Quadrant Model of Reality

ADDICING FOURTH TO PEIRCES THREE- TETRADIC THEORY

http://www.paulburgess.org/triadic.html

Thirdness, in its logical aspect, is triadic. It is a First bound together in relationship with a Second by the mediation of a Third: "The beginning is first, the end second, the middle third." (1.357) It is combination, pattern, structure, mediation, continuity. A monad can form no combination with another; two dyads can join together only to form another dyad (think of two lengths of pipe screwed together); but triads can combine into arbitrarily complex structures (think of a tinkertoy set, or the colored plastic beads in a chemistry class which can snap together to form models of molecules).9 Thus, argues Peirce, there is from the logical side no need for a category of Fourthness.10 Peirce equates Thirdness with the structure of hypothetical or "abductive" thought, and with the conjunction into an organic unity in the act of perception of imputed individual entities and the qualities attributed to them. (Cf. 1.471-520, 1.551, 2.694-794, 5.171-174, 5.590-604)

Peirce's Sign: Why Triadic?

It should now be very clear that triads and trichotomies form the warp and woof of Peirce's approach to semiotics. The debate over this triadicity has taken two divergent and incompatible avenues: proposals for a category of Fourthness which question the sufficiency of Peirce's semiotic, and proposals for a reduction to dyadicity which would render the semiotic triad unnecessary.

"One, Two, Three... But Where Is the Fourth?"

At least three proposals have been made for a Peircean category of Fourthness: those of Donald Mertz, of Herbert Schneider and Carl Hausman, and of Carl Vaught.

Mertz's proposal is the most easily disposed of. He examines Peirce's argument for the irreducibility of triads, as illustrated in "A Guess at the Riddle":

...the fact that A presents B with gift C, is a triple relation, and as such cannot possibly be resolved into any combination of dual relations. Indeed, the very idea of a combination involves that of thirdness, for a combination is something which is what it is owing to the parts which it brings into mutual relationship. (1.363)19

Mertz grants Peirce this argument, but denies Peirce's claim that "the quadruple fact that A sells C to B for the price of D" can be reduced to "a compound of two facts: first that A makes with C a certain transaction, which we may name E; and second, that this transaction E is a sale of B for the price D."20 Mertz notes correctly that Peirce would express this symbolically as R(A,C,E).R'(E,B,D) = S(A,C,B,D), but then asserts that the second fact should be expressed dyadically as E = T(B,D) so that, by substitution, R(A,C,T(B,D)) = S(A,C,B,D), an irreducibly tetradic relation. From another angle, Mertz states that no two of the triadic relations "1) A sells C to B, 2) C is sold to B for D, and 3) A sells C to D" can be conjoined to arrive at the original tetradic relationship. (It is neither here nor there, but puzzling: why does Mertz omit "A sells to B for a price D"?) Mertz concludes that Peirce was blinded to such arguments by his fascination with logical diagrams which Peirce named "existential graphs."21

Mertz's argument may be convincing on its own terms, but Mertz is manipulating Peirce's polyads in ways which bear little resemblance to any way in which Peirce ever worked with them. In his first argument, Mertz is attempting to join a dyad with a triad without identifying which of the elements in the dyad is being joined to the triad. His second argument involves joining two triads by joining two elements in one with two elements in another. The former procedure has no precedent in Peirce, and the latter under Peirce's methods ought to yield a dyad, not a tetrad as Mertz thinks Peirce would have expected. Since Mertz gives us no clue as to how his procedures relate to those of Peirce, we have no way of judging whether Mertz is doing anything really entailed by Peirce's project, or whether Mertz is simply arriving at different conclusions because he is working out of different assumptions.22

A more detailed proposal for Fourthness comes from Herbert Schneider. Schneider concedes three categories to be adequate for dealing with cognitive processes, but argues for "importance" as a category of Fourthness. He notes that, for Peirce, any purpose or good has meaning only in relation to a completely general summum bonum. "No Kantian idealist could have stated this conception of moral science more formally."23

Schneider observes this scheme does not accommodate norms which might apply "even in the absence of a summum bonum," itches that call to be scratched for their own sake. Such norms he proposes as a phenomenological aspect of Fourthness: logical import is Thirdness, vital importance Fourthness. Satisfaction may comprise either the Thirdness of achievement or the Fourthness of satiety or contentment. The moral self-control of Thirdness in pursuit of an abstract summum bonum is only an abstract "intellectual framework" until it is taken up into the "concrete universal" of the moral self-criticism of Fourthness. Fourthness supplies what depth psychology, but not the Kantian "moral law within," acknowledges.24

In logical terms, Fourthness would constitute a temporal sequence, though one which is an absolutely discontinuous string of points superimposed on the triadic continuum. Triadic semiosis is "prospective and cumulative"; tetradic semiosis adds a fourth factor which is non-cumulative, but "retrospective" along the hierarchy of categories, giving "meaningful individuality" to instances of Firstness. Since Firstness and Secondness "look 'forward'" to Thirdness while Fourthness "looks back" to Firstness, Thirdness in a sense "governs" Fourthness while Fourthness provides the steam to "drive" Thirdness.25

Carl Hausman provisionally adopts Schneider's scheme, applied to both ethics and aesthetics with "importance" or "value" as a possible category of Fourthness. Hausman notes that Schneider's suggestion rejects "Peirce's own principle that a highest good makes specific goods intelligible," but that, quite regardless of this, Schneider puts a finger both on the problematic status of value in Peirce's semiotic and on an apparent "special connection" between value and Firstness.26

Hausman turns to Peirce's categorial classification of the sciences to investigate the relations among value and the categories. Peirce categorially divided philosophy into phenomenology, normative science, and metaphysics, and normative science into aesthetics, ethics, and logic (1.186). Just as normative science rests on phenomenology and on the prior field of mathematics (within which two fields Peirce constructed his categories), so the categorial hierarchy is reflected in an order of dependence in which "ethics rest[s] on aesthetics, and logic on ethics," all three but especially aesthetics dependent on phenomenology.27

All three sciences, according to Peirce, distinguish a kind of good and bad, "Logic in regard to representations of the truth, Ethics in regard to efforts of the will, and Esthetics in objects considered simply in their presentation," with both logical and ethical goods presupposing esthetic good. Remarks Hausman: "This is why Peirce says that ethics must appeal to aesthetics for aid in determining the summum bonum."28 Considered thus, value itself is not a fourth category. Rather, it seems related but not identical to the teleological thrust built into Peirce Thirdness, and bound up with the categorial hierarchy.29

Hausman notes that the categories can be related by what Peirce calls discrimination or distinction, prescission, and dissociation (1.353). We can prescind Firstness and Secondness from value, but neither value nor Thirdness from each other. Hausman speculates that value and Thirdness could be separated by "discrimination," "so interdependent that they are co-present as mutual grounds for one another," though this is difficult to determine since Peirce is none too clear as to what he meant by the term.30

I think that Schneider and Hausman are correct to point out the problematic place of value in Peirce's thought. As Peirce himself was aware, as a natural scientist he devoted more attention to logic and less to ethics and aesthetics (2.120, 2.197).31 And I think the two are correct to note a connection between value and Firstness, though I consider Hausman's attempt to bring value in tandem with Thirdness more economical than Schneider's attempt at severing it altogether from Thirdness, and thus from a summum bonum. But I would concede Schneider's point that Peirce's account, as it stands, does justice better to the apollonian than to the dionysian side of human existence.

However Peirce, especially in his explorations of phenomenological Firstness (cf. 1.312-316) and his discussions of vagueness (cf. 6.494ff.), is far more sensitive to these issues than Schneider will grant. And a sequence of monads co-present with the semiotic "time line" which present themselves immediately and spontaneously in the interpretation of the sign ought to suggest a factor of the Peircean sign already familiar to us: the immediate interpretant.

Although Peirce is not explicit, the logical and phenomenological characteristics which Schneider attributes to Fourthness all fit well the spot the immediate interpretant occupies in Peirce's semiotic. By its relation to dynamical and final interpretant, the immediate interpretant is certainly "related but not identical" to the teleological thrust of Thirdness. And energetic immediate interpretants would "give a meaningful individuality" to Firstness. Finally, if Hausman's interpretation of "discrimination" is correct, then the immediate (unlike the dynamical or final) interpretant would indeed be related to the sign by discrimination.32

The most detailed proposal for Fourthness comes from Carl Vaught. Vaught points out that, phenomenologically, there is a similarity between things such as a left hand and a right hand which does not seem reducible to a combination of identity and difference. According to Peirce, most spatial relationships can be dealt with predominantly in terms of the Secondness of objects in dynamic interaction or relative location. Yet as Vaught notes, Peirce vacillated over whether right and left can be distinguished in terms of Secondness-- for example, indexically (2.290)-- or whether Thirdness must be invoked, as for example in what physicists call the "right-hand rule":

Thus, your right hand is that hand which is toward the east, when you face north with your head toward the zenith. Three things, east, west, and up, are required to define the difference between right and left. (1.345)33

Vaught argues that not even Thirdness suffices to distinguish right from left, since the definitions of east, north, and zenith themselves presuppose a distinction between right and left. Right and left, Vaught concludes, embody a similarity irreducible to identity and difference.34

In a parallel manner, on the logical level, Peirce insisted that analogy is reducible to a combination of univocity and equivocity, and is not a mediating third between them (3.421, 3.483-485, 1.34). Thus, a four-term analogical relation would be reducible to a combination of simpler terms. But Vaught argues that just such irreducible analogical tetrads occur in Peircean semiosis.

For semiosis gives rise to a sequence of signs and interpretants, each at a slightly different moment in time, and as Peirce's later distinction between immediate and dynamical object implies, the object itself is not static but constitutes a corresponding temporal sequence of objects in interaction with the sign/interpretant sequence. Within the vagueness inherent in Peircean semiosis then, says Vaught, lies precisely a similarity, irreducible to identity and difference, which embraces interpretant at t1, interpretant at t2, dynamical object at t1, and dynamical object at t2 in a tetradic analogical relationship which due to the vagueness is not precisely reducible to any combination of triads. As an example of this, consider a legisign which grows and develops over time: its form at any two instants can, under this proposed Fourthness, be seen as related by an irreducible analogical similarity. Likewise, the similarity between right and left can be seen only through a judgment of analogy.35

Vaught's argument is both closely reasoned and richly textured, very much in the spirit of Peirce's own approach. If similarity is not to be understood (as Peirce saw it) as reducible to some combination of univocity and equivocity, then Vaught's argument is probably correct. But I think counter-arguments can be mounted on both logical and phenomenological fronts.

Logically, if the sequence of interpretants in Vaught's argument on analogy are considered, not as a sequence of discrete frames in a movie film (as Vaught takes them), but rather as a genuinely continuous flow of interpretants (and likewise the flow of dynamical objects continuous), then the need for the four-term relationship vanishes and Thirdness suffices. The situation becomes logically similar to, and no more problematic than, an account of a continuous function in differential calculus.

On the phenomenological front, I note that mathematicians define the "orientation" or handedness of a space by dyadic and triadic arguments alone. The proof is rather technical, but it enables one to speak of left- or right-handedness in space (of three dimensions, or even more) without resort to any tetradic combinations and without any prior invocation of right or left.36