prenex forms were grammatical, but that is not going to fly -- nor should it.
    • An alternative might be to have the variable bound by its highest occurrence, not by its first, but that seems to open a nest of interpretation problems as well (I have not sorted tham all out, however).
  1. {ge tu broda gi su'oda zo'u ga da brode gi tu brodi da}. This one is surely wrong as it violates the left grouping of jeks.
    • But enforces the right grouping of scope: a quantifier turning up works like a left parenthesis at the beginning of its minimum sentence. Not a likely reading, but not excluded by CLL in any clear way (because CLL is never clear on any of this) and certainly justifiable logically.
      • I don't like this one at all, if the quantifier can jump out of one connective, it might as well jump out of two (why open just one left parenthesis), as you say this gets easily out of hand.
        • This was my least serious suggestion. It violates symmetry all over the place and creates a rash of new problems -- but it is possible and has good logical gounds; just no Lojban ones.

pc: I think that one of the best arguments that quantifiers leap out of complex contexts without regard to the status of their first occurrence is that all of the basic connectives (except U) are symmetriccal, so, for example, nP & Q should be equivalent to Q & nP, making the first appearance of a quantified variable in both P and Q be in Q and so recto (inside an even number of negations in conjunction-negation normal form). But this does take some rewriting, too. (I wonder, by the way, what the effect is of a quantifier in the ignored component of a U compound.) On this view, then, a particular quantifier, wherever it occurs in a compound sentence, so long as its variable occurs in in more than one component, leaps to the smallest part (including the whole) that contains all its occurrences and keeps its particular character. Very much like the epsilon descriptor, on a more limited scale. I suppose that {ganai ge da broda gi da brode gi ti brodi} still comes out to the front as a universal.

  • I don't understand why it comes out as a universal. Is it because of the {nai}? Would an explicit {roda} come out to the front as a su'oda? I think it is better not to let it leap out.
    • Yes because of the {nai}. The smallest subformula containing all occurrences of {da} is {ge da broda gi da brode}which comes out to {su'o da zo'u ge da broda gi da brode} but that is the first component of a conditional and thus negated, so that the quantifier when moved before the {ganai} is {ro da}.
      • Yes, of course, I was confused. I was thinking of {ganai da broda gi da brode}.

I note also that {ganai da broda gi da brode} looks to be equivalent to {naku ge da broda ginai da brode} by simple De Morgan, but pulling the {da} to the front of this changes it to {roda} again, by the same argument (this is Peirce's standard universal conditional).

  • So, if I understand correctly, you are suggesting this:

ge da broda gi da brode == su'oda zo'u ge da broda gi da brode

ge da broda ginai da brode == su'oda zo'u ge da broda ginai da brode

genai da broda gi da brode == roda zo'u genai da broda gi da brode

genai da broda ginai da brode == roda zo'u genai da broda ginai da brode

ga da broda gi da brode == su'oda zo'u ga da broda gi da brode

ga da broda ginai da brode == su'oda zo'u ga da broda ginai da brode

ganai da broda gi da brode == roda zo'u ganai da broda gi da brode

ganai da broda ginai da brode == roda zo'u ganai da broda ginai da brode

I don't like the asymmetry. "If something is broda then it is brode" may make {ganai da broda gi da brode} look natural with {roda}, but then what about "something is broda if it is brode", shouldn't {ga da broda ginai da brode} get the universal quantifier as well? And there are other cases of universal "something" in English, like "something made of gold is valuable", so would that extend to {da poi broda cu brode} forms?

pc:

Over night I have come to the position that it really is a recto-verso matter (even or odd number of negations in AND-NOT normal form)for the smallest chunk that contains all the occurrences of the variable. Thus, I would now reject numbers 3 and 4 from your list, since the container is an unnegated AND, but also 5 and 6, where the container is a negated AND P OR Q) is NOT(NOT P AND NOT Q. I have to admit that I am not completely happy about 5, but everything else seems clearly right. Of course, spotting side on the fly is no snap, so maybe some other plan would be better for Lojban, whatever its consequence logically. But, as a rule of thumb, if the whole scope is atomic or conjunction or U, then the occurrence is recto (the quantifier on the outside is unchanged), otherwise it is verso (quantifier changed). Of course, that scope may be buried and thus be further changed as the quantifier is moved to the front of the whole compound.

The "something" cases are a bit risky to use; at most they show that the universal reading of particulars is not unprecedented, but they cannot be generalized. I would suppose that "Something gold is valuable" needs {roda} in Lojban (unless "something" is /smthn/, in which case the claim is not universal anyhow).

  • So, would the recto-verso system work like this:

ge(nai) da broda gi(nai) da brode == su'oda zo'u ge(nai) da broda gi(nai) da brode

gu(nai) da broda gi(nai) da brode == su'oda zo'u gu(nai) da broda gi(nai) da brode

ga(nai) da broda gi(nai) da brode == roda zo'u ga(nai) da broda gi(nai) da brode

go(nai) da broda gi(nai) da brode == roda zo'u go(nai) da broda gi(nai) da brode

(where the nai's on the left match those on the right)?

  • Would we also have:

ga(nai) roda broda gi(nai) da brode == su'oda zo'u ga(nai) da broda gi(nai) da brode

go(nai) roda broda gi(nai) da brode == su'oda zo'u go(nai) da broda gi(nai) da brode

? These ones seem fairly unintuitive.

pc:

They do indeed, but the alternative seems to be to have {da} and {roda} amount to the same thing in disjunctions and equivalences and that is even more counterintuitive. Or come up with another rules that gets right what this one does but also gets these cases right as well (assuming we could agree on what is right).

On the other hand, if we treat "if everything is F then it is/ they are G" as we did the same pattern with "something," then that form does imply "there is something such that if it is F then it is G," just as the "something" form turned into a universal. In neither case does the inference go the other way, however. But these go beyond strictly working internal quantifiers outward, so again it is merely a suggestion that the step in question is not crazy, not a guarantee that it is right.

We could, I suppose, go to having the quantifier as given come out to the smallest unit containing all the occurrences of the variable, but that seems to ignore negation, to the point that one wonders why {da na broda} needs to be expanded to {roda naku broda}. But altering that does seem to be going too far.

  • "Quantifier comes out as given" seems to be the safest path. Otherwise we get very strange results for quantifiers other than su'o, things like {cida broda gi'a brode} become quite odd.
    • Numerical quantifiers are improper symbols and the way they unfold in logic makes "coming out as given" almost automatic, regardless of what happens with {su'o} and {ro}. I don't think they count as evidence either way -- from the logical point of view; maybe they do for Lojban.
  • The behaviour of {na} when right in front of the selbri is broken in any case. {da na broda} is supposed to be {roda zo'u da naku broda}, but {da na broda gi'e brode} has to be {su'oda zo'u ge da naku broda gi da brode}.
    • Does it? Giheks join bridi tails, so presumably the {na} has the whole sentence as its scope: {naku da broda gi'e brode}, that is, {naku ge da broda gi da brode}, which ends up a universally quantifed disjunction when all is said and done. Your proposal seems to come from {da broda nagi'e brode} and so is not an exception to the original suggestion nor to the negation rule within the "come out as presented" rule. Again, it is not clear how decisive all this is one way or the other.
      • At least for the parser, na is part of the bridi tail, so {da na broda gi'e na brode} parses as {da (na broda) gi'e (na brode)}. That proves nothing, of course, but if we are going against that parse we should have some justification. If the first na takes the whole sentence, bridi-tail connectors would not be symmetric. I could have given {da brode gi'e na broda} as the example, which is just slightly less striking. I agree that none of this is decisive one way or the other, but the more we stray from the parse structure, the more likely it is that we will run into this type of conflicts.
        • Well, the parse structure is notoriously uninformative for logic: the notion of a bridi tail to begin with and then putting the negation in it for another. In fact, {na} seems to turn out to be part of the selbri in the parser, and that is clearly wrong logically: {da na broda} is explictly distinguished from {da nalbroda}. The parser is constrained by the limits imposed by YACC and is often misinforming about crucial things. That being said, your reading has at least the parse going for it; mine has a couple questionable lines here and there in CLL -- probably meant to be restricted but not presented as such. A third possibility -- also based on CLL text -- would be to say that the original sentence in question is equivalent to {ge da na broda gi da na brode} from which the {da} apparently would be extracted as {su'o da} in spite of all its occurrence being in overtly negative context. If nothing else, all of this makes clear how much CLL needs further elaboration, which is the task that is being undertaken (Huzzah!). I suspect that any decision on some these issues can be made to work with only occasional difficulty (we don't do a lot of complex sentences really), but we need to be careful that the various decisions cohere and that they are expounded both in generality and in how that works out in particular cases. I notice the same run of problems coming up with tense and modals, even though tense is probably the most thoroughly worked out feature in Lojban. The "tuck it inside" design scheme kicks back mightily later on (though the "leave it front" is often unspeakable -- in several senses).
          • {roroi} and {su'oroi} are good "tenses" to start with, as they work like pure quantifiers. The usual interpretation is that {na roroi broda} and {roroi na broda} mean the same thing, but I don't see any justification for that, given that both the negation and the quantifier are at the selbri. And the same goes for other tenses "regularly not broda" is not the same as "not regularly broda".