The Tetrast
Sketcher of various interrelated fourfolds.

Logical quantity & research scopes - universal, general, special, particular, individual, singular

November 18, 2013.

Latest significant edit: November 21, 2015.

Some logical quantities, such as the general and the singular, pertain mainly to terms or their objects, rather than mainly to propositions, and are the occasion for the perennial argument between nominalists and realists over the problem of universals. My treatment of such logical quantities differs from tradition. I've discussed these matters in older posts. Here I will adopt some different terminology (and I've changed it again since writing this post). I will discuss: uniformity for logical quantities; definitions; the conjunctive compounds and correlated areas of research; philosophical tradition; the arts and still other areas of knowledge; and C. S. Peirce.

Uniformity for logical quantities

Now, a singular is usually taken as monadic, that is, Socrates is singular, but the dyadic Socrates, Aristotle are taken as singular separately, not also polyadically in the sense of singular, singular, polyadically such that one might want to call them "multi-singular" or some such (even if the polyad is just Socrates, Socrates). So, with some etymological defensibility, I coin the word "ingular" to refer alike to the monadic singular and to singulars taken polyadically. Unlike the idea of the singular, the idea of the ingular is on the same footing as the idea of the general in having and evoking polyadic versions as well as monadic versions. (An example of a polyadic general is the two-place (i.e., dyadic) general "__discussing__.") Such sameness of footing is desirable when one seeks to be systematic, and this pays off for example in the case of the consideration of a universe or total population and its more-or-less collective description (its parameter set) as ingulars that are also fully universal (in that universe of discourse). Traditional singularity conjoined with such full universality evokes merely a one-object universe, but that conjoint vista seems mostly barren only because the window is narrowed to the monadicity of the traditional singular. Broaden the singular into the less-restrictive ingular, and said conjoint vista broadens itself into at least a rudimentary version of the populous subject matter proper to deductive mathematics of optimization, of probability, of information, and of logic.


It seems best for the elementary definitions to be in a monadic and de facto perspective where formal considerations to accommodate polyads don't come into play (although I'll still use the term "ingular" which applies to monads as well as polyads). Additionally, I coin the word "omnial" to take the place of the informal and potentially misleading phrase "fully universal."

Suppose that there is something glad.
Question A: Is there something else glad? If yes, then I call 'glad' general. If no, then I call 'glad' ingular.
Question B: Is there also something non-glad? If yes, then I call 'glad' special. If no, then I call 'glad' omnial.

The above supposition that there is something glad (or whatever) is not axiomatic but merely a hypothetical condition, so such a thing as is glad (or whatever) may completely lack instances and thus be both a de facto non-general and a de facto non-ingular, even though the general and the ingular seem each the other's negative. The same goes for the special and the omnial.

Questions A and B may seem excessively simple. For example, one thinks of the general term not just as actually true of something else, but as potentially or purportively true of something else, indeed of various other things, perhaps indefinitely many. Still, it seems best to keep the elementary definitions crude but refinable to suit the occasion, and to remember them in that light.

Now, Question A ("Is there something else glad?") and Question B ("Is there also something non-glad?") do not depend on each other at all. The four answers can be conjoined without contradiction in four ways.

Simple & conjoined positive logical quantities for terms or objects
General: 1. General-cum-omnial.
Simple example: two (things among many).
3. General-cum-special.
Examples in practice: blue, resilient, melodious, etc.
Ingular (monadic singular, polyadized singulars, etc.):2. Ingular-cum-omnial.
Gamut, universe of discourse, total population, its parameters.
4. Ingular-cum-special. Monadic, polyadized, etc., singular(s) in a larger world.

The ingular-cum-omnial, when monadic, is the object in a single-object universe, but is polyadic for a larger universe and can be much less boring then.

The conjunctive compounds and correlated areas of research

Now, I have coined some further terms for brevity. I hope that I haven't erred in making trade-offs between conventionality and evocativeness of the coinages.

1. The general-cum-omnial, or etceteral.

The general-cum-omnial is that omnial which, given a monadic or polyadic instance, is also instantiated by further monads or polyads that don't share all the same members. Roughly speaking, it's the fully-universal that is not the whole universe at once. Consider in a first-order logical sense the idea of two such that "two" is true collectively of any x⁠y such that x is not y, and consider the case where, besides x⁠y, there are also z, w,..., etc., that are not x or y and are distinct from one another. Now, this predicate version of number won't get us to Peano arithmetic but, analogously, first-order logic's singular predicate or subject doesn't get us to empirical science; I'm just trying to treat the various logical quantities on the same common elementary level. The term "two" will be true of everything in that universe, each thing not monadically but instead in some combination or other, and indeed in every dyad of distinct things and in every polyad of just a one and an other, be they mentioned soever many times under soever many designations (unless it is mentions or designations that are being counted as objects themselves). It does not depend on particular positive qualities or characters of things, or on distributions of such qualities, and it does not depend on the positive thisness or haecceities of things (e.g., it doesn't matter if one is talking about Socrates and Aristotle, only that one is talking about two distinct objects). For any given non-zero whole number, this works in any sufficiently large universe. The perspective is that of two things (or three things, etc.) such that there is still another two of things (or three of things, etc.), or indeed indefinitely many twos, threes, etc. It's an idea of universality combined with an idea of further instances, even unto "infinity, or the miraculous jar of mathematics." Hence my coinage "etceteral." Particularly natural expressions of the general-cum-omnial, a.k.a. the etceteral, are mathematical operations and (lambda) functions, as well as mathematical one-to-many relations and many-to-many relations. Now, one often thinks of The Number Two, etc., as abstract singulars rather than as etceterals. One may think of it as a collection of two units or as the class of all sets of two elements, or in some other way (not my expertise). By abstraction and imaginational machineries such as set theory one revives the logical-quantitative variegation of overall experience, and gets numbers like Zero besides.

2. The ingular-cum-omnial, or solipsular.

The ingular-cum-omnial is the logical quantity of a single object in, and only in, a one-object universe. A more populous case would be that of the universe of a plinker's notes c⁠d⁠e⁠f⁠g⁠a⁠b, with an idea of ceteris paribus, "the rest staying the same," the rest coarse-grained out, summed over, to the extreme of ceteris non existentibus, i.e., the rest (of one's world) not existing. (I got this idea of total populations and universes of discourse as just various ways of coarse-graining the same grand world from somewhere in The Quark and the Jaguar by Murray Gell-Mann. The ingular-cum-omnial generally seems to me the logical quantity with a tinge of The Twilight Zone — which in turn is a reminder that a universe of discourse can't be a mere coarse-graining of the real if that universe harbors fictional elements.) So this is the logical quantity for a total population's or universe-of-discourse's members taken polyadically and more-or-less collectively, and quite collectively when one considers probabilities. Hence my coinage "solipsular." One could consider the ingular-cum-omnial, a.k.a. the solipsular, that specifies sequence, the kind that does not (e.g., "The Three" in a three-object universe), and mixed cases. A frequential distribution (such as '30% of the total population') of a characteristic across a total population is a solipsular. Natural expressions of the solipsular include information as a quantity, probabilities, and feasibility and optimality as mathematically studied. An abstract total population's members will not usually be spelt out with singular constant designations, but, if spelt out at all, then with singular dummy letters perhaps regarded as singular veiled constants, or with variables; one could have an ingular term that looks monadic but is to be construed as polyadic, but probably people prefer to express these things with sets. The solipsular is a total population and also parameters of attribution or distribution of characters, said parameters as belonging to the total population. It lends itself to abstraction and formalization such that the particular qualities distributed do not matter, but only their samenesses and differences, as well as the samenesses and distinctnesses among individuals, and not their 'thisness' or haecceities, their individual "identities" in the sense of everyday English.

3. The general-cum-special, or transcernal.

This is the logical quantity most natural for monadic & polyadic positive qualities and characters of things, which are such that we expect further and even indefinitely many instances and also some and even indefinitely many counter-instances. Hence my coinage "transcernal," evoking an idea of sifting through. This is the perspective of inductive fields like inverse optimization, statistics, information theory's inductive areas, and (I think) philosophy, concerned with positive phenomena in general but not, except in applications, with individual positive phenomena in their thisness or haecceities.

4. The ingular-cum-special, or obstular.

This is the logical quantity most natural for concrete individuals taken monadically and polyadically, but not as a total population in the abstract with ceteris non existentibus. This is the usual sense about individuals and singulars — that they're not only individual or singular, but also not absolutely alone, as if each one or each handful were a universe unto itself. Instead they're individuals in a larger world. Hence my coinage "obstular." It's the perspective of sciences and studies of concrete phenomena, what C. S. Peirce called "idioscopy, or the special sciences." Aristotle said that there is no epistêmê (often translated as "science") of the individual, but by epistêmê he meant something deductive, or nearly so, and not including concrete experimentation. The subject matter of idioscopy is concrete individuals in the larger concrete world, but the objective is to learn about their individual connections and tapestries, their positive qualities and characters, their parameters and laws, and their applicable mathematics.

Simple & conjoined positive logical quantities for terms or objects.
Researches correlated thereto by scope.
General:1. General-cum-omnial, i.e.:
Subject matter of pure mathematics — fields for deductive conclusions typically through equivalences & equipollencies.
3. General-cum-special, i.e.:
Subject matter of studies of positive phenomena in general: inverse optimization, statistics, information theory's inductive areas, and (I think) philosophy — fields for inductive conclusions from parts or samples to larger wholes.
Ingular (monadic singular, polyadized singulars, etc.):2. Ingular-cum-omnial, i.e.:
Subject matter of so-called "applied" yet abstract and significant maths drawing from ideas of total populations, universes of discourse, etc.: deductive mathematics of optimization, of probability, of information, and of logic — fields for deductive conclusions typically "forward-only," from wholes to parts or particular cases.
4. Ingular-cum-special, i.e.:
Subject matter of sciences & studies of concrete phenomena: sciences & studies of forces & motion, matter, life, and mind — fields for (soever cogent) hypothetical conclusions.

In discussing pure mathematics, I said, "By abstraction and imaginational machineries such as set theory one revives the logical-quantitative variegation of overall experience...." I'll go out on a limb here (I'm no mathematician) to give examples, not of how, for example, some mathematical ideas are, in their way, obviously more general than others, but of how logical-quantitative properties in mathematics are associable with systematic purposes analogous to those outside mathematics. Ordinals are like singular obstulars for systematically capturing, if not concrete thisness, still a kind of "whichness." Think of series and summability, theory of limits, structures of order, conditions for mathematical induction. A function's derivative, and an arithmetical calculation's result, are like transcernals for systematically classifying together the various functions or various sets of numbers or letters that result in them. Combinatorial enumeration and mathematical integration are like solipsulars for systematically analyzing a number or an area into its constituents. Topological forms and graph-theoretical graphs are like etceterals for systematically determining traversals and transformations. Well, that last assertion is a bit too vague, but I hope to improve it in time.

Philosophical tradition

(The sections on tradition were originally near this post's start but I reorganized the post so as better to 'cut to the chase'.)

Of the logical quantities exhibited by things as represented by terms, only two sorts have been regarded as noteworthy by most philosophers, and the two have gone under at least two pairs of labels: universals & particulars; and generals & singulars. (C. S. Peirce emphasized three such logical quantities.) By "a universal" or "a general" is usually meant, by philosophers, a thing of which there are AT LEAST TWO INSTANCES (actually or, for some philosophers, at least potentially), and often indefinitely many instances. By "a singular" is meant a term that has or is defined to have just one object, or a thing that has accidentally or oftener intrinsically JUST ONE INSTANCE (if any at all). Sometimes such a thing is called "an individual" or "a particular" (for example by E. J. Lowe, who classes as particulars not only individual substantial objects but also individual monadic and relational property-instances, a.k.a. tropes).

Despite perennial philosophical attention to the problem of universals — the question, disputed between realists and nominalists, of whether universals (a.k.a. generals) are real or merely verbal ("nominal") — philosophers have mostly ignored the structure of such logical quantities. The terminology is threadbare.

The tradition's edge and beyond

The idea of something universal to simply everything is involved in the idea of being itself, also in such tautologous ideas as known-or-unknown, and in the Scholastic idea of the transcendentals of being (unity, truth, goodness). The Aristotelian categories are sometimes regarded as summa genera, highest genera.

Now, a quality such as blue is typically regarded as a universal but not as being fully universal, universal to everything; one wouldn't expect any positive quality to belong to everything. Rudimentary ideas of one and two do seem fully universal to everything monadically or polyadically, in the sense that anything x is one, and any x⁠y such that x is not y are two, and so on. Of course "three" is not true of two things per se but it is true of them in combination with any still other thing. I think that that is a viable idea of reasonably full universality, even if it is not the fullest imaginable universality, and that it is more fruitful in that, unlike the fullest imaginable, it is populous, indeed infinitely so, with non-equivalent examples. Such ideas as two and three do not depend on things' positive qualities, much less on things' being any individual this such as Socrates or Bucephalus, but only on their selfsamenesses and distinctnesses, which are abstractibles that pour themselves into formalization in ways that individual and qualitative positive phenomena do not. So one has notions of the (reasonably) fully universal and of the not-fully universal but special like blue and Socrates.

What about the affective arts (those of music, dance, sculpture, drawing, painting, language, story, theater, cinema, etc.), and still other kinds of knowledge?

Here I seem to have used up the logical quantities as perspectives, scopes of subject matter, just to map, so to speak, the main classes of more-or-less theoretical research. So maybe one should do likewise for each of the other knowledge disciplines (such as the affective arts), but that seems a daunting task. Still, I'd say that there seems:

A. across theoretically-oriented research, an overall tendency, a kind of overall urge, toward etceteralities where individual cases or facts harbor enhanced lessons.

Maybe the other disciplines of (fallibilistic) knowledge share the overall tendency toward the etceteral, yet vary insofar as they are cognitive disciplines not of cognitive bases but of decisional impetuses, competential means, and affective effects. I am unsure about this. But at any rate, there seem:

B. in the affective arts, some overall tendency toward the solipsular, the selective composing, the selective combining, of things, modes, etc., into worlds in which qualities harbor enhanced values;
C. in the productive arts/sciences (know-how — engineering, etc.), some overall tendency toward transcernalities where domains harbor enhanced reliabilities, norms; and
D. in the so-called ruling arts (design, architecture, community planning, education of personal character, etc.), some overall tendency toward obstularities where regimes (rules and constraints, in some sense) harbor enhanced optima, with the obstulars concretely (multi-)individualized and customized into the larger world or even, in some sense, imposed upon or against the larger world.

But I probably should save such talk for my Speculation Lounge blog.

Let me note a still bigger picture, where questions of correlations to logical quantities may arise. The above are areas with a kind of upper or second-order level on which the prevailing element — not the only element, but the prevailing one — is that of

A. cognitively assessing (i.e., as to lessons) — and (fallibilistic) knowledge (to wit: ruling art is knowing from what impetuses one decides things; know-how is knowing by what means one achieves things; affective art is knowing in what effects one affectively feels things; math, science, etc., is knowing on what bases one knows things; I use "knowing" vaguely here; often it's better to say and think "(cognitively) learning" instead, and "coming to feel" instead of "feeling," etc.);

but the aforementioned bigger picture also includes areas with upper prevailing (though not exclusive) elements of:

B. valuing, appraising — and sensibility, appreciation, devotion (including but not limited to religion);
C. practice — and skill; and
D. struggle / striving — and (volitional / conational) strength, dedication (in conflict, competition, rivalry, dispute).

For a big table, see "A periodic table of aspects of humanity." For associated methods of learning (cognitively and otherwise), see "Methods of active learning by basic faculties" (at The Tetrast2: Speculation Lounge).

Charles Sanders Peirce

The collective STRUCTURE of such logical quantities as the singular and the general has been barely studied in philosophy except, as far as I know, by C. S. Peirce. Yet philosophy perennially pursues the problem of universals, the question of what sort of being or reality belongs or can belong to that which is not a concrete individual object (where, again, 'universal' refers to that which characterizes more than one thing, at least two things, and, in some contexts, possibly indefinitely many things). The terminology has varied: "universals and particulars," "generals and singulars," and so on. Among major philosophers as far as I know, only Peirce has introduced a more-than-two-way distinction, for which I don't know where to send the reader for a brief sketch, so I will supply one here. He made a three-way distinction, a trichotomy, of:
  (1) the vague, the indefinite, such as a quality as contemplated without reaction or reflection,
  (2) the individual, determinate, and
  (3) the general.
Said trichotomy
(A) is based by him in his three respective phenomenological categories:
  (1) Firstness, quality of feeling (more as quality of a sensation than of an affect such as pleasure or pain), essentially monadic,
  (2) Secondness, reaction/resistance, essentially dyadic (individuals, brute facts, etc.), and
  (3) Thirdness, representation/mediation, essentially triadic (rules, habits, norms, dispositions, etc.); 
(B) reflects three traditional affirmative logical quantities for propositions, respectively:
  (1) the existential particular (Some food is good),
  (2) the singular (This food is good), and
  (3) the hypothetical universal (All food is good). This hypotheticality (as in "each thing is, IF food, THEN good") is important in Peirce, since he usually treated Thirdness as involving conditional necessities, conditional rules, etc.

Peirce made a distinction (to which he did not always adhere terminologically):
Singular individuals, or singulars for short, "occupy neither time nor space, but can only be at one point and can only be at one date" (i.e., point-instants).
General individuals, or individuals for short, do occupy time and space and "can only be in one place at one time."
(See "Questions on Reality," 1868.)

Now, Peirce defined the real as the object (the topic or subject matter, not necessarily a concrete thing) of a true proposition (whether actually expressed or not), such that anything real and any truth are what they are irrespectively of the opinions of particular minds and particular communities (fallibilism) and would be discovered by investigation if such investigation were to be pursued sufficiently (cognizabilism, opposition to radical skepticism). Thus he held that there are real generals, the objects of true general propositions. He held that this is a logical presupposition which, in its turn, metaphysics, which he held to be based on logic, not vice versa, fleshes out as robust and nontrivial. Thus Peirce was that which is called a realist in metaphysics. He was quite anti-nominalist. By "actual" on the other hand, Peirce meant the individual, the this, i.e., the concrete individual objects that nominalists take as the only real things. Thus Peirce held that rules, qualities, and individuals can be real, but rules and qualities can't be actual (strictly speaking) since they are not individual things. In particular, Peirce held that indeterminacy is real and that there is spontaneity, absolute chance.

I'll probably add to this post later.

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