logic Language Draft 4.1
We have met worlds already: a set of things and an interpretation which assigns the things to fixed terms and the supersets to appropriate predicate and relation expressions. In this abstract form, nothing is said about what the things in the set are. Indeed, they are -- from the object language of view -- chunks of prime matter, since all their properties are added later by the interpretation and the same set may be at the heart of many worlds, subject to many interpretations.
For AFOPL all that matters about a world is how many things basic there are in it. Validity can be attributed to all the worlds of a given size: some wffs will be 1-valid but not 2-valid
("Vx Fx => (x)Fx," for example) and so on. Validity pure and simple means valid in all worlds, regardless of the size of the domain (though, in ordinary First Order Logic, validity in the lowest level infinity, aleph-null, is sufficient for validity tout court). Of course, in finite domains, some objects will receive more than one name and, since the number of basic objects limits the number of sets of them, so will many sets and relations.
Clearly in the original understanding of worlds, the basic things do not include sets, since there are always (even in an empty world ? which is generally excluded from these discussions) an unreachably infinite number of these -- in some sense (most set theories are done in First Order Logic and therefore have models in an aleph-null domain). But as soon as worlds come to be applied, the possibility that sets are among (even all of) the basic items has also to be allowed.
The height of abstraction for worlds would have it that there is only one domain ("this one") and that logicians create the various worlds merely by changing the interpretations (or the language, as it were). For a First Order System, that domain could be just the terms themselves, with the various apparent sizes achieved either by restricting consideration to some subset of the whole ("the real things") or by introducing as identity something other than the set of ordered pairs with the same element in both positions. Another possibility is to start with a set of properties (this is clearly getting somewhat more applied) and then get the domain as the set of all collocations of properties, that is, there would be an individual with (exactly) each possible combination (and maybe impossible ones as well) of properties.
Typically, attempts to specify the content of worlds are part of either a metaphysical presentation (there are only the instantaneous perceptions that compose the skandhas as a start on Buddhism) or a metalogical claim (logic is only about words, not things, or set theory does not require sets). But another source is from generalization of situations. Situations arise in logic primarily as refutations, ways of showing that an argument is invalid without going through a whole world. It is enough to show a small case where all the premises are true but the conclusion is false. The case can be a real, taken from life, with appropriate interpretation of the symbols, or it can be manufactured, with the symbols left uninterpreted or interpreted only to give vivacity to the case. A situation, then, typically deals with only some predicate symbols and only some terms, not the whole language. For some purposes, a situation does not even have to be consistent. However, the consistent ones can be extended to full worlds in a variety of ways (in fact, doing so is a part of most proofs of completeness and consistency for First Order Logic).
And, of course, in a language for communication among humans, it is situations, never whole worlds directly, that we deal with. And so we come, when talking about such a language, to populate our worlds with the characters of the situations that we deal with in our communications. We shall come back to the effects of this move on the logical worlds directly.
When we move beyond the standard logic into modal logics, logics that take more than one world into account at a time, we incorporate all the worlds into our field of interest. In this area (or, better, areas), in addition to the size of each domain, the relations among domains are important. Indeed, modalities turn out to be shorthands for claiming that certain relations hold between worlds of certain specifications. To say, for example, that something is possible in one world is to say that it holds in a world related to the given world by a relation of a certain sort.
Four kinds of such relations have had some investigations in logic: corresponding to alethic, deontic, temporal, and subjunctive modalities. Alethic modalities are the familiar "possible" and "necessary." The relations in this group tend to be reflexive -- every world is reachable from itself, so whatever is is possible (true in some reachable world). Major systems add that the relation has to be transitive (what is reachable from a world reachable from here is reachable from here directly, what is possibly possible is actually possible) or symmetric (here is reachable from what is reachable from here) or both. But there are countless systems that fall outside these or fit between members of this collection. In addition to the relations -- or as part of the definition of them -- there might be internal restrictions on the worlds reached. They might have to have the same or very close domains, and within those they might have to have virtually the same distribution of properties among things. Or they might have to reflect the same laws of some sort. These latter restrictions correspond to the intuitive notions of different kinds of possibility: logical (non-contradiction), physical (laws fulfilled), technological (state of the art maintained) and so on.
Some of these notions of possibility clearly involve time as well as other worlds: we do not expect that technically possible things will come to be instantaneously; they take time. In fact, our travel through time (at the rate of 60 seconds a minute) is our most immediate practical experience of other worlds, for the world now is different -- however slightly -- from the one when you began this sentence and from the one when you end it. So another type of relation considered in modalities are the temporal ones, which order worlds into sequences of some sort, "earlier than"-"later than" relations. These use two weak modals ("possible"), one for looking up the sequence ("future") and one for looking down ("past"). Systems then arise by various restrictions on the relations and so the orderings involved. The ordering can be, for example, discrete (there are events which are future but not future future since there is no intervening world), or dense (contrarily) or continuous. There may be a single ordering for all the worlds ordered, or it may branch toward the future or the past or both. Ordering may curve back on themselves in one direction or both (so that an event occurring implies that it has occurred before -- the Qoheleth option -- or will occur again. And so on through innumerable variations of how time (or history) might be structured.
Many of the special alethic modalities come from defining "possibility" in temporal terms, as several ancient philosophers (and not a few moderns) do. Given a definition like Diodorus -- that the necessary is what always has been, is now, and ever will be (or possibility as what once was, is now or will be sometime), variations on the structure of history give rise to a variety of modal systems that are hard to characterize by looking at the relations among worlds directly. Or, in another sense -- necessity as what is certain, working from Peirce's definition of necessity as what has been is now or definitely will be (is somewhere on every future path) generates a variety of system that fall well outside the classes discussed above (an occurrent event is necessary, for example, not merely possible), so much so as to be considered by some as a totally different concept. Again, internal features of the worlds involved may play a significant role in these orderings -- contents of the domain ought to change regularly over time and once something leaves the domain it does not return (or, more usually, once it goes into the "dead" class it does not leave it unless it drops out of the domain altogether). And the content of the denotations of various predicates also shifts slowly over intervals on the path(s) through time.
These kinds of considerations are also apparent in deontic modal systems. Here the worlds down relation from a given world are morally (etc.) idealized worlds from the original. To say what one ought to do in a situation (world, for now) requires that the idealized worlds be as much like the original as possible, up to what one does (notice some temporal notions coming in here, at least in the decision making use of such modalities). The relations are clearly not reflexive: what one does may not be permissible (true in some idealized world) and what is obligatory (true in all idealized worlds) may not be what one does. The base world is (by definition?) not idealized. Besides the temporal elements using deontic modalities to discuss decisions, they may be involved also just in the notion of idealization: the right thing is what leads eventually to the greatest happiness for the greatest number (utilitarian morality), say, or what is most likely to lead to success (prudential advice).
All of the modalities so far considered are represented in logic by one-place connectives, their syntax paralleling NOT. Possibility is usually represented by a diamond ("L" in Polish systems) and necessity by a square ("M"). Temporal logic uses "P" for "is truth as some point on some past path" and "F" for the corresponding future. "G" is then for "true at all points on all past paths" and "H" for the corresponding future. The "true at some point on each path" has not been developed to the point of having a symbol. Deontic logic usually uses "O" for "obligatory," true in all idealized worlds, and "P" for "permitted," true in at least one. Alethic and temporal modal systems have also been constructed with primitive 2-place connectives, from which the one-placed can be derived by definition.
Subjunctive modality is fundamentally relational however. It is about what happens in another world or, better, class of worlds defined as those in which a certain claim holds. In natural languages it often takes the form of a conditional to introduce the world(s) and then some implicit marker to continue in that world as long as need be. It is usually assumed that these new worlds are as much like the present one as is compatible with the changed state that defines them. What those difference might be is usually the point of this modality: what will be true in all such worlds, what will usually be true in them (i.e., true in most of them) and so on. Clearly, working out the truth of such claims is not going to be easy. The usual convention of logic that all assignments of properties to classes are independent of one another would mean that the change made no difference. Having the currently known laws that connect certain predicates carry over to the alternate worlds helps -- with the provision that the defining assumption can be that one of those laws does not hold, so that only the remaining ones still apply. But many of the interesting questions to be posed in this way are precisely not lawlike but about character, say, or some other intuitive but largely unanalyzed notion.
Subjunctive modality is clearly related to temporal ones in many uses. We often ask "What would happen /have happened if such-and-such were the case?" where the question is largely about things that happen after such-and-such coming to be the case, generally at a particular moment in time. Indeed, many natural languages -- English for one -- construct their subjunctive forms out of the material of their tense representations. The general map is roughly going back to a time when the defining state might have occurred but did not (the Confederacy winning the Battle of Gettysburg -- maybe more specifically Pickett's Charge succeeding) and then -- assuming time to be branching to the future at each point -- proceeding along a different set of paths. Viewed in this way, much of the requirement that things be as much like the present as possible takes on some specific content: all of history, all of character, all of the laws up to the crucial time are set and, barring that a change is what is called for by the defining assumption, will continue in natural development. We may argue about what that natural development is in the altered circumstances, but the question has some solid content. On the other hand, when subjunctives get away from this sort of question, when they cannot be tied down to a point in this world, their truth become pretty airy speculation. We can deal with "What if Socrates had escaped from prison?" pretty well, because we have a vast amount of background information about the people and society involved. But "What if Socrates were an 18th century Irish washerwoman?" leaves us with nothing to hang on to: there are twenty-two hundred years of history without Socrates to deal with, for one thing, so Lord knows what Ireland may be like (if it is at all) or the status of washerwomen or.... Of course, the question might involve a rebirth in the regular history as of now -- one with Socrates from the fourth century BCE on -- and with the notion that being Socrates, in whatever guise, means have a certain character and set of propensities ("every name is a disguised description"). In any case, it is easy to see why subjunctive modalities are the least developed of the lot.
It has to be admitted that, even for the thoroughly developed alethic modalities, satisfactory development does not extend much beyond the propositional level. That is, modal sentences that involve names or quantifiers or descriptions are not settled. While there are a number of systems, each of them poses serious problems: some are incomplete, some are not proven consistent, most give counterintuitive results at some point, and the most nearly successful ones break most sharply with what has gone before in one-world-at-a-time logic.
The focus of the problems is things. Can the same thing be in more than one world? If so, must it be called by the same name in all the worlds it is in? Conversely, must what are called by a given name in several different worlds be the same thing? Can there be things in one world that are not in another (or, if you want to insist that nothing can be in more than one world, that are not counterparts of things in another)? Is there just one domain and quantification then restricted different subsets in different worlds, with perhaps different treatments of the remainder as unreal objects in that world? And so on. The upshot of all these questions is that, when we come to natural language modalities and quantifiers or terms, we have little guidance beyond the automatic ones that define intensionality: without further qualifications, terms in the scope of modals cannot be generalized outside, nor terms outside be used to instantiate inside, and identities from one side cannot be used for replacements on the other side.
A set of problems that have not yet been dealt with in these logics, at least not directly are those that have to do with properties. Does a thing in one world have properties in that world that depend upon properties it has in another world? Are dispositional properties real properties or just disguised subjunctives? Are abilities real properties? Does a person carry around his whole history or some large part of it? What about his future (or is this part of the asymmetry of time)? Can an object then be identified just by its property (and so its name be definite descriptions)? Or is part of what makes something a counterpart a shared earlier history ? along a different path?
One extreme here is to say that no dispositional properties are actually had at the moment, that all attributions of dispositional properties are disguised modal sentences of some sort?typically subjunctive. This line has the problem that most properties turn out to be dispositional to some extent, to make claims about what would happen if some unmade test were made. And that, of course, leads to endless regresses, since the upshot of the test is also a dispositional property looking to further tests. The other extreme is the classical one that every dispositional property ("capability" or some such) is rooted in -- and so can be identified with -- some occurrent property -- however hard it may be to locate. Between is the relativist position, which holds that, in a given context, we take some properties as occurrent, relative to which we take others as dispositional, even though, in another context, the roles might be reversed. This last seems to be the way we usually act, but is hard to work into a formal system, so logic has no answers and little in the way of developing alternative theories.
When we turn to Lojban, we discover systems sometimes richer, sometimes poorer, than those which logic offers. Let us consider the various types of modal systems separately: alethic and temporal in some detail, deontic and subjunctive more cursorily.
Lojban does not have an alethic modal system in the usual sense. It has two predicates sarcu “necessary” and cumki “possible”, which take abstract arguments and so can function more or less as operators. Both of them have a place for conditions, making them more like two-place operators -- but not the sort usual in modal logics. sarcu also has a place for a purpose, thus moving away from the open modality toward a narrower one. It is possible, of course, to zero out these additional places or to fill them with tautological arguments. Or to use them to distinguish among the various kinds of alethic modalities noted earlier: logical, physical, technological and the like. Thus, while we do not get a natural parallel to the logical systems – which would expect that the sentences involved were on the first level, not removed into an abstraction – we end up with a potentially richer system in expression. It is, however, one that presents many problems in application; in particular, the role of quantifiers (including names) is difficult to express and deal with.
What Lojban has of the expected sort, operators on the sentences in which they occur, is a system about abilities rather than about possibilities. Abilities are, of course, related to possibilities of some sort, as well as to subjunctive and temporal notions. The Lojban system makes the connection to temporal notions explicit: the lexed members of the system are “actually is, ” “inherently able, ” “can and has,” “can but has not,” and “has not.” As the translations suggest, ability is usually taken to reside in objects, not in propositions and, thus, the connection apparent in the grammar of CAhA, the selma’o of such operators, makes with selbri must be taken as belonging in fact to some sumti, presumably the first. The connection between the Lojban sentence and the logical is thus the reverse of that for the usual modalities: the operators are best understood as logical predicates and the overt sentence is to be taken as one abstracted to an argument of that predicate (or, in most cases, if all the operators are to be treated alike, of several predicates). Alternatively, the system here could be compared to the little developed notion of epistemic modal logic, which has an operator which takes both a term (the knower) and a sentence (the known) and which also corresponds to a Lojban predicate with propositional arguments.
The fact that kakne has a place for conditions (though CAhA does not, of course), again allows it to be used to distinguish a number of different types of abilities. The basic one is given as “is inherently able to,” which is sufficiently vague to be philosophical about. It appears to depend upon categorizations: a blind person is inherently unable to see as blind, but inherently able to as a human being -- though now incidentally unable, and perhaps able again given some technological advance or even a miracle. And so on. All of the logic of these notions is little developed in modern times (though Aristotle’s comments are still pretty reasonable, as are some of the Medieval remarks). It would seem that the Lojban systems, properly understood, are an adequate language for most of what a logic of this sort might develop (presumably within subjunctive logic, but maybe as an easier system on its own). The ka’e format does, however, present the usual problems with negations and distinguishing between not being able to do something and being able not to do something. The predicate forms avoid this easily, and the corresponding quantifier problems as well.
These problems arise as well for the temporal modalities (as, indeed, for everything in the preselbri packet), for which Lojban is otherwise overly well equipped. Lojban has operators corresponding to both the logical P and F, pu and ba respectively. It is less clear that it has G and H, though something of the sort can be fashioned out of roroi, perhaps. What is not clearly possible is the direct representation of the usual definitions of G as ~P~ and the like (though na pu na is grammatical and even brackets correctly – as near as is likely to happen). It is also not clear how to distinguish between VxPFx and PVxFx (one contrast between English past and perfect, for example).
These matters aside, the Lojban tense system is capable of everything else that the logical one does and more beside, and clearly goes beyond that of any natural language. Like the logical language, Lojban can have any number of tense operations in a row, while no natural language gets beyond PFP in a systematic way. In contrast to logic, Lojban has an explicit present tense, where logic makes do with only the absence of any tense marker. As a result, Lojban could distinguish between a present past and a past present, in the way that natural languages often do, but logic cannot. Lojban has, however, not actually used this potential and, indeed, explicitly denies it: pu ca = ca pu = pu officially. This removes one possible solution for some quantifier problems from the board (quantifiers typically go with the axis = present tense marker).
Putting aside quantifiers and negation for the moment, Lojban can directly represent any logical formula in the obvious way, pu for P and ba for F. Surprisingly, G can be represented by pu roroi and H by ba roroi. Any combination of these chunks will apparently pass the parser, though the bracketing will not always be what is intended.
And of course Lojban can do much more. Logic has yet to do anything significant with aspects, for example (though there are some fuzzy logical suggestions on these issues – putting inchoative and initiative, terminative and perfective in the penumbra and umbra of fuzzy values for the central claim). Nor is there much in the way of topological logic, the logic of place and direction, for which Lojban has a considerable – though still improvable – vocabulary (it is still hard to aim artillery in Lojban, where you need simultaneous surface and elevation directions, for example). The fact that, in Lojban, these are treated in the same mold as temporal modalities, suggests one line of development for such logics. Since temporal modalities have already considered circular cases, for example, the peculiarities of the NEWS directional system will not be a shock (except for the poles). The need to deal with three dimensions – at least sometimes – will complicate matters somewhat, especially in the need to work with the intersections of them (Lojban is not too good yet on these – see the gunnery example above or try boxing the compass in Lojban).
Most Lojban tense operators are hidden quantifiers over times (or spaces) and derive much of their character from this fact, which they share with the logic forms. Like the logic forms, they are not really up to the complexities of branching time: F or ba means at some time along some future path and H and ba roroi means at all times along all future paths. The most interesting item from branching time: on each path at some time (what we often mean by future necessity) is not available, even though there are often reasons for thinking that Lojban, like most natural languages, takes time as branching to the future (or, what amounts to the same thing, the future as generally not already fixed). As noted, the logic of this has been little developed, but it would be useful for Lojban to have (as it would be for logic generally). Note this is neither “future necessarily” nor “necessarily future,” both of which reduce to things sayable in the existing system.
The tense system in Lojban is better than most at marking the scope of its operators. Assuming that tenses begin at the beginning of the bridi in which they occur (in order in the pre-selbri pack), they run until explicitly reset or to the end of the context (new topic altogether). There is some room for uncertainty (alias “flexibility”), especially in story time, but the overall picture is remarkably clear. There is the question of what to do with new tenses in the scope (apparently) of other tenses: do they take off from the established time or start anew from the present. Lojban seems to go (as seems wisest) with the first, requiring an explicit nau to get back to the moment of speaking reference point.
Like the regular modal logic, deontic logic is handled in Lojban with brivla rather than CAhA, with all the advantages and problems that entails: placement of quantifiers is easy, dealing with complex situations hard. bilga and curmi, the obvious brivla to use, are not, in fact, a set, since the obligation is abstract, based on a principle, while the permission is personal, granted by someone (possibly a principle, to be sure). bilga has the subject-object split of the English “for… to…” construction, while curmi does not. curmi, on ht other hand, is conditional, while bilga is not (although we could build conditions into the principles, Kant to the contrary notwithstanding). The best developed of deontic logics uses both an obligation and a permission of the split sort (are operators which connect a term and a sentence), presumably requiring the sentence to have the term (or equivalent) as subject (or taking a lambda’d sentence as second argument – not at all common, alas).
It is important to note that the bilga or curmi system is factual; that is, the truth of a sentence involving one of these depends upon whether the indicated event actually is obligatory or permitted, as the case may be. This contrasts with an apparently similar items in UI: e’a and ei. The first of these clearly marks a kind of performance, the granting of permission (or laying down an explicit prohibition with -nai). The second is less clear, as are so many UI, but seems to be again a performance, in this case judging or recommending. In either case, the truth of the sentence in which it is embedded does not depend upon whether the person is permitted or obligated. Indeed, the sentences are non-assertive; that is, don’t have truth-values at all. It is true, however, that after the utterance is made, the designated person does have the permission of the speaker to do what described – whether the speaker has any authority to make that permission viable is another matter. The same is not the case with the recommendation, which, first of all, is clearly conditional – “if you want to achieve such and such a goal then you ought…” and, secondly, is clearly subject to factual constraints. The fact that the speaker judges something to be the best course and therefore recommends it does not mean it actually is the best course. But these questions are beside the point when considering the role of these UI in speech acts.
Subjunctive logic is not very developed and its representation in Lojban is also controversial. It is clearly conditional and just as clearly not material implication (ganai gi). The Lojban operators are pretty surely da’i and da’inai, but the exact constructions for their use are controversial. The operators are UI, essentially grammarless, so there are few constraints – but also few guidelines -- for making a pattern. The safest course seems to be to take a sentence containing da’i anywhere as setting up a subjunctive (typically contrary-to-fact) environment that then runs until da’inai or the end of discussion. The da’id sentence is non-assertive, merely stage setting. The da’inai sentence is assertive about this world. The intervening sentences are assertive about the world(s) in which the da’id sentence is true. This is needlessly complex for the stereotypical subjunctive case, the counterfactual conditional. For this, there is the possibility of starting the conditioning sentence with ida’i, but finding a device to introduce the apodosis is not yet resolved: the usual connectives are all wrong logically and the other likely choices seem wrong functionally (this is not a “therefore” or “and then” connection nor some kind of vocative as “this continues what went before” suggests). iji’u is a real possibility, I suppose, up to grammatical constraints.
Whatever that connection may be, da’i needs to be distinguished from another UI, e’u which marks the performance making a suggestion, a weaker version, perhaps, of ei, though the latter calls for a justification and the former only for discussion. ei is also more obviously action oriented, while e’u might more easily play a role in theoretical discussions, a certain kind of “maybe.” To be sure, the discussion around an e’u might quickly go over into da’i as the results of the suggested situation were examined, but they play separate roles.