logic Language Draft 77.1
- As the number suggests, this page is from way late in the program. It is presented now because parts of the topic are of current interest. It is not a final version, as it may need to be modified by intervening pages -- and discussion.
In logic, the variables stand for individuals, but logic makes no prescriptions about what sorts of things these individuals are. Ordinary languages similarly do not start by defining what an individuals is to be. However, we can – or at least linguists and philosophers have thought they could – extract from the way that users use the language what sorts of things they are talking about: objects, processes, individuals, kinds,... . In most languages, speakers talk of several kinds of things and the task is then to figure out which one(s) the language takes as basic and how talk about the others is derived from talk about these.
Following this (iffy, tenuous) process for Lojban gives something like the following result. The basic items to be talked about are objects: spatio-temporally continuous and delimited things. And, of course, first in order of usage are those of medium size (visible to the naked eye at earth bound ranges). Such objects have parts, which in turn have parts and so on down to subatomic particles and perhaps beyond. They also are parts of larger objects, which are also parts of still greater ones, reaching eventually to the universe (codimensional with space and time). All of the steps along this way are still objects, arrived at later in the developing use of language.
Beyond the simple mereological (part-whole) relations, language – but not logic – recognizes a special part-whole relation: constitutive, unified, or some such term. The difference between a person’s arm and a hydrogen atom in an amino acid molecule in that same person displays how this latter relation differs from simple part-whole. Part of the difference can be summed up in the claim that the careful observer can analyze what the whole does or is in terms of what the constitutive parts are or do. A private saluting involves (or just is) certain characteristic movements of the right arm and hand and fingers. That motion can, in turn, be analyzed into the contraction and expansion of certain muscle fibers, the rotation of various bones at various joints, and, perhaps, even the firing pattern of certain neurons (the level to which this sort of relation extends seems to be constantly expanding downward; I doubt the atom will ever be counted, though the amino acid molecule might).
The other characteristic of this relation is that it continues even with spatial separation. Joe’s arm is still Joe’s arm when lying on a table in the lab while Joe (the rest of him) is several floors above (transplant surgery has muddied this claim considerably: is Joe’s kidney still Joe’s when nestled by Jack’s spine and filtering blood flowing through Jack’s veins?). Atoms at least, and probably an array of higher parts of something, come and go without keeping their connection with that thing. Probably every breath we take contains an oxygen atom that was in Caesar’s last breath, but we contain no part of Caesar.
The constitutive relation goes upward from middle-sized objects as well as downward. But more interestingly, it extends by analogy from natural objects to intentional things, things that are created by “some objects being taken as going together” (we carefully avoid saying who does the taking and in what it consists). Let’s call these things groups, as a convenient term. Basically, the objects involved relate to their group by the membership relation (set-theoretic epsilon or its unsystematized origin), but – mereology and set theory resembling one another so much – a constitutive relation also arises (intransitive like both the mereological form and epsilon itself). That is, the members of a group play the same sort of role as the constitutive parts of an object. Spatial separation is a given here, the members are discrete from one another. But the doing and being of the group is (is analyzed into) the doings and beings of the members -- at least initially. As with parts and natural wholes, what the members do may be quite different from what the group does; yet the action of the whole would not exist without the actions of the members, which account for the group action.
Groups can be nonce, lasting only long enough to account for a single event (itself intentionally defined) or they may endure for a long time, spread out over a great area (with lots of spatial gaps, of course) and even undergo many changes of members while remaining “the same:” the Smith Family, New York Giants, General Motors, AF&AM Lodge 73, the neighborhood and so on. (In retrospect, we see this sort of constitutive relation explaining the problem case in the natural setting: the change is like adoption or picking up a contract: last years Tiger is this years Giant, Nero Ahenobarberus becomes Nero Claudius, Joe’s kidney is now Jack’s.)
Groups not considered constitutively, but simple under epsilon, have less explanatory value. They provide convenient targets for functions, when we would strain having somehow to express clearly how to go around and deal with every member separately. We can count the members of them. We can relate one of them to another: contained, containing, overlapping, or totally separate and count the overlaps if any.
At some point, groups come to be treated by a language as objects – syntactically at least, maybe even semantically and pragmatically. Still, anything that can be said by reference to a group can also be said by reference to the underlying objects alone – in theory; the practice is sometimes unresolvedly difficult. Of course, this becomes even more difficult when the “objects” involved include groups already. Some constitutive groups even come to have names as the examples above show, though the simple sets generally do not (unless you count “the set of all …” as a name). This change is important at least in logic because, even in the guise of objects, some moves that work with basic objects do not with groups – or at least not in the way expected from the results with basic objects.
This sort of problem becomes even more pronounced when a language hypostatizes (takes as an object) the results of other, more clearly explainable, mental processes, like generalization or abstraction or some combinations. If there are 4n As and 3n As that are also Bs then the probability that an A is a B is .75 and so the average A is a B. The first “the” is usually not problematic; the second may be, if we were to think that, because there are 12 (n=3) As and the sum of their heights is 68 feet, the average A is 5’8” tall – as well as being a B. If we don’t count, we may say that As are typically Bs – and so that the typical A is a B. And so on through a variety of generalizations of various sorts that might be hypostatized (“archetypical,” “stereotypical”). Of course, none of the 12 may actually be 5’8” tall and even if one is, it need not be one of the Bs. “The average A” is not an A but a generalization about the A group (the things taken together as that group). Putting it in almost any logical place as an object will give wrong results (see substitution on identity and particular generalization disproved above).
And suppose that C is colored a lot like D. Then we might say successively that they are the same color, that the color of the one is the same as the color of the other and, supposing that they are both bluely colored, that the blue of one is like the blue of the other, that the color of each is blue, that each has blueness. And so on. Again, putting any of these in as an object will result in strange results. For one thing, most of the predicates that make sense for objects will not with these – and conversely. So they are at least different kinds of things (most predicates that make sense of rats don’t of argon gas). But they behave differently in other ways as well.
Objects can be represented in more than one world; each has an object concept that maps each member of some subset of worlds onto a member of the domain of that world. Groups don’t map, though they can be reconstructed in other world if the defining conditions for being taken together still apply to things there (but it may apply to different things, not the representatives of the members here). Of course, averages and the like don’t carry over -- there is surely a world in which the stats on As are very different from this one, in which the average A is not a B nor 5’8” tall. And in such another world, C may be differently colored from D and neither bluely.
Curiously, properties – blueness or blue, for example – do have concepts going with them. Corresponding to each is a function from worlds to subsets of the argument worlds’ domains, the extension of the property in each world. Here we see sets being used as a convenient target for a function, when, of course, what is wanted is not really the set but the members. One might better take the value for each world to be a function that assigns each object in the domain either a “yea” or a “nay” – the characteristic function of “is blue,” for example. This suggests that properties in this sense also deserve to be treated as individuals – or at least special cases – in a language.
But there is also a different sense of “property,” which is more clearly world-limited, not carried over from world to world. This notion takes (sticking with our example) the particular of blue of a particular object (on a particular occasion from a particular angle…) and generalizes on this to all these instances of blue, though exactly how this generalization works is unclear – or perhaps several notions lie behind this description. Some things said about this type sound like a set, others like a constitutive group, and other like a function from objects (with whatever conditions need be added) to hues.
Other notions in this area also seem to have several different types of things in mind. Kinds look sometimes like the constitutive group of all the whatses but at other like just the set and at yet others like the property (in one or another sense). Stuff is similarly obscure, although it most often seems to be the constitutive group of all the parts (down to some vaguely specified level, perhaps) of all the whatses – ladling out of the product of the universal grinder.