Recentest edits: August 5, 2012.

E.J. Lowe, also known as Jonathan Lowe, proposes a quantificationally arbitrary square of ontological categories — Kinds, attributes, objects, and modes (property-particulars, a.k.a. tropes, e.g., this redness). There's a kind of double half-way correlation to my fours.

E.J. Lowe's four-category ontology

	Substances		Properties
Universals (a.k.a. generals)	Kinds	Kinds are characterized by attributes	Attributes
	Kinds are instantiated by objects	Attributes are exemplified   by objects	Attributes are instantiated by modes
Particulars (a.k.a. singulars, individuals)	Objects	Objects are characterized by modes	Modes (property- particulars, a.k.a. tropes, e.g., this redness)

Tetrastic version of logical quantities, categories, & classes of research, at a glance

Conjunctively compounded logical quantity:	Category with affinity for the logical quantity in the first column:	Class of research with corresponding subject matter (objectives may be broader):
Universal-cum-general. • •• ••  •• ••  ••     ••  •• …	Correspondences / variances ("another than," "double of," "inverted order of," "sum of," "anti­derivative of," etc.).	'Pure' mathematics.
Universal-cum-(monadic-or-polyadic)-singular (e.g., total popula­tion & para­meters).	Modes of attributability ("indeed," "not," "if," "novelly," "with a probability of 37%," "feasibly," "optimally," etc.).	Applied yet mathema­tically deep/nontrivial mathematics (maths of optimization, probability, information, logic).
Special-cum-general (i.e., neither singular nor universal). # # # # # … * * * * * … # # # # # …	Accidents, attributes, properties, etc. ("strong," "weak," "dependable," "healthy," "brave," etc.).	General ('domain-independent') studies of positive phenomena (inverse optimization, statistics, inductive & descriptive info theory, philosophy).
Special-cum-(monadic-or-polyadic)-singular. A. B-C. D. E-F-G. H. …	Substances (primary substance: "this man," "this horse," etc.).	'Special' sciences (sciences of forces, matter, life, mind).

Basically, Lowe considers, as comprising his first categorial dichotomy, the particular (the singular or individual) and the universal (by which he means that which has more than one instance, and which I instead call the general). Logically there are four such quantities, not just two, arising simply and naturally. But Lowe is eclectic in a traditional way and doesn't consider as categorial divisions (1) the sweepingly universal in the sense of that which is true monadically or polyadically of everything ("one," "two," etc.) and is therefore extremely formalizable, or (2) the special in the sense of that which lacks such extremely formalizable universality ("blue," "elastic," "Jack," etc.).

Perhaps the biggest impediment in taking inventory of logical quantities has been that we don't usually consider both the monadic singular and the polyadic singular (or polyad of singulars) as being, both of them, singular, just as we consider both the monadic general and the polyadic general as being general. A related impediment has been a common initial veering into regarding the sweepingly universal only as a highest genus, strictly monadic, trivial for most purposes, and confined to a narrowly gabled attic, so to speak, of the house of logical quantities, an attic with room for just one such universal, logically equivalent to every such universal. Perhaps a third impediment has been some sort of neglect about defining logical quantity for terms through some same question or questions asked in all cases.

Given a term "H" true 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?
If no, then "H" is singular. If yes, then let us call "H" general.
- and -
Is there something which isn't that thing x and of which the term "H" is instead false?
If no, then let us call "H" universal. If yes, then let us call "H" special.

The twin questions stand mutually independent and resolve into four answers, conjoinable in four ways (see the table "Tetrastic versions..." above), 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.

One may think at first glance that one of the conjunctions, the universal-cum-singular, enframes a nearly blind window, looking out only on the case of a one-object universe. But let us practice consistency of conception, avoiding special wrinkles and complications, and classify the singular and the singulars-in-a-polyad together as singular in logical quantity, just as we class both the monadic general and the polyadic general as general. Then the monadic-or-polyadic singular-cum-universal comes forth naturally as a logical quantity corresponding to a gamut, a total population and its parameters, a universe of discourse, etc., supporting for example a collective predicate such as "30% (are) blue." (Those collective predicates are pretty hard to get without polyads.) Usually when we think of the universal, we think of something like a law, with many, even indefinitely many instances. That is actually a compound quantification; such a universal is also non-singular, i.e., also general.

Of course, in the sense that two are not three, "two" is not universal. But "two," "three," etc., are universal in the sense of being true of anything in some polyad or other; the qualities, the particular attributes, of the counted objects don't matter; only each object's being other than the others matters. Then we abstract the numbers and think of them as singulars. Thanks to its imaginative apparati, mathematics can re-create the world's logical quantitative diversities and variegation on abstract levels.

Now, since a universe-encompassing polyadic subject fully spelt out in sequenced monadics is sometimes daunting, consider a universe-encompassing relative or collective predicate; consider a universe-encompassing expressionally streamlined polyadic subject; and consider also a predicate-formative functor such as "with a (frequency) probability of 35%."

The question of variety among exhaustive sequences of the same total population's members is not a vexatious complication (raising the question of whether a total-population term is general when it applies to various sequences albeit of the same objects) but instead a good complication and part of the solution to the question of what might be interesting about the monadic-or-polyadic singular-cum-universal logical quantity as a perspective. It goes to show that one should check to see whether one has defined parametric options in a consistent manner, especially in order to avoid jumping to conclusions about seemingly trivial or seemingly near-empty compounds of parametric options. That seemingly almost blind window turns out to view a populous class of research, including, and not limited to, probability theory and deductive theory of logic.

If one defines logical term quantities such as the universal, the general, and the special such that the terms may be either monadic or polyadic, then one should likewise define the singular, even if it means giving the singular another name, so as to keep the parameter of monadicity/polyadicity consistently independent of the parameter of logical term quantity. If one is proceeding exploratorily, then one's logic should not be given special wrinkles in order to prejudge such questions as whether there’s any point to defining a monadically-or-polyadically-singular quantity. Such an anti-pre-judicial consistency, in the exploration of logical quantities, matters especially when one is interested in grasping logical quantities in a general way (general like statisticality and information) as perspectives characteristically emerging, even without formal articulate ado, as scopes in research and intelligent decision-making, performance, affectivity, cognition, etc., of whatever kind.

Because of the common philosophical failure to differentiate the singular as non-general sharply enough from the single as monadic, one easily fails to notice that a polyadic version of the singular could be a whole universe, and therefore sweepingly universal (in its universe of discourse), without being general and non-unique (again, in its universe of discourse) and is not a trivial almost-blind window onto a mere one-object universe. In other words, in missing some of the simple quantities, one misses their conjunctive compounds. Thus one also misses the fact that the singular as usually understood actually involves a conjunctive compound of singular and special (or non-universal) in the sense that the singular, as usually understood, is not a total population. If "H" is a singular predicate, one usually assumes not only that there is some x of which "H" is true such that there exists nothing else of which "H" is true, but also that there exist, distinct from x, things of which "H" is false. That assumption is actually an option with a significance that becomes clear if by "singular" one means only "decidedly non-general" and not also "decidedly monadic." One should do that for conceptual consistency in considering logical quantity and term "adicity" or "valence" as separate dimensions.

I didn't start out hoping to find some way to include "total population" or "universe of discourse" as a logical quantity on a par with "general," "singular," etc. I hadn't given any thought to the idea of totality or universe as logical quantity. I simply followed the logic out consistently and tried to understand where it led. It leads to an old philosophical desideratum, a correlation of logical quantities to major classes of research subject matters. It even makes the old nominalist-realist wrangle seem less interesting, because now one is not confronting the same old stark dichotomy again and again. A universal such as "two" differs as much from a non-universal general like "healthy," as either of them differs from a non-universal singular like "Jack."

So I recognize four quantificational divisions, conjunctively compoundable in four ways, where E.J. Lowe recognizes only two logical quantities. Mine are logically more systematic and turn out to correlate to the subject matter perspectives of the major classes of research. If, for instance, one considers singulars polyadically as well as monadically, it is more natural to regard the 'special' sciences as being about singulars in a larger world. Before crying "there is no science of singulars," one should also remember that the subject matter of a science can differ from the object or objective of a science, and that the special sciences seek, as their objective, to discover laws, distributions, kinds, and individual histories of the subject matter, concrete singulars sometimes one by one and sometimes together in various ways. Even laws in physics take on the singular aspects of giant events, for instance the signal speed limit, which may have changed over time in relation to other fundamental physical quantities.

Insofar as more traditional categories such as substance and property are arrangeable in a pattern of nonbinding affinities with logical quantities, there again, I have four where E.J. Lowe has two. (Skip tables ►)

Simple & conjoined logical quantities for terms (subject, predicate, etc.)  Skip ►

	Universal:	Special:
General:	1. Universal-cum-general.	3. Special-cum-general (neither singular nor universal)
(Multi-)singular (monadic, polyadic, etc.):	2. (Monadic, polyadic, etc.) Universal-cum-singular (gamut, universe of discourse, total population & its parameters)	4. (Monadic, polyadic, etc.) special-cum-singular (monadic, polyadized, etc., singulars in a larger world).

Tetrastic categories  Skip ►

		Positive phenomena:
	1. Correspondences/ variances (another than, sum of, inverted order of, anti-derivative of, etc.).	3. Attributes, properties, accidents etc. (firm, unsound, well, ill, steady, irregualar, strong, weak, etc.)
	2. Modes of attributability (indeed, not, if, novelly, probably, feasibly, optimally, etc.)	4. Substances (primary substances: this man, this horse.).

Tetrastic classifications of research  Skip ►

	Typically, the premisses deductively imply the conclusions:	Typically, the premisses don't deductively imply the conclusions.¹
Typically, the conclusions deductively imply the premisses:	1. 'Pure' mathematics: equations, topology, graphs, integration, measure, enumeration, functions & derivatives, algebra, limits, order	3. General ('domain-independent') studies of positive phenomena: Inverse optimization, statistics, inductive & descriptive areas of info theory (& their math formalisms), philosophy.
Typically, the conclusions don't deductively imply the premisses¹′:	2. Applied yet mathematically deep deductive theories of: optimization, probability², information³, logic	4. 'Special' sciences: sciences of motion & forces, matter, life, mind, intelligence, intelligent life
¹, ¹′ Notwithstanding inferences within imported formalisms. ² Or, more generally, uncertainty theory. ³ Mathematics of information overlaps significantly into pure math, especially abstract algebra.

Now, such quantificational divisions are no more to be eschewed for parsimony than corners of the Square of Opposition; they are so systematic that it takes more information to eclectically select a few than to take them all. Occam doesn't raze exactly one or two corners of the Square of Opposition. To fail to recognize this leads to arguments over how few angels can dance on the head of a pin. If instead one listens to that which the logical structure is "trying to tell" one, then one may avoid the excessive foreshortening of the world's divisions that is echoed by the classic Saul Steinberg cartoon. For another instance, logical connectives can all be done in terms of negative alternation, and can all be done in terms of negative conjunction. But this means that the negative alternative and negative conjunctive are particularly versatile logical connectives; it does not mean that one or the other of them is really the only logical connective. Now, Lowe intends his ontology for the natural sciences, but I don't know what the natural sciences gain by minimizing or ignoring the difference between "blue" (true of this and of that, but not of some third or of some fourth thing) and a numberish universal like "three" (true of everything in one or another polyad xyz where xyz are all of them distinct objects and in a universe with more than two objects). I suspect that the real issue is an avoidance of evoking or suggesting further mountain ranges of those entities (beyond the generals or "universals" which Lowe already countenances) which nominalists dislike. Thus does the addictive battle between realism and its opponents distract from other interesting issues, distort and crop straightforward logical formalisms and their potential applications, and prevent philosophers from doing justice to the ideas to which they commit themselves in adopting a logical formalism such as that of logical quantity. For my part, I generally take the involvement of questions of a subject matter's ontological status 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.

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