The Tetrast
Sketcher of various interrelated fourfolds.

Why tetrastic?

March 31, 2005.
Reader: “Never mind this blather.
Let me glance at the tables
(ones not still ‘in progress’).”
I: Scroll downward to see tables of
fourfolds and sixteenfolds.
(Latest significant edit: Monday, July 31, 2006 January 20, 2014. Old BlogSpot-style formatting repaired, at least mostly.
  As I revised this post over time, it became too long, too strident in some places, and also somewhat repetitive. It became unwieldy, I didn't know what do with it as a piece of writing. And now I've changed my mind about at least a few things in it, too, but I don't feel like doing much rewriting.)

Point at image.
Various fourfolds which are discussed here. Because there are marked patterns of philosophically relevant fourfolds.

It’s simple logic in some cases, as when two recurrent plain questions mirror each other logically and still stand logically independent of each other, e.g., “is there risk?” and “is there opportunity?” To them beckon four combinations, of course, of simplest answers: Y, YY, NN, YN, N.

(Pronoun table in progress)
Three Parameters
Broken Out
Addressor (s)
Non-addressor (s)
Addressee (s)
We (inclusive),
(in self-address:)


Non-addressee (s)
We (exclusive).

Not dividing
I'm unsure which (if any) shuffling of the above cells best correlates with other fourfolds. A "good" fourfold has not two but three seemingly meaningful two-value parameters though any two render the third one technically redundant. One will usually seem a bit more abstract than the other two, but maybe this depends on the labeling of the parameters ("addressor [Y/N]," "addressee [Y/N]," and, along the diagonals, "dividing participants [Y/N]").
No deeper than that, likely, dwells much of the reason for marked patterns of philosophically relevant fourfolds.

Yet, some philosophically relevant fourfolds echo each other in ways for which I have not yet managed, at least to my satisfaction, to uncover the reasons, even when the fourfolds separately from each other have seemed clear enough. Turn a sign this way, then that, align it with others, the world seems to crack open, and the chase may be on. The chase after a recurrent logical pattern in philosophical issues has led through Aristotle’s Four Causes, Thomistic beauty, categories, logical quantities, semiotics, elementary aspects of information theory, the special-relativistic light cone, philosophical classification of research areas, philosophical classification of human & social activities, etc., as reflected in tables below. I’m less interested in “ultimate” philosophical questions than in tracing out this pattern of fourfold structures. Since anybody still reading at this point will probably be wondering, I may as well say now, that four was indeed my favorite number — when I was four years old; five my favorite when I was five; and thereafter I ceased having a favorite number. I started attempting categorization a few decades later, trying first with three, then four, then six. By the time I learned of and started reading the trichotomist C.S. Peirce, I was already an inveterate “fourist.”

Keep scrolling down for still more tables.
I (rs)↔A (rs)
There's R, all or
none of it S.
I (rs) & A (rs)
There's R, all of it S.
O (rs) & E (rs)
There's R but no RS.
I (rs↓ A (rs)
I (rs) & E (rs
[Tautologously false]
I (rs) &F& A (rs)
Square of
A (rs) v E (rs)
There's no RS or there's
no R that's not S.
I (rs)←A (rs)

A (rs)
There isn’t R that isn’t S.
All R is S.
E (rs)
There isn’t R that’s S.
No R is S.

A (rs) & E (rs)
There isn’t R.
I (rs)' & A (rs)
I (rs) v O (rs)
 There’s R.
I (rs)→A (rs)
I (rs)
There’s R that’s S.
Some R is S.
O (rs)
There’s R that isn’t S.
Not all R is S.
I (rs) & O (rs)
There's RS & there's
R that's not S.
I (rs) &~ A (rs)

(rs) v E (rs)
[Tautologously true]
 I (rs) v Tv A (rs)
(rs)↔O (rs)
If there's R, then: there's RS &
there's R that's not S.

I (rsexclusive-or A (rs)
I (rs) v A (rs)
If there's R, some of it is S.
O (rs) v E (rs)
If there's R, some of it isn't S.
I (rs) | A (rs)
(on Boolean
...(maybe unnamed) ...positive
...(maybe unnamed)
"exclusive" or "eclectic"
might work.
(validity of the biconditional)
CAN’T be, 1st true, 2nd false
CAN’T be, 1st false, 2nd true
p. p. T. T. F. F.
(validity of the conditional with
nonvalidity of reverse * conditional)
CAN’T be, 1st true, 2nd false
CAN be, 1st false, 2nd true
p. (p v q). p. T. F. p. F. T.
(validity of the reverse * conditional
with nonvalidity of the conditional)
CAN be, 1st true, 2nd false
CAN’T be, 1st false, 2nd true
p. (p & q). T. p. p. F. T. F.
(nonvalidity of the conditional
in either direction)
CAN be, 1st true, 2nd false
CAN be, 1st false, 2nd true
p. q. p. (q v ~p). p. (q & ~p). p. ~p.
Material logical compounds table * Trivial in themselves are the differences between forward & reverse conditionals and between forward & reverse strict implications.
Non-trivial in itself is the difference between non-reversible (a.k.a. “strict”) deduction & ampliative induction.
CAN both be true
CAN both be false
CAN both be true
CAN’T both be false
CAN’T both be true
CAN both be false
CAN’T both be true
CAN’T both be false
(nonvalidity of
(validity of positive
alternation with
nonvalidity of
(validity of negative
alternation with
nonvalidity of
(validity of
I came to direct my categoristic impulse at fourfolds since (a) the “field” of logically four-way philosophical distinctions remains largely unplowed, and since (b) four nevertheless both is small and is often the number of terms in a general logical alternative (e.g., the Square of Opposition and such foursomes as that of subalternation, subcontrarity, contrarity, and contradiction),— and, since, as a practical matter, (c) I found myself making eventually more headway with fours than with threes, fives, sixes, etc., in tracing out divisions having a family resemblance and having either logical analyzability or at least a realistic prospect thereof. So, now I call myself a “tetrast,” as if to make philosophy with a threadbare motif. I don’t rule other numbers out; four just seems to work peculiarly well.

It’s philosophically helpful that one cultivate an intuition for common transparent logical fourfolds, since philosophy is a science or study of reason (among other things, including reason’s crackups). As a study of inference processes, philosophy is specially allied, on the concrete side, with the human and social studies, and, on the abstract side, with deductive study of logic and, beyond it, on inference’s ‘pure’-mathematical side, with the study of ordered structures. Yet, philosophy neither draws its conclusions deductively nor seeks (like the human and social studies) to establish or explain special phenomena by surmise, special experiment, and inductive syllogism, drawing conclusions which would be, formally, neither truth-preservative nor falsity-preservative. Almost by process of elimination it seems that, if one wishes to see philosophy in terms of how it could be distinctly yet still securely and capaciously anchored and related to research in general, then one needs to see philosophy as drawing inductive generalizations (falsity-preservative, not truth-preservative) as conclusions, somewhat as in statistics, yet having for its allied inverse not probability theory but instead deductive logic, and drawing conclusions about reason and experience in general. In that case, it should be philosophically helpful that one cultivate an intuition or instinct for common transparent logical fourfolds present or echoed in intriguing and somewhat opaque structures of alternatives, alternatives involved at a general level in complex sign- & evidence-supported processes of character, competence, sensibility, and intelligence including philosophy itself and all research, to say the least. And it is also then helpful that one seek to abstract pairs of mutually relevant logically two-valued parameters and to systematically explore all their combinations of values, not just some.

I know of no major philosopher who’s been a thoroughgoing — or, more reasonably, a thoroughseeking — tetrast.[1]

[1] I learned of a philosopher E.J. Lowe who is something of a tetrachotomist and, thanks to Steven Ravett Brown at Ask A Philosopher, I felt it worth looking further into Lowe, found his paper “Recent Advances in Metaphysics” online, but confirmed an earlier disappointment. Lowe’s ontology tops off with two logical quantities, the universal and the particular, corresponding likely to what I call the general and the singular, not that it matters, the point being that the elementary logical quantities characterizing terms are logically not two but four (and involve four conjunctive compounds), while Lowe, with his structure reflecting an undertaken but arbitrarily incomplete account of the logical quantities, veers off in another direction, to double his logical-quantity-defined categories prematurely with categorization according to other considerations.
Update: I go into a bit more detail in my E.J. Lowe entry in "What of these other fours?"

E.J. Lowe's four-category ontology

(a.k.a. generals)
Kinds characterized by Attributes


(a.k.a. singulars,
Objects characterized by Modes
a.k.a. tropes,
e.g., this redness)

À la monist, dualist or dyast, triast — tetrast. I’ve found that the word “tetrastic” already exists, though mostly as a variant, as Gary Richmond pointed out to me, of “tetrastich,” a four-line poem, stanza, or epigram.

“Tetrastic” here means fourfold as either tetradic (four-element) relation or tetrachotomic (four-way dividing). I don’t mean that the world is four elements, e.g., land, wave, wind, sun, or that the mind-matter dualism should be recast as a tetrasm of mind, life, matter, forces in the sense of four mutually alien elements enigmatically combined. (“Tetrasm” sounds bawdy, ergo this Website has at least one thing that maybe, just maybe, would have been undispleasing unto Quine.)

Some items in these fourfolds have mattered amply in philosophy, so it seems significant that they fall not uncomfortably into a common and sometimes intriguing pattern.

Beginning, Middle, End, Check, shown in typical arrangement: Col 1 Row 1, Col 1 Row 2, Col 2 Row 1, Col 2, Row 2.
Starts (at t)
(isn’t till t,
is since t).
Stops (at t)
(is till t,
isn’t since t).
Continues (at t)
(is till t,
is since t).
Continues-Not-To (at t)
(isn’t till t,
isn’t since t).
By an odd turn of mind, I fail to understand why one wouldn’t expect any recurrent logically analyzable pattern at all in philosophical questions. Anyway, it’s not a bad thing that the pursuit of such patterns requires tracing out the full sets of terms of the logical alternatives underlying various philosophical conceptions. For my part, faced with an alternative such as that between means (middle) and end, I feel lost until I have fleshed out correlatively applicable conceptions of a beginning and a checking or holding (off) (or a staying-ended). In human affairs regarded such that happiness is typically the end, what is typically the middle or means? The beginning? The check or checking? Or do you think that in this and many other matters, there are only middles and ends? Odd on the face of it, at least. If you are a philosopher, you may consider it worth more than a moment’s thought.

I wonder why philosophers don’t trace out such full sets — for another instance the full set of elementary term-quantities such as the singular and the general. This, given the importance which the “question of universals” has had in philosophy, seems as curious a case of philosophical neglect as I have encountered in my narrow pursuit. Some philosophers will improvise responses mentioning Occam’s Razor, as one can well imagine, but here it cuts against them. Application of half or three quarters of a logical square is less simple and less consistent than application of all of a logical square. Strange would be the museum that, with little or no explanation, displayed many a quadruped running on only three or two of four apparently good legs. It is philosophically needful to satisfactorily explain the applicability or inapplicability of every option in a broad philosophically relevant alternative (in the case under discussion here, a logical square of options) that was initially invoked for some not all of its options. The refusal to attempt such a completion renders suspect the seeming success of the given square’s partial application.
The four-structure of lightcone-style times (with some non-technical wordings) allowing causal relations. Consider it as generalized & fuzzied to reflect various communication modalities. Also shown are associated turns of:
a. will (human agency, making, deciding); c. affectivity (humanly being affected);
b. ability (human bearing, carrying (out), sustaining); d. cognition (human supportedness, borneness).
1. (beginnings)
“Lightlike” or “improvisational” future or audience-like becoming-present, a kind of “would-be” or almost -present
(whatever is in or along one’s outward- fanning info’s potential or actual path at the path’s moment).
a. will:
c. affectivity:
b. ability:
d. cognition:
imaginative impression.
3. (ends, culminations)
The Barely Present, the Just-Now. “Lightlike” past or apparent present, just-now or
barely -present
(whatever is in or along one’s convergently
incoming info’s potential or actual path at the path’s moment).
a. will:  
c. affectivity:  
pleasure, satisfaction.
b. ability:  

d. cognition:  
2. (middles)
The Later.Later or developmental future,
entrainment, concatenation, or unification of temporally successive almost-presents.

a. will:
as a logical entrainment of successive
c. affectivity:
as a logical entrainment of successive
b. ability:
preparing or approaching
as a logical entrainment of successive
testings or
d. cognition:
expecting or anticipating
as a logical entrainment of successive
fancyings or
imaginative impressions.
4. (checks)
The Earlier.Earlier, or settled past,
entrainment, concatenation, or unification of temporally successive barely-presents.

a. will:
as a logical entrainment of successive
takings or
c. affectivity:

as a logical entrainment of successive
pleasures or satisfactions.
b. ability:
or skill
as a logical entrainment of successive

d. cognition:
as a logical entrainment of successive
noticings or

Light cone shown labeled with info from the table. Light cone shown in a more 3-D manner, with labels 'So be it?' on the future LC surface, 'So be it.' inside the future LC, 'Is it?' on the past LC surface, & 'It is.' inside the past LC.
  Will, Conation Ability, Handling Affectivity Cognition
Almostward Trying (out)/(for) Testing, devising Desire Fancy, immed. impression
Laterward Seeking Preparing, approach. Hope, Confid. Expectation, anticip., etc.
Barelyward Taking, picking Achieving Pleasure, Satisf. Noticing, discernment
Earlierward Adhering, habit Maintaining, skill Attachment Memory
To complete a structure of alternatives like that of the singular versus the general? About the logical quantity of a monadic or polyadic predicate true of a given thing (or things in a polyad), one asks two elementary questions, (1) whether there be some thing(s) else whereof the predicate is true (monadically or polyadically), and (2) whether there be some thing(s) whereof the predicate is false (monadically or polyadically). They are a pair of mutually independent questions, each with two answers, for a total of four answers conjoinable in four ways. To complete a structure of alternatives like that of the deductive versus the non-deductive? About inference modes one asks the elementary question, does it add information (i.e., is it ampliative or is it deductive and automatically truth-preservative)? One should also ask, does it remove information (i.e., is it precissive (that would be the technical term; I’d prefer “refinitive”) or does it automatically preserve falsity)? They are a pair of mutually independent questions, each with two answers, for a total of four answers conjoinable in four ways. About abstractions or abstractional intensions, one can and should ask the same two elementary information questions, each with two answers, for a total of four answers conjoinable in four ways. Physics2005 logo showing stylized light cone, but with colors transposed to accord with color 'code' here. As to time and causation? Special relativity’s light cone is a structure of four especially time-relevant answers to elementary causality questions, a structure of a kind which philosophers failed to discern before Einstein, a structure which they might wish, now, a century later, to start taking seriously as the ubiquitous physical instance of a kind of general phenomenal structure which appears insistently albeit in less exact forms in communication, perception, etc. It’s a shame that phenomenologies and linguistic analyses have missed all these things subtle but not too far from elementary, failed to address them even if only to dispute their philosophical importance.

Also, too seldom (or so it seems) has a philosopher (Peirce is an exception) done, and kept in view, inventory of the items to which a structure of logical alternatives will be applied, inventory toward which one shouldn’t scorn Roget’s thesaurus.

Mendeleev in his cabin labored over various arrangements of pieces of a chart, in search of the structure which became his periodic table. Yet, in philosophy, incompletely explored logical structures are applied to incompletely inventoried items, and Occam’s Razor is invoked to call the job finished. Rules of thumb about logical structure might help to avoid some of the omissions, conflations, and foreshortenings involved in this. More or less off the top of my head, for instance:

1. The smallest sets of multiple deductive logical options (notwithstanding the eight implicitly provided by the Aristotelian Square of Opposition) are usually 2, 4, and 16. (I regard philosophy as tending to draw ampliative inductions as conclusions, but also regard it as a setting for the development of domain-independent deductive formalisms based on “contingent” or phenomenologically based assumptions, albeit such that the proceeding as a whole is utimately an ampliative induction guided by standards of cogency and illuminativeness regarding phenomena in general.)

2. Check that one has defined one’s parameters in a consistent manner. (E.g., 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 prejudge which philosophical views one should adopt in such matters as whether there’s any point to defining a monadically-or-polyadically-singular quantity. Such an anti-pre-judicial consistency, in the logical exploration of logical quantities, matters especially when one is interested in grasping logical quantities in a general way (general like statisticality and information) as mental perspectives characteristically emerging even without formal articulate ado as scopes in research and intelligent decision-making, performance, affectivity, cognition, etc., of whatever kind. The singular is a case in point (and, by now, obviously a case on my mind), understood, as it usually is, as being in two oppositions, one versus the general, and the other versus the polyadic.)

3. Do not jump to conclusions about seemingly trivial or seemingly near-empty combinations of parametric options. That’s an occasion rather to review and check one’s parameters for consistency of independence, and to review and check one’s inventory of the items to which one hopes to apply the logical or categorial structure. Listen and scout around for what the logic is “trying to tell you.”

Now, I usually think of a fourfold in the square 2x2 arrangement, not unlike the simplest typology in morphological analysis. Obviously sometimes it’s easier just to lay it out 1,2,3,4, especially when it supplies column or row headings.

A few of the fourfolds shown in this post are obvious, at least when laid out. For others I have argued or expect to argue in various posts here. One of the 4x4 tables below involves the philosophical classification of research fields. I don’t know of any philosophical attempts to do this with research fields as currently evolved, but would be interested to hear of any. Birger Hjørland has written: “There is not today (2005), to my knowledge, any organized research program about the classification of the sciences in any discipline or in any country. As Miksa (1998) writes, the interest for this question largely died in the beginning of the 20th century.”

From post “A periodic table of aspects of humanity which lend themselves to social compartmentalization.”
A periodic table of the maths, sciences, & areas intermediate between them. See post “Logical quantities, categories of research, and categories.”

. . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . 1st level


...w.r.t. 1. Decision- makings, power.
...w.r.t. 2. Abilities, means.
Production, processing,
...w.r.t. 3. Affectivity, ends.
...w.r.t. 4. Cognition, checks.

Each row:
a research ‘family.’

|||| Each column:
a research ‘coterie of friendly cousins.’
1. Mapping, pathfinding (objectification-like processes): 2. Counting, measure, apportionment (representation-like processes): 3. Calculation, coding, appraisal (interpretation-like processes): 4. Inference, reason (& reason’s crackups) (inferential & (dis-) confirmatory processes):
. . . .
1. Decision- makings, power...↑


Affairs of power (polit., milit., etc.).

Business, trade, economic affairs.

Show, sports & games, fashion, etc.

Case-building, debate, dialogue, etc.

1. Pure Mathematics:
Typical conclusions:
via equivalences
Subject in scope:
universals that aren't universes
Many-to-many (graph theory, topology, simult. eqs., extremi­zation, etc.). One-to-many (enumeration, measure, integration, etc.). Many-to-one (algebra & groups, deriva­tives, etc.). One-to-one (math-induction applica­bility conditions, structures of order, limits, etc.).
. . . .
2. Abilities, means...↑
Practices (cooperation, tolerance, etc.):

Management, compliance, etc.

Work, labor, skill-plying, etc.

Recreations, hobbies, entertaining, etc.

Study, communication, etc.

2. Applied yet mathematically deep mathematics:
Typical conclusions:
forward-only deduction (except in historical overlaps with pure maths)
Subject in scope:
totalities, universes
Deductive maths of optimization. Deductive maths of probability. Deductive maths of information. Deductive maths of logic.
. . . .
3. Affectivity, ends...↑
Valuings (communion, joining, etc.):

Ruling / governing valuings (relig., morals, etc.).

“Care-how,” valuings re: work, production, competence, etc.

Affective and gratificational valuings.

Cognitive valuings, curio­sity, love of knowledge, rigorism, etc.

3. Abstract yet positive- phenomenally deep sciences/studies:
Typical conclusions:
ampliative induction
Subject in scope:
non-universal generals
Inverse optimization problems [Google on them ]. Statistical theory. Descriptive & inductive information theory. Philosophy.
. . . .
4. Cognition, checks...↑
Disciplines (supports, checks, & balances):

Ruling / governing arts.
Object(ive) in scope: (multi-) singulars among more singulars

“Know-how,” practical / productive arts/scis.
Object(ive) in scope: non-universal generals

Affective arts (music, literature, visual, dramatic, etc.).
Object(ive) in scope: totalities, universes

Sciences, maths, & disciplines intermediate between them.
Object(ive) in scope: universals that aren't universes

4. Concrete empirical (“special”) sciences/studies:
Typical conclusions:
Subject in scope:
(multi-)singulars among more singulars
Forces. Matter. Life. Intelligent life.

Tetrastic structures several but similar.
The 4 turns of becoming: 4 rough tags: The 4 phases of code-bound communic. 4 phases of intelligent communication (code-unbound enough to try out & evolve codes): The 4 stages of creative process (Helmholtz & Poincaré).
~ The 4 stages of aesthetic apprehension (revised Joyce).
~~ The 4 aesthetic properties (Aristotle plus Aquinas). :
4 categories: 4 sets of kinetic & associated mechanical conceptions: The 4 Causes’ revised principles , & the 4 Causes a little revised: 4 kinds of complex systems: 4 levels of concrete pheno­mena: The human forms of revised prin­ci­ples of the 4 Causes:
Starts (at t)
(isn’t till t,
is since t).
Main cause, driving cause. Source. Semiotic object
“the agent”).
~ Arrest.

~~ Due directive magnitude, due force
Object(s)- to- object(s) relator. System’s external momenta or forces. Agens, beginning, effector (Aristotle: "initiating source of change or rest"). Sensitive dependence on initial conditions (await inverse-optimizational treatment?). Forces. Will, conation, character.
Continues (at t)
(is till t,
is since t).
Further cause, intermediate cause. Encoding. Sign
“the patient”).
~ Fascination.

~~ Due proportion, due rhythm, harmony, etc.
Whetherhood (Y/N, pro­bability, infor­mative­ness, logical condi­tioning, etc.). System’s internal energy or power (down thru the elementary particles)
(rest energy :: rest mass)
Sustinens (patiens), middle, means, matter/process (rest mass :: rest energy) Fluctuation-smoothing depen­dence on intermediate-stage conditions (stochastic processes). Matter. Ability, competence.
Stops (at t)
(is till t,
isn’t since t).
Main effect,
culminant effect.
Decoding. Interpretant (interpretation)
“the act”).
~ Takenness, enrapturement.

~~ Radiance, vibrance.
Attribute System’s external energy or power. Actum,
, culmination.
Corrective (feedback) depen­dence on output conditions (cybernetic processes). Life. Affectivity, sensibility.
Continues-Not-To (at t)
(isn’t till t,
isn’t since t).
Further effect,
“side”/ “after”- effect.
Destination. Recognizant (recognition, verification, etc.). Verification.
~ Attachment.

~~ Wholeness, (structural) integrity.
Substance (this man, this horse) (or hyposta­tization). System’s internal momenta or forces (including down thru the elementary particles) (statically or otherwise balanced relative to an observer at rest). Sustentum, check, form, structure, evidence, etc. Fortificative dependence on sign-&-evidence conditions (intelligent processes). Intelligent life. Cognition, intelligence, knowledge.

The four conjunctive compounds of simple term-quantities including a seemingly near-blind window.

Universal Special
“H” as
universal -cum -general :
Given Hx,
there are both
nothing non-H
(“H” as universal) &
something H
(“H” as general).
“H” as
special -cum -general :
Given Hx,
there are both
something non-H
(“H” as special) &
something H
(“H” as general).
“H” as
universal -cum -singular *:
Given Hx,
there are both
nothing non-H
(“H” as universal) &
nothing H
(“H” as singular).
“H” as
special -cum -singular :
Given Hx,
there are both
something non-H
(“H” as special) &
nothing H
(“H” as singular).
* The universal-cum -singular is not restricted to a one-object universe when one allows a polyadic “singular” (i.e., polyad of singulars) just as one allows the polyadic general and universal, though maybe it should be called something other than “singular.” For a discussion click here.
Research families and typical modes of conclusions
Reversible deduction
(neither adds nor removes info).

Pure mathe­matics
(abt. the object(s)-to-
object(s) relators adduced thru the imagination or its extensions by both removing & adding info).
Non-reversible deduction
(only removes info).

Applied yet mathe­mati­cally deep maths
(abt. “whetherhoods,”, universes, etc., info-increasingly imputed through the intellect or its extensions.
Amplia­tive induc­tion
(only adds info).

Abstract yet positive- phenomenally deep sciences/ studies
(abt. attributes info-decreasingly abstracted thru the sensory & intuitive faculties or their extensions.
Surmise, abductive inference
(both adds & removes info).

Concrete empirical sciences/ studies
(abt. the concreta accessed thru commonsense perception or its extensions without info-changing abstraction, at least in intention).
Ampliative inference:
Adds info,
doesn’t preserve truth.
Adds no info,
preserves truth.
Surmise, abductive inference:
Adds info, removes info;
preserves neither truth nor falsity.
Conclusion tends to be to causes, reasons, entities, laws, “puzzle pieces,” on principles of SIMPLICITY, NATURALNESS, EFFICIENCY.
Compare versus OPTIMALITY
Non-reversible deduction:
Adds no info, removes info;
preserves truth but not falsity.
Conclusion tends to be to a not actually recognized though already formally implied thing sifted out on principles of NOVELTY OF ASPECT.
Compare versus INFORMATION
w.r.t. MANY-TO-ONE
no info,
Compare versus PROBABILITY
w.r.t. ONE-TO-MANY
Conclusion tends to be to tendencies, trends, extensions of a quality, etc., on principles of LIKENESS of sufficiently representative portions or samples to a whole.
Adds info, removes no info;
preserves falsity but not truth:
Ampliative induction.
Compare versus TRUTH
w.r.t. ONE-TO-ONE
(all true propositions are materially equivalent-cum -subcontrary).
Conclusion tends to be to something formally equivalent to the premisses but in different form, on principles of DEPTH.
(In mathematical induction, the conjunction of minimal case with heredity is equivalent to the conclusion.)
Adds no info, removes no info;
preserves both truth & falsity:
Reversible deduction.

Tetrastic structures seem to have been neglected by philosophy, at least in comparison to dyads and dichotomies and even to triads and trichotomies. Yet they seem to work rather well to bring a little more system to some traditional philosophical structures, and it’s useful to cultivate an intuition for them. In deductive logic, there are the logical connectives, the quantificational forms, etc., and a logician gets to know those structures inside and out. I would argue that philosophy is the field which deals with the inverse problem to that of deductive theory of logic (or as it’s sometimes called nowadays, just plain “logic”), rises to concerns of a general kind with complex processes of the intelligent mind and heart, and of society, and tends to draw ampliatively inductive conclusions.[2]

[2] Of course, if philosophers do not so conceive of it, then in an important sense philosophy is not the field which deals with the inverse problem to that of (deductive theory of) logic, rises to concerns of a general kind with complex processes of the intelligent mind and heart, and of society, and tends to draw ampliatively inductive conclusions. But what other secure place will persist for philosophy as primarily a research discipline, and a major one at that? Philosophy — cast by some in philosophy’s linguistic analysis school as the analysis of arguments, in contradistinction to logic as the theory of arguments (by which I take them to mean deductive theory of arguments themselves deductive, inductive,
What is philosophy?
Which are its sibs and
which its close cousins?
or whatever) — philosophy would be reduced, if people took this seriously, to a subdiscipline (explicit arguments encountered in the world) within applied deductive theory of logic, itself an interdisciplinary subdiscipline, and a fine one it is (if it indeed exists as a discipline), serving as liaison between the world and ‘pure’ deductive theory of logic, a fine role but not the kind of thing to merit a separate department, much less a separate building, at every liberal arts college or university in civilization. The phenomenologists and existentialists seem uninterested in connecting philosophy with research in general (rather than with human & social studies only) — even Merleau-Ponty seemed to find something distinctly sinister in science, sinister, that is, in some way that philosophy seemed not to be for him. The analytic linguistic turn and the phenomenological epoché alike, either one taken as philosophy’s basis, seem to estrange philosophical schools from each other (who cares, right?) and to estrange philosophy from other research (there, philosophers should care), leaving philosophy taken mostly unseriously by non-philosophers. Meanwhile it proves more quixotic or cynical, than fruitful, to make a virtue of seeming necessity and define philosophy as some sort of perplexity research, sheer intelligent conversation, literary expression of theoretical problems, or other such half-intentional jests. Philosophers seldom, since the system-builders’ time, have tried to take the obvious course of investigating philosophy’s connections, resemblances, interdependencies, etc., with other fields, in order to help conclude where philosophy belongs. As for Rorty, who actually proposed redefining philosophy as some sort of literary field (as if literature didn’t already have enough problems), he seems some sort of self-deprecating, ingratiating philosophy-is-empty-so-don’t-be-intimidated-by-it type destined for salons if he is not already a fixture there, even as he seeks gingerly to replace truth with the glass gem of warranted assertion and with the politically loaded mantling of the democratic exchange of views as a supreme value over (and inevitably superseding) that of the value formerly known as “truth.” C.S. Peirce seems to have been the last philosopher to make real headway in the classifications of research and philosophy's place and role.

Meanwhile, the “children” — really, the sibs and cousins — have grown up and the mansion has become a city where philosophy retains a place for the time being. Amateur and professional alike in philosophy delight in complex reasoning processes, and philosophy’s reflexivity ever tempts them to go at it with might and main. But, again, no real place, no secure and capacious place, remains for philosophy in modern research, except in the family of fields drawing inductive generalizations as conclusions, researching phenomena in general (e.g., statistical theory), where philosophy would fit reasonably well as the ampliatively-inductive inverse of deductive theory of logic, likewise as statistical theory is the inverse of probability theory. Recognition of this as philosophy’s locus means attributing to philosophy a distinctive and affirmative core which broadly, deeply, brightly, and firmly anchors and relates philosophy to research in general. (In that place, philosophy remains capable of producing essentially domain-independent deductive formalisms based on “contingent” or phenomenologically developed assumptions.)

It might be a good idea to work out a reasonable elementary level of philosophy, just as there are elementary levels of communication theory, cybernetics, statistical theory, etc. E.g., in a sense, philosophy studies reason : philosophers treat, as subject matter, things (or stuff or whatever), and signs and evidences of them, and interpretations thereof, and recognitions / (dis)confirmations thereof. And, in a sense, philosophy studies experience and phenomena in general : Philosophers do not, qua philosophers, make cogent surmises to entities and laws subject to special tests; nor qua philosophers do they draw deductive and mathematical conclusions, tried though some have so to base philosophy. So, if philosophical conclusions are by ampliative induction (as distinguished from surmise and from any kind of deduction), then — philosophy’s sibs are its fellow areas of research into phenomena in general, and the family consists of: (1) the young field of inverse optimization problems; (2) statistical theory; (3) the descriptive and ampliatively-inductive areas of information theory; and (4) philosophy itself — and philosophy’s inter-family band of friendly cousins includes its fellow areas of research into reason and reason’s crackups, and consists of: (1) order and mathematical-induction applicability conditions; (2) deductive mathematics of logic; (3) philosophy itself; and (4) areas much pitied and envied — for their conclusions, as cogent as they can make them, are by that power of surmise which is a wonder at their subject matter’s heart as well —: the social and human studies. Philosophy, as one of the studies or sciences of reason, is an especially reflexive discipline, so by all means the Socratic Apology and related dialogues should remain in introductory courses.

This is to say that philosophy stands to formal and mathematical logics, as statistical theory stands to probability theory. (A nice thing about tetrastics is that they lead one to trace out patterns of so-called inverse relationships.) Tetrastic structures do recurrently peek out or emerge in philosophy but go undiscussed as a theme, appear recurrently with sometimes less transparency than has been brought to those in deductive logic. So, it’s good for the philosophically minded person to become familiar with some of the more obviously structured ones (not to mention with at least some elementary deductive-logical structures!).


The semiotic tetrads which I have pursued echo the tetrachotomies. Also, no element of such a tetrad can be reduced to any of its fellows; nor can such a tetrad discard one of its four elements without losing its integral semiotic or, if you prefer, integral semiotic-cum -observationalistic, character. I do not aspire to frame a Peirce-style reduction thesis about an irreducibility and sufficiency of tetrads generally. But the semiotic tetrads do align with the tetrachotomies, and more than one such tetrachotomy is conceived in terms of a given pair of closely related independent two-value parameters generating exactly four possible options of the given kind. That sort of sufficiency and irreducibility may prove extendible to the semiotic tetrads themselves. So tetrastics are not entirely the potential mere uncontroversial workaday philosophical tool that I was starting to make them seem, and they’ve led me into some arguments (usually amicable!) with Peirceans.

Charles Sanders Peirce’s triads and trichotomies seem to me to be better when reworked as tetrasms, either by addition of a fourth element or by “dis-conflation” of one element into two, or by deeper reworking. But Peirce’s “triastic” structures are not generally such that I would say that there is in every case some same element missing which you could just pop back into place. Photobucket. Peirce's categories: 1stness, quality (reference to a ground): ground; 2ndness, relation, reaction, resistance (reference to a correlate): [1st] relate, [2nd] correlate; 3rdness, representation (reference to an interpretant): [1st] sign, [2nd] semiotic object, [3rd] interpretant. Still, insofar as his categorial conceptions of reaction, quality, and representation are involved, and insofar as the category of representation is the category of meaning (in a broad sense), I hold that what’s missing as a separate category is that of legitimacy and of one thing’s counting as another legitimately — particularly for evidentiary and interactional purposes. However, I see this extra category as the logical or semiotic category; so what do I do with Peirce’s Thirdness as Representation? Insofar as it characterizes a “communication” system but not a verificatory system, I think of it as value or importance, perhaps in a kind of information-theoretic sense. Now, to emphasize ideas like those of value or importance is, I think, to emphasize the third communication stage, decoding (1st, source; 2nd, encoding; 3rd, decoding; 4th, destination), or the third semiotic stage, the interpretant (I see a fourth stage in semiosis: collaterally based recognition). Such is the kind of threeness or thirdlike-ness that I find there.

Now, Peirce’s own writing shows a constant awareness that knowledge and reasonable belief are reached through adequately explored experience and indeed research. This awareness is the outer essence of his Pragmaticism (the inner essence being his recommendation that, in clarifying one’s conceptions, one best does so in terms of conceivable experience that would conceivably have practical relevance).

But I think that Peirce’s Pragmaticist view is not adequately built into Peirce’s basic semiotic structure which is object-sign-interpretant (sign = not necessarily a linguistic symbol, but anyway something interpretable as saying something about something; interpretant = interpretation in the sense of product, rather than activity, of interpreting; there’s an old word “interpretament ” from Medieval Latin interpretamentum (see Charles Short’s definition via Perseus Tufts) which means an interpretation as a product; Peirce probably knew the word well but perhaps thought it too lengthy).
Peircean triad,
augmented to a tetrad
by incorporating
a semiotically determined,
object-observant subject.

Object. Sign. Interpretant. Recognizant. Peircean triad, augmented to a tetrad by incorporating a semiotically determined, object-observant subject.

Absent, from the object-sign-interpretant triad, is the semiotically determined relationship whereby the sign and its interpretant (which is a mere construal) are brought to the experiential test against the object which they merely represent. Pragmaticism itself agrees that there is no way but the passing of such a test for sign and interpretant to merit recognition as legitimate and truthful. A sign is “almost” its (the sign’s) object and conveys information about the object, but is not the object, so familiarity with the sign is not familiarity with the object. The interpretant is the sign’s meaning clarified, such that the interpretant itself is a sign (a) of the object and also (b) of interpretant’s “predecessor” as a sign of the object. Peirce, unlike so many before and since, saw that there’s much more to signs as a general phenomenon (general like statisticality and information) than “signifier” and “signified.” Not only does a sign require and address itself to interpretation, but the interpretant itself is a sign, a night’s womb to a further interpretant dawn, just as a translation is into something itself further translatable, a ramification has ramifications, and meaning means, means ceaselessly and sometimes to our chagrin (Merleau-Ponty said “we are condemned to meaning”) — and so the interpretant is a sign, promoting and provoking further interpretation. But the interpretant, though it’s a sign, is not an object’s “mere” sign which one would never guess is also a sign about a previous thing-as-sign about the same object, instead the interpretant is a sign having reference to an interpreted sign as well as to the object, and in fact practically all signs are like this in the interpreter’s perspective, links in chains stretching both fore and aft, just not always with clarity (so usually it’s a relative question, a role question — “is it the sign or the interpretant?” — just like the question of which codings are encodings and which are decodings), and Peirce unswervingly conceived the interpretive chain as operative all the way down to the level of the infinitesimal and the truly continuous which he ultimately regarded as beyond all multitudes or Cantor’s alephs. Actually so continuous or not, perpetual interpretation is sometimes to our chagrin, yet it also, for instance, lets us see around the bends of planets and hearts. But the caveat, the string attached, is this, that, the mere fact that you interpret, understand, take some appearances, some events, some words, as signs about some object, doesn’t mean that you really know or really reasonably believe or learn anything about that object, i.e., and, as Peirce states, sign and interpretant contain none of the needed familiarity-dependent understanding of the object, and such familiarity and experience must be had collaterally. Therefore, (a) the collateral experience is no mere sign or interpretant in the sense and in those relations in which it is the collateral experience, and (b) only through changes of semiotic frame of reference can such experience be analyzed into signs and interpretants. Likewise (and only likewise) can interpretants be analyzed into pre-interpretant signs, and signs into objects — every supposed reduction of such experience to signs and interpretants marches onward all too powerfully, to reduce away the classic semiotic triad itself. To say that we can shift frames and regard the recognition as a sign or interpretant of some other object (as it likely is) is to say that we can simply bar our eyes from the question of what is the semiotic status or role of the collaterally object-observant recognition of the interpretant and sign as corresponding to the object already in question. We could likewise ignore, as some do, the questions of what is the interpretant, even what is the sign. The sign’s object is not per se the recognition, for then signs and interpretants would never be needed. Nor is the verificative recognition some representational, qualitative, or reactive aspect or relation of interpretant or sign or sign’s object; if that were so, then one or more, among object, sign, and interpretant, could somehow, or in some combination or relation, contain conveyable familiarity with the sign’s object. In sum, it is completely ruled out, that the recognition is interpretant or sign or object or any representational, qualitative, or reactive aspect of any of them or relation among them. Could the recognition still be some kind of relationship among object, sign, and interpretant? Yes, if and only if that relationship contains familiarity with the object. Object-observation-based recognition of the sign and interpretant as corresponding to the object ipso facto contains familarity with the object and conveys such familiarity across the memory and experience of whatever mind (or “commind” or “quasimind”) which it inhabits. There are chains not only of interpretation but also of recognition and verification which are not mere interpretation, mere construal.

Left to its “own devices,” semiosis could not learn the difference between sense and nonsense; it could still, as it does, lead to hopeful monsters, like they say of biological evolution, but hopeful monsters which go untested, unchecked.

Instead there is to philosophically acknowledge as a semiotic phase or element the observational / experiential recognition formed as collateral to sign and interpretant in respect of the object and to explore it as a further complex of semiotically determined relationships. It might be a recognition of a hat being worn as one expected, or a recognition of one’s hopes as unexpectedly fulfilled; or a recognition of a given interpretant as resting soundly enough upon recalled experience. Beyond the category of value, import, meaning, etc., is the category of consistency, truth, validity, soundness, and not just in the sense of the shallow or trivial but even and especially in the sense of the challenging deep. Beyond the end or culmination or actum as actualization, comes the check or checking, the sustentum, alitum, or altum as borneness, sustention, a system’s agencies’ balancement and stabilization. “Beyond” in some logical sense. The check doesn’t shed the end or leave it behind.

Edward Dahlberg & Charles Olson. Conn. U's Charles Olson Research Collection.
Edward Dahlberg & Charles Olson
Conn. U's Charles Olson Research Collection
Interpretation and recognitive observation cannot culminate and solidify except as the stuff for more of themselves. They, along with objectification and representation, keep on into, through, out, around, continually renovating and occasionally reforming. Charles Olson said that Edward Dahlberg pounded it into him as a poet that “every perception leads DIRECTLY and IMMEDIATELY to another perception.”

Within the semiotic triad of object, sign, and interpretant, there is no way to understand how a sign can verify or (dis-)confirm anything — no way to capture the common idea of evidence as being a kind of sign. Only if a sign has not only purport or meaning clarifiable into an interpretant, but also observational legitimacy or authority solidifiable into a recognition, a “recognizant,” is it possible to understand the evidentiary and verificatory character of a sign — for a “mere” sign can after all, on the basis of experience with the object, count observationally in some regard as the object itself (and do so without being confused with the object). Without all this, signs would require constant collateral observations comprehensively checking them. Signs would fail to expand our horizons and to let us peer, so to speak, around corners near and far. If signs always failed in this way, they would succeed in little else. And, for its part, experience would be impoverished of all that continually leads it beyond itself.

Now, an observation or experience, for its part, is semiotically determined, enriched, informed, fortified — how? How else? than by being formed as collateral to sign and interpretant in respect of the object.

To be able to clearly conceive and express such things — this welcome slackening of a counterproductively over-strict need for direct experience of the object in order for the sign to work at all, and a simple statement of the relationship through which semiosis determines and informs experience — are gains reached through recognizing the collaterally based recognition as a stage and element of semiosis. Now, Peirce classes the mathematical diagram as an icon, defined by resemblance to its object(s), in spite of the chasms of apparent dissimilarity across which the mathematical diagram is applied through imaginative and logically supported bridges of equivalences. Peirce says that the mathematical diagram is a sign subject to mathematical observation and experiment, but still classes it as an icon rather than as a sign defined by object-observational-&-experimental legitimacy, the recognition which it would merit on the basis of (collateral) experience with its object(s) (I call such a sign a “proxy”). Peirce does not recognize observational legitimacy as a dimension of sign power in general nor as a way to define a class of signs defined like (a) index, (b) icon, or (c) symbol by the sign’s relation to a semiotic element such as (a) object, (b) sign itself, or (c) interpretant (respectively). Peirce does not do so, since collateral observation or collaterally based recognition are not, for Peirce, a semiotic element such as object, sign, or interpretant. Yet collaterally based recognition is the most characteristically logical or semiotic stage if indeed semiosis and logic transcend calculation and information theory by being sufficiently u n b o u n d to code in order to be about verification and (dis-)confirmation of signs and of interpretations and of their systems and codes. After the culminative interpretation, comes the solidificative recognition.

To hold instead, as a last resort, that verification is not a semiotic or logical stage but is part of an “outside” of semiosis, an “outside” into which semiosis merely characteristically joins or bonds, is to deny the verificational heart of logic’s guiding research interest; it is also to deny, that the characteristic verificative joining, bonding, or anchoring of objects, signs, and interpretants to recognitions is characteristic of semiosis. It is to deny the depth and rootedness of the world’s logical or semiotic aspect and to make the wrong kind of distinction between logic and facts. In the idiosyncrasy-shunning spirit which it cultivates, one should just as well exclude surmise from the kinds of inference.

It is also to be noticed that the sign qua pre-interpretant may saliently have arisen in, and as, observation and experience of the particular or general object (object-experience by another mind (or quasimind) or by the same qua other or by the same qua same), and that this is really a distinctive perspective of the pre-interpretant sign — that of a judge or judgment, measurant or measurement, describer or descriptor witness of the object, perhaps being an aspect or part of the object itself, anyway something which (a) may, separately, be interpreted as a sign implying said object when the object is unavailable (at least in the relevant regards) (e.g., “The car is long and orange ? It could well be Jack’s”) and (b) means more than, ramifies beyond, that which it patently is. Furthermore its familiarity, such as it may be, with the general or particular object may be the interpreter’s own but is generally unconveyable (qua familiarity by the sign qua sign) and is at any rate generally less far-reaching than the (further) meaning; anyway the thing, in those regards in which it is a sign about the object, does not “contain” familiarity with the object, i.e., such that the interpreter could imbibe of or absorb the familiarity, though for instance a person familiar with an object may for that very reason be a sign to another person about that object. With the kind of generalization possible through such Peircean conceptions as the quasimind, it is probably possible to legitimately analyze every case of a sign both as such a measurer/describer and as already-an-interpretant now under further interpretation. These two perspectives are distinguishable analogously as are encoding and decoding in information theory. However the distinctively pre-interpretant perspective is that of such a (possibly other) measurer/describer (the “encoding”) and in this sense a complete pre-interpretant sign stands as a judgment which tends to persist beyond its origin. The judgment itself may tend to stand less in question, than its meaning or ramifications do, but whatever to any extent stands in question solicits to that extent interpretation.
1. Objectification.
Conception, percept, image, etc.

Correlated with: graph theory & maths of many-to-many relationships, deductive theory of optimization, the (young) field of inverse optimization problems, and sciences of motion & forces.
3. Interpretation.
Inference to a conception, percept, image, etc.

Correlated with: algebra & maths of many-to-one relationships, deductive theory of information, ampliatively-inductive information theory & cybernetics, and biological sciences.
2. Representation.

Correlated with: enumerative combinatorics & maths of one-to-many relationships (measure & integration, etc.), probability theory, statistical theory, and the material sciences.
4. Recognition.
Inference to a judgment.

Correlated with: maths of order and one-to-one relationships, deductive theory of logic, philosophy, and the sciences/studies of intelligent life.
Aquinas. The Common Sense & the Proper Senses. Grasping appearances & combining & distinguishing them across sense modalities. Concept. ← Imagination. (Avicenna, a.k.a. Ibn Sina, divided imagination into two à la reminiscence & memory. Aquinas disagred.) Mentally storing, retrieving, distinguishing, & combining appearances. Judgment. ← Cogitative power. (In animals, 'estimative' power, i.e., instinct.) Grasping the _intentiones_ (meanings). Reasoning. ← Reminiscence & memory. Mentally storing, retrieving, distinguishing, & combining (things in terms of their) _intentiones_.
This happens to bring me to the question of how I rework Peirce’s alignment of sign-object-interpretant with the classic conception-judgment-inference trichotomy rooted in Aristotle. An inference to a conception (or more generally, to a conception, a percept, an image, etc.) is to be distinguished from an inference to a judgment, just as conception and judgment themselves are to be distinguished. For instance, a equational inference from conception to conception deserves, I think, to be called a calculation, whereof the mathematical theory is algebra, a math of many-to-one relationships, which is distinguished from maths of order, one-to-one relationships, and mathematical-induction applicability conditions, à la (calculative) conception inference from (math-inductive) judgment inference, respectively. Algebra and maths of order philosophically deserve such distinction at that level where one is distinguishing two wings of combinatorics: distinguishing graph theory (many-to-many relationships) from the theory of counting, a.k.a. enumerative combinatorics (one-to-many relationships), à la conception from judgment, respectively. Indeed there is no philosophical reason to generally conflate the many-to-one with the one-to-one at that level where one generally distinguishes, from them and from each other, the many-to-many and the one-to-many. Now, I suspect that conception (or percept, etc.) and judgment, qua pre-inferential, can be distinguished from the same qua inferential, as being reached through a process for which the term “abstraction” may fail only by insufficient generality. A percept impresses itself on a mind, for instance. The point is the “extractive reception” of the conception (or percept, etc.) and judgment from an object as experienced.

I discuss some of the above semiotic matters at greater length in the post “Semiotics: collaterally based recognition, the proxy, and counting-as.”

As for semiologists, some of them are no Peirces and seem simply to ignore the critical dimension in the phenomenon under study and to blandly assume that the people are the sheeple manipulated by signs. Whatever one thinks of people generally (I’ve muttered a few times about “the sheeple” myself in a bad mood), this ignoring is simply and uneuphemizably stupid for a student of signs to practice. One may think that the people are sheep deeply asleep, but to discuss them as if they flatly lacked any critical and confirmation-seeking attitudes or approaches is to ignore a basic dimension of the phenomenon, the very dimension from which science itself rises, the dimension which, at bottom, distinguishes intelligent life from vegetable life. Systems and processes such as minds, human users of language, etc., sufficiently u n b o u n d to codes and sign systems in order to regard them critically and with a need for trial, testing, confirmation, revision, etc., are the subject where we look beyond calculation theory (algebra), information theory, and biology, to logic, philosophy, and the human and social sciences. The sufficient unboundness-to-code and the structuring of the intelligence to deal with questions of factuality, legitimacy, authority, etc., give, I think, to human meaning, linguistic and otherwise, an extra richness, a cornucopia of extra dimensions, like it or not. This is another one of those senses in which the intelligent being is evolution increasingly self-aware and self-possessed, be it soever much a child in unknown bigger pictures. Of course one’s views of these things will be warped if one views things like truth, legitimacy, etc., as nothing but some pretentious “narratives”; such views are self-hanged in the drama of logic, as shown by Plato and Aristotle in their exposures of sophistry and wishful thinking.


Object(s)-to-object(s) relator. Whetherhood (Y/N, probability, informativeness, logical conditioning, etc.). Attribute. Substance (you, me, the lamp post, etc.). In one of the charts above appears the sequence (1) object(s)- to-object(s) relator (I’m thinking of things like mathematical operations, functions, antiderivatives, analytic and simultaneous equations, etc.) (2) “whetherhood” (Y/N, informativeness, probability, etc.), (3) attribute, and (4) substance (this man, this horse). If you think that they should at least be in the opposite order, indeed maybe they should, but the numbers serve at least as tags so that sequences can be permutated together. I’m not convinced that I should simply call object(s)- to-object(s) relator a “relation” for it may be too general. Also, even seemingly layman-accessible discussions of mathematical category theory on the Internet are few and far between. John Sowa, in an email Jan. 19, 2006, told me that a morphism has to be one-to-one or many-to-one; antiderivatives (which are one-to-many) and analytic equations (which are many-to-many, e.g., x^2 + y^2 = 1) are not morphisms).

Anyway, the conceptions in that sequence constitute my rough inductively generalized take on the issue. My conceptions of the four broadest categories are meant to, among other things, reflect the four broadest categories of things dealt with in all research — the respective subject matters of (1) pure mathematics, (2) applied yet mathematically deep mathematics, (3) abstract but positive-phenomenally deep sciences/studies, and (4) the “special” or concrete empirical sciences/studies. The objects of all such studies — as opposed to the subjects — may tend to the universal wherever possible, but that is a different question. The research families and the categories correlate to four typical logical-quantity-perspectives which are logically based, in such a way as gives them a kind of symmetry and exhaustiveness. They are tetrachotomies, four-way logical divisions of given wholes of logical possibilities.

In other words, I am less interested in a general theory about categorizing, than in pursuing broadest categories. One might say that, notwithstanding the non-idiosyncrasy of logic, I want to get the geography going before I worry too much about the geology and the plate tectonics. To put it another way, there’s set theory, with its infinite hierarchies of infinite sets, and then there are the infinities actually often encountered in mathematics generally — the countable infinity of discrete points on a line, the uncountable infinity of the points in the continuum, and an even more populous infinity, that of curves or functions in the continuum (add discrete finite sets into the mix and you have a foursome). It might be added, that a physical geography may find patterns that an ealier-conceived geology had not thought of. In particular, there seems a pervasive pattern of inverse relationships which I’ve hardly addressed on this Website. I’m starting to need that “plate tectonics”!

Peirce seemed a bit reluctant to use the word “accident” among the categories. Why not an alternative like “property” or “attribute”? Maybe because: “Property” originally meant something like "idiosyncrasy" (not in the sense of “quirk,” but simply of “unique/distinctive but non-essential”); and, through Aquinas and others, “attribute” acquired a sense of “essential attribute.” I long used “accident” in accordance with Peirce. I’m not quite happy with “attribute” which I’ve started using now. I still like it better than “property” it seems less ill-suited to items which we would not often regard as properties, e.g., a thing’s market value or its being within earshot of something loud. I regard the apportionment or “copulation” of attributes to substances as forming still another category, the category which I call “whetherhood,” a kind of variety-enriched version of the olden anitas. Whetherhood’s modes include formal and material truth values, logical dependences, probability, information or informativeness (novelty, distinction or difference that makes a difference), etc.; without them you could not relate attribute to substance and they themselves are not directly affirmable of substances nor are they substances complete with attributes. As apportionments of attributes they are a way of relating attributes to attributes, with regard to relative frequency, overlap, etc. I regard as a further category oneness, otherness, etc., among substances (or among attributes treated as substances, i.e., hypostatized attributes), which I call “object(s)-to-object(s) relators” but which, if I were more daring, I would call “ofhood” or “thanment” (as in “double of..,” “three more than..,” “inverted order of...,” etc., i.e., I’m thinking of operations, functions, antiderivatives, etc.). Whetherhoods apportion explicitly or implicitly in terms of a totality, a universe of discourse, etc., such as to amount to valuations of predicates or propositions or outcomes or states of affairs, etc. Whetherhoods are a way of dealing with attributes simply as divisions of a universe, while object(s)-to-object(s) relators are a way of dealing with substances simply as generic objects endlessly multipliable and rearrangeable. Actual research involving these things necessarily goes far beyond the elementary conceptions outlined here — e.g., apportionment among universes instead of a division of a universe, and of course the development of the imaginative apparatus of set theory. I discuss categories, logical term-quantities, etc., in the post Logical quantities, categories of research, and categories. (There one may learn why the heck I keep saying “polyadic singulars” in my logical-quantity tables.) The following table mostly repeats things from the tables above in this post, but shows some of the alignments more plainly.

Research family Typical logical-quantity perspective (in spirit) Essential subject matter category Correlated grammatical form Correlated kind of abstraction Typical mode of conclusion drawn
1. Pure mathematics. Universal-cum -general,
i.e., perspective of the universal that isn’t the universe or totality.
Object(s) -to-object(s) relator,
e.g., operation, function, antiderivative, etc., e.g., “inverse sequence of...,” “double of,” “multiplicative product of...,” etc.
Subject-formative functor. Info-adding-&-removing abstraction,
4. Reversible deduction
(preserves info;
preserves both truth & falsity)
2. Applied-yet-deep mathematics
(deductive theories of optimization, probability, information, logic).
Universal-cum -monadic- or-polyadic -singular,
i.e., perspective of a universe, collective totality, gamut.
probability, info (qua novelty, newsiness), logical conditioning & compounding, etc., e.g., “not,” “with a probability of 57%,” etc.
Predicate-formative functor. Info-increasing abstraction,
3. Non-reversible deduction
(decreases info; preserves truth but not falsity).
3. Abstract-yet-positive- phenomenally-deep sciences/studies
(ampliatively-inductive theories of inverse optimization, statistics, information, philosophy).
Special-cum -general,
i.e., perspective of the non-universal general.
Attribute. Predicate. Info-decreasing abstraction,
2. Ampliative induction
(increases info; preserves falsity but not truth).
4. Idioscopy, the special sciences/studies
(physical, material, biological, human/social).
Special-cum -monadic- or-polyadic -singular,
i.e., perspective of singulars that aren’t the only ones.
e.g., you, me, the lamppost, etc.
Subject. Concrete (info preserved),
1. Abductive inference, surmise
(adds & removes info; preserves neither truth nor falsity).

Sundry remarks

In some cases (e.g., the modes of inference) I had arrived at my tetrachotomical version of a Peircean trichotomy before I had read Peirce or learned anything about him beyond his having been a philosopher, though in those cases I tended not to understand my tetrachotomy nearly as deeply as Peirce understood his trichotomy. Peirce has been a boon to me (and I hope he wouldn’t mind my applying to his structures the word “triastic” which he never used). I never even considered tetrads or any kind of polyads until I had read Peirce for a while.

Aristotle’s Four Causes seem a slightly rough tetrachotomy, in some ways a patchwork selection from corresponding terms of a set of closely related tetrachotomies, but certainly much more than a mere list of four items.

I also see four-folds and cosmic quadratures in: the classic info-theoretic source-encoding-decoding-destination setup; the special-relativistic lightcone; special-relativistic kinetics; the classification of research; the classification of areas of human social activity; and sundry. In pursuing these subjects, I’ve found that there’s not exactly a forest of us “fourists,” of the dusty-shelf philosophical variety anyway.

Given the series:
(1) many-to-many — graph theory — optimization — distance, difference (with direction)
(2) one-to-many — enumeration — probability — ratio
(3) many-to-one — group theory — information — logarithm & base
What is the next step?
(4) one-to-one — order, relations — deductive logic — (fill in the blank) base, root (arity, adicity, valence, etc.

Update January 20, 2014: I've cut out a bunch of speculation about the roles of tetration's inverses (hyperlogarithm, hyperroot). I remember years ago speculating via email with Chris Lofting that the "next step" mentioned above would be one of tetration's inverses (hyperlogarithm or hyperroot). At that time I was thinking of arity only in regard to truth values. For a little more discussion, click here. End of update.

Indeed, logical issues grow potentially so complex so rapidly, and sometimes at least seemingly tetratively, that it’s hard to believe that the mind deals with them in anything like the way it deals consciously with logical issues. The conscious mind gets confused and, as a solution, ends up abstracting the logical structure which becomes another chunk or three in the soup. Often many little chunks floating randomly. Same scene, different metaphor, the bicyclist through excess of self-consciousness falls off his bicycle. It’s natural for the mind to shun such crackups. But tetrastic structures are only a little beyond the kind of starting-gate logical complexity (dualistic) with which most people prefer to deal. And if there’s something “more,” something irreducible, in whatever sense(s), beyond the dualistic, beyond the triastic, about a given tetrastic structure, it will be most hard to glimpse if one always starts by chunking down to twofolds. Also, the chunking may be arbitrary in terms of a given foursome. There’s no pre-eminently optimal way to simplify the foursome “starts continues stops refrains” to a twosome; instead, a term usually ends up omitted from common quasi-formal thinking — usually it’s something like “start-continue-stop.” “Continues not to” doesn’t even have a fully general word in English; the nearest that English comes are the volition-suggestive words “refrains” and “abstains.” Which is kind of ironic, considering how miserable or happy people make themselves in pondering what keeps not being but used to be or would have been (a real and often neglected economic concern in the form of opportunity cost) or still! hasn’t happened yet or etc. Hardly anybody even notices that, though English has words for creation, preservation, and destruction, it has no plain word for keeping something from existing. (As for why “refraining” should be regarded as coming after the other turns of becoming rather than before, the reason seems to arise in applications. Logically of course the refraining slips in smoothly either before or after the other three; something continues not happening for some period until it starts happening just as well as it continues not happening for some period after it stops happening. But it’s hard to regard an initial refraining as a staying-ended in parallel to a continuing as a staying-started. That’s an abstract half-aesthetic concern in application. The concrete concern comes when one seeks that which specifically embodies or represents a stage or turn of a thing’s becoming. What pertains more specifically to a thing, that which precedes its beginning, inception, conception, or that which follows its end and culmination? After the end comes the wake, the trail, the track, the shell, the husk, the structure abandoned or inhabited, the evidence, what I call the check, that which embodies the continued endedness and having-culminated of a thing (Aristotle's conception of entelechy is a special case of this conception). Such check stands in relation to the end or culmination or actualization of any given moment, as the refraining stands in relation to the stop. A KGB head once said, nothing moves through time and space without leaving a trail that gives it away. Any pre-beginning or precursor which represented a thing so specifically as the thing’s check does would tend to be regarded as part of that thing’s beginning already. Nyet? )

In one tetrachotomy, a fifth term emerges as the common element of which certain pairs of the first four terms are pairs of complementary forms. It seems no violation of any reasonable tetrastic thesis, instead it seems a natural enough expression of an aspect of a logically fourfold structure, like introducing a quantificational T (for “quantificational truth”) as the tautologous alternative between opposite corners of the logical Square of Opposition
. I would take such a fifth term to be usually formulable as the generic term specially applicable to all of a given four. Overall the patterns which I find seem tetrastic. I’ve tried to hew to the question of “why tetrastic?” but find that this has become a sort of overview. Perhaps my other posts here may serve as more-specific discussions of “why tetrastic?”.

Quite a few of my tetrastic structures have either emerged as, or taken on the cast of, revisions of Peirce’s triastic structures. Paul Burgess, in a 1988 paper “Why Triadic? ” available at his Website, reviews past attempts to add fourth elements to Peirce’s threes. As far as I can tell, my efforts don’t resemble those.

At the opening of his fragment “Trichotomic” (EP I: 280), Peirce evocatively says:
TRICHOTOMIC is the art of making three-fold divisions. Such division depends on the conceptions of 1st, 2nd, 3rd. First is the beginning, that which is fresh, original, spontaneous, free. Second is that which is determined, terminated, ended, correlative, object, necessitated, reacting. Third is the medium, becoming, developing, bringing about.

If the means-end dichotomy is useful or illuminating, such that any philosophical thought is built on it, then it is both necessary (by the standard of information-theoretic simplicity) and certainly tempting, to at least try to extend it in correlations to all FOUR turns of becoming: starting, continuing, stopping, and abstaining or “continuing not to.”

1st is the BEGINNING: forceful, leading, ice-breaking, vying, trying, exploring, deliberating, decision-making, conceiving, designing, optimizing, adoptive, appropriative, sensitive, magnifying, exaggerative, creative, “so be it?” 2nd is the MIDDLE OR MEANS: resource, fostering, patient, nourishing, tempering, mediating, harmonizing, normalizing, finessing, So be it? So be it. Is it? It is. executing, cooperating, adaptive, processing, “so be it.” 3rd is the END: converting, rewarding, motivating, consuming, affection, communing, expressing, interpreting, precising, picky, adjusting, enlivening, perfecting, sharpening, culminating, “is it?” 4th is the CHECK OR CHECKING, solidifying, establishing, ruminating, assimilating, confirming, integrating, cognizing, grounding, staunch, structured, supports, checks, and balances, building, renovating, evolving, “it is.”

Here is a down-to-earth example.

In human affairs such that happiness, satisfaction, etc., are the end and the investing of work, preparation, etc., the means, what is the beginning? It might be interesting for the reader to stop for a moment and think of an answer to that before reading on.

Beginnings, Archaí. Middles, Mésa. Ends, Télê. Checks, Íchnê, Élenchoi, Entelecheíai.Trying, attempting, deliberating, undertaking, trying out, are, I think, the beginning in the same sense that happiness and satisfaction are the end and rational work the means. What is the check (or the checking)?

Knowledge, familiarity, recognition, etc., are the check. One often checks because one knows that nothing is guaranteed, that one is fallible at best, and that success usually involves some luck. One rightly feels at least somewhat lucky in success. Luck, for its part, favors those who try.

Closely coupled yet mutually independent two-value parameters seem quite common. Again, a reason to look for extended or recurrent tetrastic patterns in philosophically interesting structures is that it seems that very few have tried or, at any rate, pursued it far. And related to that is another reason why I like Peirce — he saw his triads and trichotomies in proliferant pattern, pattern whereby he navigated seven seas of metaphysics and indeed all the philosophically journeyable worlds that that astounding polymath could find.

HOME || Deductive vs. ampliative; also, repletive vs. attenuative || Plausibility, verisimilitude, novelty, nontriviality, versus optima, probabilities, information, n-ary givens || Logical quantity & research scopes [...] || Telos, entelechy, Aristotle's Four Causes, pleasure, & happiness || Compare to Aristotle, Aquinas, & Peirce. || Semiotic triad versus tetrad. || Tetrachotomies of future-oriented virtues and vices. || What of these other fours? || Fantastic Four. || Why tetrastic? || The Four Causes, their principles, special relativity, Thomistic beauty. || Logical quantities, categories of research, and categories. || Semiotics: collaterally based recognition, the proxy, and counting-as. || A periodic table of aspects of humanity [...]