Sketcher of various interrelated fourfolds.
July 17, 2007.Recentest (mildly) significant change: January 2, 2009 (third such change since August 6, 2007). This post is much less speculative in style than the others on this blog. But my other blog "feels" filled up, I can't quite say why. Still, maybe I'll eventually move this post to there.
We tend to consider the logical quantity of the term and not only that of the proposition, especially when a logical quantity such as the singular gets involved. Yet tradition has kept the spotlight on propositions (or sentences, etc.) because of the interest in valid argumentation involving them. That seems to be why logical quantity from the term's viewpoint has lain largely unexplored by philosophy. Philosophy hasn't stopped and smelt the roses long enough to see what vistas might spread thence. Given a term "H" predicated (truly or purportively truly) of something (call it "x"), the question of its logical quantity then depends on quantification over the rest of the universe of discourse: Is there something which isn't that thing x and of which the term "H" is also true? -- and -- Is there something which isn't that thing x and of which the term "H" is instead false? The twin questions stand mutually independent and resolve into four answers, conjoinable in four ways (notwithstanding issues of term purport which multiply relevant options). For the polyadic case, incorporate criteria requiring one-to-one correspondences as needed and slackening as needed to compensate for sequence variety. None of the four conjunctions enframes a blind or almost blind window as long as we class the singular and the singulars-in-a-polyad together in logical quantity, just as we class both the monadic general and the polyadic general as general. One such conjunction, the monadic-or-polyadic singular-cum-universal, is a logical quantity corresponding to a gamut, a total population and its parameters, a universe of discourse, etc. The eventual result of a systematic approach to logical quantity from the term's viewpoint is a surrounding scene of various categories of the 'essences' -- attributes/modifications, modes of attributability, and forms of mathematical correspondence -- whereto nonsingular terms are often allied, 'essences' categorially as different each from the others as they are from the scene-completing object -- this man, this horse, etc. -- of a typical concrete singular term.
The "problem of universals" is a philosophical perennial. Now, before one does a metatheory about, say, the theory of geology, one needs first to do theory of geology. And, before that, one needs to do physical geography. The "geography" of logical quantity (singular, general, universal, etc.) seems to have lain largely unexplored by philosophers. Aristotle and C.S. Peirce are exceptions.
On July 17, 2007, I searched on Google for the two phrases problem-of-universals logical-quantity. Only two results came up, both mine -- the first version of this post and a similar thing which I posted to peirce-l some weeks ago. I searched for problem-of-universals logical-quantification and found few results, half of them mine. (My own earlier post on the topic at The Tetrast doesn't come up, and of course the problem of universals isn't always called that by name, but it still seems fair to take the paucity of Google results as significant). The lack of an adequate systematic terminology is another sign of how little attention philosophers have given to the topic of logical quantity, despite their long interest in the problem of universals.
The problem of universals gets its standard name from the noun "universal" in the sense in which one finds it used in translations of Aristotle -- that which is true of more than one object, a sense for which the word "general" is now sometimes employed as a noun in philosophical discussion and is in any case usually so employed here.
Singular and general in standard 1st-order logic
Now, in the standard terminology of first-order logic, a "general term" is a term which does not purport as to logical quantity (or has only a "default" purport to the existential particular affirmative when the term is true of something). If the monadic general term were to purport, when true of an object at all, to denote more than one object, then a proposition claiming in effect that the term were uniquely true of some given object would be formally false. Instead such a proposition is merely contingent. In other words, a so-called general term in standard first-order logic is vague in logical quantity and is 'general' from a kind of second-order viewpoint -- one might call it "general" across various possible logical quantities. On the other hand, a "singular term" in standard first-order logic is a term (and indeed a subject term rather than a predicate term) which does purport as to logical quantity, and purports to singularity, so that a proposition which claims in effect that a monadic singular term corresponds to two different objects is formally false. I am speaking of constant a.k.a. definite terms such as "blue" and "Jack." Constancy versus variability is a similar yet distinct issue or dimension which complicates an elementary discussion.
Generality more generally
In speaking philosophically of generality, not adhering to the linguistic habits of standard first-order logic, we may mean neither vagueness in logical quantity nor a purportive (or still some other de jure) generality; instead we may mean a de facto generality, for instance that of a monadic term like "blue" which happens to be true of more than one object. In speaking of singularity we may likewise mean a de facto singularity. I'm not sure what there is for all this, except to get used to the distinction between purportive and other sorts of de jure, and de facto. It seems difficult to limit one's discussion to examples of just one kind or just the other. The distinction does not seem so hard and fast to intuition. "Blue" -- as term or as idea or as quality -- is the kind of thing which one would not expect to be true of just one single object.
In order to distinguish the sense of "general" as that which corresponds to more than one object (in the monadic case), I will speak of the coaliant general. (I could just call it the "coaliant" per se but I wish it remembered that I'm speaking of a kind of general. I coin it from co- + aliud + -ant.) The coaliant general corresponds, purportively, etc., or de facto, to something but not to that thing alone but also to something else. In the polyadic case, consider it to correspond to polyads whose intersections lack objects from each polyad. (As for re-orderings or re-sequencings of the same polyad, they are another issue which complicates an elementary discussion.)
A bustling floor under generality
Since one thinks in terms of greater and lesser generality, there arises an imagery of limits. Such imagery is itself limited in usefulness but inevitable in its way.
Now, the coaliant general (monadic or polyadic) encounters something like a limit, closure, or bound, at the "low" end, in the singular or singulars in a polyad. A polyadic version of a singular is not strictly to be called "singular" in that it is not monadic, and "plural" already has specialized meanings in logic. One might say only loosely that it is a polyadic singular. The word "singular" isn't quite right for a logical quantity definable by its opposition to the general -- the mind places "singular" opposite not only to "general" but also to "plural" and thus also to "polyadic." In order to unglue term adicity a.k.a. term valence from logical quantity and instead to treat all logical quantities on the same plane, I'll call any monadic-or-polyadic singular transingular. The coaliant general encounters an excluded or external limit, at the "low" end, in the transingular. A transingular term can be a subject but also can be a predicate or other things.
A bustling ceiling into generality
The coaliant general, if it has an upper limit in some sort of "most general," will include it in a way that it does not include the transingular, since the coaliant general is definable as the determinately non-transingular. What would non-arbitrary utmost generals be? They would be something like the Scholastic transcendentals (unity, truth, goodness) which are true of each and every thing automatically, in sheer virtue of the thing's existing at all -- the given thing is one thing, a true thing, and a good thing, at least in respect of its existence if not of its character. That seems to make of the utmost general a rather narrow window, while other logical quantities at the same level of analysis are rich and, in their way, panoramic. Is the world's symmetry really that deeply broken? A systematic understanding of logical quantity does not foster a view of the world as arranged mainly into genus-species type relations, strict inclusions, etc., with one or a few utmost generals monotonous at the top. Confronted with the Scholastic transcendentals one, true, good, one may ask, what about two things? Aren't any two things two in sheer virtue of their being things xy such that ~(x=y)? Now, if one views collections in such a way as to see othernesses and unities among selected parts as definitive attributes of the whole, then, since obviously not every such collection consists of exactly two things or of exactly one thing or of etc., in that sense such numberish predicates are not utmost generals. However, any object (in a large enough universe) will fairly belong among polyadized objects whereof "two" is true collectively. Keeping this in mind, we have a notion of universality reached by utmost generality, universality which can be extended to sequence schemata, etc., and which seems, as a "window," practical and cornucopious like the singular or transingular. If we "arbitrarily" declare a given predicate term universal, equivalent to a predicate like in "Tx" or in "Hxyz v ~Hxyz", it can be refined by formal schemata. (Via a richer formalism such as set theory or the like, mathematics can treat these universals as more or less general and even unique properties of various sets or the like, and mathematics can re-generate the world's wild variegation, while building imaginative, metamorphosic bridges of equivalences across the greatest disparities of outward appearance.) The point is that the 'accidents' or 'modifications' of the objects xyz in the above example don't matter. All that can matter is their othernesses and unities, relationships defined within the formalism (of first-order logic with equality a.k.a. ...with identity). On the other hand, with things like "blue," we're getting into modifications of objects. Such terms or ideas or qualities as "blue" and "Jack" befit (at least in a realistic universe where not everything is blue or Jack) that which I will call the special, or contraliant special to ensure clarity as to just what sense of the polysemic word "special" I mean. ("Contraliant" from contra- + aliud + -ant). The contraliant special term is (or purports to be) true of something (or things in a polyad) but decidedly not of everything.
Yet the universal can be either transingular (as in the case of a total population, its parameters, etc.) or (coaliant) general (or indeterminate about that alternative in the case of a term's de jure applicability). So the universal is better pictured as a ceiling into generality than as a ceiling in generality.
The universal supplies the upper limit of the coaliant general, and is a kind of extreme to which the coaliant general reaches, like a line segment which includes its endpoint adjoinment with something else (a universal may be general or instead transingular). In the other direction, generality's "line segment" includes everything till the transingular but not the transingular itself, like when a mathematician replaces an endpoint with a little bubble. A coaliant general is either universal or contraliant special (or indeterminate about that alternative in the case of a term's purport, its de jure applicability, or the like).
- The (coaliant) general has two limits -- an excluded limit, the transingular, and a partly included limit, the universal (a universal is not necessarily general).
- The (contraliant) special has two limits -- a partly included limit, the transingular (a transingular is not necessarily special), and an excluded limit, the universal.
- Should the general-cum-special be considered a fully included "limit" of both the (coaliant) general and the (contraliant) special? Here we seem to approach a limit to the usefulness of the imagery of limits.
A transingular may be universal too. If the transingular is a total population, a universe, a gamut, then it is also universal, at least in the relevant universe of discourse. When it is not the universe, the transingular is (contraliant) special. (In the case of term purport, the transingular may be indeterminate about that alternative.)
Universals & universes
Basically one ends up with two kinds of (coaliant) general and two kinds of universal. Now, in the universe of a plinker's distinct notes cdefgab, that gamut is the universe. It is both unique and universal. In its universe of discourse there's no polyad that contains notes uncontained in the gamut. "The gamut" is true of cdefgab and there's nothing else of which "the gamut" is true. In that sense it is not general. Yet it is universal, it is the universe and, in that sense, it is not (contraliant) special. A gamut, a universe of discourse, a total population is a transingular universal. Also universal is a monadic or polyadic term which does not exhaust the universe's population in a single predication yet which, like "one," is true of each object distributively or which, like "two," is such that every object is among some objects whereof the term is true collectively. Such a universal is also general, since there is more than one instantiation of it in its universe. One the other hand, "THE one" and "THE two," etc., are not general, insofar as they are true of the one object in a one-object universe, the two objects in a two-object universe, etc., respectively.
So we have two kinds of universal, one a transingular and the other a (coaliant) general. A universal which does not exhaust its universe in a single predication is (coaliant) general, not transingular, and is closer to the kind of thing which one usually has in mind with the word "universal," something like a rule, with more instances than the given one, indeed sometimes infinitely more, as with the "miraculous jar" of positive integers.
To be in the world
A transingular which does not exhaust its universe in a single predication is much closer (than the universal transingular) to the sort of thing which one usually has in mind with the word "singular," a singular or singulars-in-polyad among still more singulars in a larger world. Such a transingular is not its universe, it is not universal. It is (contraliant) special.
So a transingular may be universal or special. Likewise, a general may be universal or special. Just because a term is general, having more than one instantiation, doesn't mean that every object is covered one way or another in its instantiations. For instance, "blue" is, eclectically, true of some things and false of the others. So now we have four comparatively simple logical quantities -- universal, (coaliant) general, (contraliant) special, and transingular -- and four conjunctions nameless except for such improvised unwieldy names as "universal-cum-general," "universal-cum-transingular," "special-cum-general," and "special-cum-transingular."
To be systematic
Any pair of statements are TT, FF, FT, or TF. We define logical binary compounds in that way. Formal logic wouldn't even think of not systematizing the four mutually exclusive and collectively exhaustive cases -- the four conjunctions based on truth conditions. And we get "and," "neither-nor," "no, but," and "and not."
In the same inevitable way, any term true of something is, de facto: -- (1) universal & (coaliant) general -- or (2) universal & transingular -- or (3) (contraliant) special & (coaliant) general -- or (4) (contraliant) special & transingular. All that's being done is to answer two mutually independent logical-quantity questions, which bring us --
To the heart of it
In the monadic case, the two logical-quantity questions are:
"Given that there's a thing (call it 'x') which is H, is there a thing (call it 'y') which isn't that thing x and which also is H?" If yes, then "H" is (coaliant) general. If no, then "H" is transingular.
"Given that there's a thing (call it 'x') which is H, is there a thing (call it 'y') which isn't that thing x and which is not H?" If yes, then "H" is (contraliant) special. If no, then "H" is universal.
The mutual independence of the twin questions needs to be appreciated; they result in four possible conjunctions. The result is not simply two separate extremes of universal and singular with the somewhat-general somewhat-special as a third, in between. The habitual swerve of thinking of the singular only in monadic terms even while thinking of all three of its kindred logical quantities (special, general, and universal) in both monadic and polyadic terms, leads to thinking incorrectly of the universal singular as a trivial combination (if one notices it at all), a nearly blind window, confined to a one-object universe. In fact the window's vista is quite populous. A grand boat gets missed there, that of a logical quantity corresponding to a gamut, a total population and its parameters, etc., along with a whole class of research, research starting from given parameters of a total population, universe of discourse, etc., to draw deductive conclusions.
(There are even more than four options for term purport, de jure applicability, or the like, 16 including the formally false option, mostly since indeterminateness becomes an option in various alternatives. Such options for de jure applicability seem to become 2^16=65,536 if we admit options for objective indeterminateness and an option for objective inconsistency.)
Now, in a large enough universe, the general-cum-special will be mostly vague in range. In the monadic case it could be true of just two things or it could be true of all but one thing or it could be anywhere in between. It is so much like logic's "general term" as to be barely distinguishable except under certain near-the-limit conditions. For similar reasons, one might question at least the utility of some of the other combinations. One might say, instead of column A, why not column B?:
|General-cum-special||Logic's "general," logical-quantitatively indeterminate like the predicate term letters in logical schemata.|
|Transingular-cum-special||"Just plain" Transingular (be it universal or (contraliant) special).|
|General-cum-universal||"Just plain" Universal (be it transingular or (coaliant) general).|
|Transingular-cum-universal||Transingular-cum-universal (a universe, total population, gamut).|
Now, if we're defining kinds of terms by purportive logical quantity for the purpose of a formalism or grammar, then Column B seems the more convenient way to go. However, Column A is logically "nicer" and more consistent in its criteria; its four logical quantities are on a par with each other. In any case Column A girt by the simple logical quantities as shown in A.1 is the completed relevant picture (almost completed -- one could also devise terms for the diagonals). And if one is interested in logical quantities as characterizing typical mental perspectives distinguishing classes of research, Column A is the way to go, and even a pair of terms for A.1's diagonals would be useful. Now, I speak of the perspective as represented by the given subject matter, not the object(ive) or goal which, for instance in the special sciences, may include finding generals true of multitudes of singular objects and events.
|Perspective in typical |
|Class of research:||Typical inferential character |
|Transingular-cum-special.||The special sciences a.k.a. idioscopy. Human/social, biological, material, physical.||Surmise (ampliative-cum-precisive).|
|General-cum-special.||Sciences of positive phenomena in general, rather than of special classes.|
Philosophy, cybernetic theory*, statistics, and inverse-optimization theory.
|Strictly ampliative induction.**|
|Transingular-cum-universal.||Deductive math theories of logic, information***, probability, and optimization.||Strict (precisive) deduction.|
|General-cum-universal.||'Pure' mathematics. Ordering, calculation, enumeration/measure, graphing/topology.||"Reversible" deduction.****|
|Kinds of |
or in related posts.
don't formally imply
|Strictly ampliative |
don't formally imply
** That's notwithstanding the internal properties of the 'domain-independent' deductive formalisms with which these fields sometimes occupy themselves.
*** Deductive mathematical theory of information considerably overlaps into 'pure' math, abstract algebra in particular, because of the pure-mathematically deep treatment of laws of information, laws which also turned out to be equivalent to some principles of group theory.
**** In mathematical induction, the minimal case and the heredity, conjoined, are equivalent to the conclusion, given the well-orderedness of the relevant set. The proof of the minimal case or of the heredity is sometimes not reversibly deductive, especially when inequalities or greater-than or less-than statements get involved. More generally, pure maths are rife with inference through equivalences and equipollencies.
Update August 6, 2007: Am I analytic?
Thank you to Enigmania for including me in the 51st Philosophers' Carnival. In answer to his implied question: Well, I don't take the analytic linguistic turn, and I went through a Merleau-Ponty phase, but I like C.S. Peirce more and don't regard science as sinister to some great extent that would distinguish science from the humanities. Indeed, as "Enigman" says, my stuff "seems to be more analytic" than Continental, "but who can say?" and this is also partly because I'm an insufficiently disciplined amateur, not a professional philosopher. If wishes were horses, and so forth. To date, I've engaged in discussion mainly with Peirceans (at peirce-l), which has been good for me and, I hope, not bad for them. I've read some of the important early papers in analytic philosophy and some books by Quine, but I haven't engaged in discussions with analytic philosophers, so I've lacked the benefit of criticism from them. I don't know how to rectify that but, if I'm lucky, the Philosophers' Carnival will help.
• I regard philosophy's best bet to be to define itself (A) as having, as its subject matter, positive phenomena in general in their inferential issues, and (B) as properly tending to draw, as its conclusions, inductive generalizations to or toward totalities -- all in all, sort of like statistical theory, but tackling the inductive inverse of the problem of deductive theory of logic rather than of probability, and thus lacking the quantitative-measurement emphasis and having multiplicity of levels, reflexivity, and so on, pursuing problems of estimating, interpolating, extrapolating the logical structure of a universe rather than the parameters of a total population, and rising to consider general processes of experience, mind, heart, society, etc., and complex inference processes including all mathematical and scientific research, to say the least. (Note: The kinship between statistical theory and philosophy isn't very close -- they're still far apart like, say, matter science and human/social studies.)
• I certainly don't oppose deductive formalisms (not to mention deductive arguments) in philosophy, any more than a statistician opposes probability formalisms. Statistics' normal curve of distribution is a way of looking at Pascal's Triangle extended indefinitely. A piece of logical formalism transits the heart of the ideas in this post.
• Still, recognition of its underlying kinship with inductive, totality-targeting fields like statistical theory could help philosophy manage and temper its own aspirations to a "God's eye view" (pace Rorty, who, complaining of its aspirations, essentially gave up on philosophy), help philosophy reduce attendant hyperbole and disillusionment, and help it be more pragmatic about vagueness, discriminate in hyperbolic doubt, fallibilistic, etc., without tending to substitute some idea of utility (not to mention power) in place of the idea of truth be it ever so slippery. My 2¢ worth. End of August 6, 2007 update (Edited, January 2, 2009).
A few informal assertions about the problem of universals.
Areas of research can be ordered according to their appeals to principles of how we know things (ordo cognoscendi, the order of learning or familiarity) and, in pretty much reverse order, to principles (entities, laws, etc.) whereby we explain things (ordo essendi, the order of being). The order of being is often preferred in the special sciences (physics first, etc.), while the order of learning and of the verificatory bases on which we know things is sometimes preferred in maths (where such preference tends to put logic and order theory first). Maybe those researches which I call "sequenced in the order of being" you would call "sequenced in the order of abstractness." Still could well be the same ordering. I'm not saying that the ontological questions are unimportant, to the intellectual climate, the human spirit, and the ultimate bearings which people take in their decisions. But for my part I generally take their involvement in questions of math and science classification as an intrusion signifying that the classification is either deficient in firm and fertile constraints or just plain nebulous. And, if people argue over whether some sciences should be ordered by increasing concreteness or increasing abstractness, and if it's essentially the same ordering forwards versus backwards,
On various topics I prefer compatibility with a range of ontological viewpoints, but I do I have my own ontological views. Generally, when people deny the reality or ontological legitimacy of generals in any usual sense, I don't know what to think but that they regard Scholastic Realism as "secretly" believing that generalities like redness and threeness exist like lamps and chairs. As if we might expect to hear a news bulletin, "Blueness, as such, has been finally been found, orbiting a house in New Orleans." Now, if "blue" is not itself a real individual object like a blue thing, still the real individual object is really blue. So blue has really-ness. But that extrapolates to coming up with syntactically complicated words for variations of "real" and you know that sooner or later we'll find some general word for them all. I foreshorten the process and take that word to be the word "real" itself and will merrily consider in what senses and what universes Santa Claus, Planet Pluto, and Cthulhu are real. Sure, some things are "realer" than others. Indeed even with reality we can admit graduality, etc., if we don't try to live always in the armor of a flat first-order logical universe, as interesting a challenge as that can sometimes be, and as needful as it may be for those whose sense of reality is unfortunately shaky. Coarse is what it is, like that browser Safari which should instead be called Tour by Tank. Anyway, Peirce's definition of the real as that which is what it is, and indeed in some sense persists, independently of that which you or I or any finite community thinks of it and which would be discovered by research adequately prolonged, suffices for a definition of "real" which takes things like blue in and is a critically unfolded version of the common-sense interpretation of the word "real." Now, if somebody, Quine or Stuart Rankin or whoever wants to come along and define "real" as "singular object" or as "Scottish" or as whatever, they can do that, but only the Peircean kind of definition has earned the force and feeling of the everyday word "real" which everybody in the discussion prizes. I certainly don't know what would be a "naturalistic solution" to generals and mathematicals and I see no germane practical significance in the idea.
The transingular subject is a this, or a this, this, that, yon,, etc., and, as a more or less haecceitous rest point or useful stopper to analysis, is also a hook or polyad of hooks on which, to borrow Peirce's phrase, to hang the hat of a predicate, it is a point of general indetermination and freedom regarding how the predicate relates to components or sections or durations (and so forth) of the singular subject(s). For instance, it is left to the definition, context, etc., of the predicate "blue" whether "something blue" means something entirely blue or mostly blue, etc.; one is not automatically forced to quantify over parts or stuff of the described subject. Many a natural thing, through such characteristics as forcefulness, endurance, vigor, and firmness/integrity, lends itself to treatment as a singular. As Peirce argued persistently, some things impose themselves on us, whether we like it or not. The haecceitous thing may come crashing in through a hundred windows. And things could not be alike in their bare singularness -- they could not all be singulars -- but for generality. And the general would not be general but for ranging over more than one thing.
The singular seems just as mysterious as the general to me, and neither one of them makes sense without the other. I can't see anything in the limitation of the real to the singular but a kind of fetish arising from the fight against the unmoored generalities so involved with causing chaos and destruction to people and society.
To go on being systematic
Also, to be concerned with the singular and the general and not also with the universal and the special seems unsystematic, unthoroughgoing, and illogical to me. The possibilities of a term's being true or false of objects besides that of which the term is predicated in the given instance don't play such favorites.
There's plenty in all that to examine philosophically. As the transingular-cum-special term lends itself to use as a subject term, and as the general-cum-special term lends itself to use as a predicate term, so a transingular-cum-universal term lends itself to adaptation as a predicate-formative functor such as "with a probability of 75%," and a general-cum-universal term lends itself to adaptation as a subject-formative functor such as "double of". There is a parallelism which runs among logical quantity, grammatical form, and philosophical category such as substance, attribute/modification, mode of attributability (modalities and "indeed," "not," "if," "novelly," "probably," "feasibly," "optimally," etc.), and correspondences/variances (such as "another than," the combinatory "Inv," "double of," "product of," "antiderivative of," etc.) The parallelisms, as non-binding affinities, seem to help empower thought.
|Logical Quantity:||Grammatical Form:||Philosophical Category:|
|Transingular-cum-universal.||Predicate-formative functor.||Mode of attributability.|
|General-cum-universal.||Subject-formative functor.||Mathematical correspondence/variance.|
Whatever one thinks of the problem of universals, still for inquiry on the problem of universals to get off on the right foot, it's a good idea to develop more than a nodding, dozing acquaintance with logical quantity. For really what there is is not simply a problem of universals but instead, from the start, a systematic complex of issues of the (comparatively) simple logical quantities and their conjunctions.
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