6 Jun 2016

Peirce (CP1.294–1.299) Collected Papers of Charles Sanders Peirce, §4, “Indecomposable Elements"


by Corry Shores

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[The following is summary. Boldface and bracketed commentary are mine. Proofreading is incomplete, so please forgive my typos.]



Summary of


Charles Sanders Peirce


Collected Papers of Charles Sanders Peirce


Volume 1: Principles of Philosophy


Book 3: Phenomenology


Chapter 1: Introduction


§4: Indecomposable Elements [1.294–1.299]




Brief summary:

A phenomenon [or “phaneron” in Peirce’s terminology] can be classified on the basis of the way its indecomposable parts relate to one another. What makes the parts indecomposable is the fact that they are the most elementary possible constituting relations on the basis of which more complex arrangements can be composed. For example, were we to analyze something into simpler and simpler parts, we would need to arrive upon parts that stand unto themselves [atoms of a sort]. But what constitutes an elementary part as being something standing unto itself? That would be its self-affirmative self-relation that needs no further external relations to be what it is. Thus a decompositional analysis will take us down to parts that are constituted by a self-relation which cannot be made any simpler and thus this monadic self-relation is itself an indecomposable element. But if we have many monadic elementary self-relations, we are missing how they are parts. For, as such they are merely wholes and not parts. To be a part means to be in a relation to some other part. Thus there is another indecomposable relation, the dyadic relation of part to part. It is not decomposable into monadic relations, because those are self-reflexive but dyadic ones are external. Yet still, we have not done our analysis properly if we only have parts related to other parts. For, supposedly they are parts of something greater that we have analyzed. Hence we need a third kind of relation, that between parts taken together as having a combined relation to a third thing, the whole they compose. So such a triadic relation as this is another indecomposable element. We cannot decompose it into monadic and dyadic relations, because neither of those involve externally related things being taken together in relation to yet another entity. But, any larger relation, like a tetradic one for example, is decomposable. This is because, having established the first three kinds of relations, any further one would need to repeat those. We cannot say for example that in the tetradic relation, the four parts are taken together to relate to the still larger whole, thereby creating a new sort of relation. For, this is no different than the triadic one just mentioned, which takes parts together and relates them to another thing. And if we say that in the tetradic relation part 1 is related to part 2 which is related to part 3 which is related to part 4, we just have three dyadic relations and no new relational structure. Any further complicated sorts of relational structures within a tetradic formation would still be nothing other than dyadic ones. Suppose for example part 1 is related to parts 2, 3, and 4, and each one of these in turn is related to each of the others. Still, these are no more than sets of dyadic relations. Thus the triadic is the most complex of the indecomposable relational structures, and Peirce considers this to be the most important philosophical concept.






[A element is something that is taken as a unitary simply thing, even though it could be further decomposed. Were we to decompose anything, we would find that the smallest parts would have to be self-related, and thus be constituted by a monadic relation. Other decomposable elements would furthermore be other sorts of relations that cannot be conceived as the bare sum of lesser relations]


Peirce now explains why he is using the term “indecomposable element” [since it might seem like a redundancy in terms. Would not an element be indecomposable by definition?] [I do not follow what Peirce says next, so you can skip to the quotation below. Specifically, I do not understand how what he says explains why “indecomposable element” means something different from simply “element”. Let me still pass through the points he makes. The first point is that to perform a logical analysis does not mean decomposing something complex into its existing elements. (But what it is instead I am not sure. From his description, it would seem still to be a sort of decomposition into existing elements.) Instead, doing logical analysis means taking a concept, and then taking with it its negation, and furthermore, taking with it all other things with which it may relate (with those other things being its “correlates”). With how he exemplifies correlates, it seems to be like the “places” of n-place predicates. One example he gives is ‘to love’ and ‘to be loved’. In these cases, there would be two correlates, the person loving and the person loved. He makes the further point that the different arrangements of this relation are not to be considered as having any order of priority. So to love, to not love, to be loved, and to not be loved are instances of the same concept. The next example will explain how concepts can be built up with other concepts. So consider the concept ‘A is parent of B’. What happens when we combine two of these relations to get ‘A is parent of B who is parent of C’? This now gives us the concept of ‘grandparent’: ‘A is grandparent of C’. Or what if along with ‘A is parent of B’ we also have ‘X is spouse of Y’? So now let us make this combination: ‘A is spouse of B who is parent of C’. We now have the concept ‘A is stepparent of C’. But suppose instead we begin with the concept of ‘A grandparent of C’. Following Peirce’s first mode of analysis, we would take with this ‘A is not grandparent of C’, ‘C is grandchild of A’, and ‘C is not grandchild of A’. But this concept can also be analyzed through decomposition. We could decompose ‘A is grandparent of C’ into two concepts, namely, ‘A is parent of B’ and ‘ B is parent of C’. So that is the analysis. But I am not sure what to say about indecomposable elements. They might be such relations as ‘A is parent of B’, which are analyzable into no other simpler relations. In that case, perhaps Peirce uses the term “indecomposable element” because were he to simply use “element,” he might also be referring to persons A, B, and C in our examples. And maybe the problem with this is that A herself can be analyzed as being made of body parts or other definable characteristics, and in that sense is not indecomposable. However, were we even to decompose person A, we would eventually have to arrive upon indecomposable elements that constitute the person on the most fundamental level. (In light of what Peirce says below, possibly these most basic sorts of relations are the monadic type.) So I am just guessing, but perhaps Peirce’s idea here is that we need to say “indecomposable elements” to distinguish the indecomposable relations between elementary parts from those elementary parts themselves, which are potentially subject to their own further decomposition. And were we to decompose any such decomposable part, we would eventually arrive upon things constituted by indecomposable simple relations like bare self-relation. Let me quote:]

I doubt not that readers have been fretting over the ridiculous-seeming phrase “indecomposable element,” which is as Hibernian as “necessary and sufficient condition” (as if “condition” meant no more than concomitant and as [if] needful were not the proper accompaniment of “sufficient”). But I have used it because I do not mean simply element. Logical analysis is not an analysis into existing elements. It is the tracing out of relations between concepts on the assumption that along with each given or found concept is given its negative, and every other relation resulting from a transposition of its correlates. The latter postulate amounts to merely identifying each correlate and distinguishing it from the others without recognizing any serial order among them. Thus to love and to be loved are regarded as the same concept, and not to love is also to be considered as the same concept. The combination of concepts is always by two at a time and consists in indefinitely identifying a subject of the one with a subject of the other, every correlate being regarded as a subject. Then if one concept can be accurately defined as a combination of others, and if these others are not of more complicated structure than the defined concept, then the defined concept is regarded as ana- | lyzed into these others. Thus A is grandparent of B, if and only if A is a parent of somebody who is a parent of B, therefore grandparent is analyzed into parent and parent. So stepparent, if taken as not excluding parentage, is analyzed into spouse and parent; and parent-in-law into parent and spouse.





[We use the term ‘Priman” in reference to the indecomposable logical relations in which the part is what it is regardless of reference to other things.]


Peirce then says that there is no a priori reason preventing us from saying that there are indecomposable elements in the phaneron that are what they are regardless of any relations to other things. Peirce uses the term Priman to refer to these things and to things of their sort. [This seems roughly equivalent to firstness with regard to logical relations.]

These things being premised we may say in primo, there is no a priori reason why there should not be indecomposable elements of the phaneron which are what they are regardless of anything else, each complete in itself; provided, of course, that they be capable of composition. We will call these and all that particularly relates to them Priman. Indeed, it is almost inevitable that there should be such, since there will be compound concepts which do not refer to anything, and it will generally be possible to abstract from the internal construction that makes them compound, whereupon they become indecomposable elements.





[“Secundan” designates phenomenal parts that are constituted just by relations to only one other thing, for example, the idea of ‘otherness’. ]


Indecomposable elements of a phaneron can also be said to be what they are on account of their relation to just one other thing, regardless of some possible relation to a third thing. We would use the term Secundan for these cases. An example is the idea of ‘otherness’.

In secundo, there is no a priori reason why there should not be indecomposable elements which are what they are relatively to a second but independently of any third. Such, for example, is the idea of otherness. We will call such ideas and all that is marked by them Secundan (i.e., dependent on a second).





[“Tertian” is the term for the relational structure involving a relation not just of one thing to another but as well of both together to a third thing, as for example the idea of composition (which has a part-part Secundan relation as well as a parts-whole tertian relation).]


And finally, there are indecomposable elements of phanerons that are what they are on account of both a relation to a second thing in addition to a relation to a third, and for this situation we use the term tertian [I do not know why it is uncapitalized.] One example is the idea of ‘composition’. [For something to be composed, it needs at least two parts. So there is the relation of the one part to the other, and that is the Secundan component. But there is also the relation of each part to the whole, which is a third thing that is part of this concept. That would be the tertian component.]

In tertio there is no a priori reason why there should not be indecomposable elements which are what they are relatively to a second and a third, regardless of any fourth. Such, for example, is the idea of composition. We will call everything marked by being a third or medium of connection, between a first and second anything, tertian.





[There are no indecomposable relations beyond the triadic, because anything further is decomposable into other simpler relations.]


Peirce says that there can be no indecomposable element that involves relations to more than three correlates. This is because were there to be one, it would be decomposable by means of the dyadic relation. [I suppose the idea is the following. A monadic relation is not decomposable, because there is only one relation. We cannot have half a relation, for example. A dyadic relation is not decomposable either. Suppose we decompose it into two monadic relations. Putting them together does not give us a dyadic relation. We just have two monads. In order to have the dyadic, we need an additional structure, namely, an exterior relation of one thing to another. Now it gets tricky. Why is the triadic relation indecomposable? Consider this. For one thing, it is not decomposable into dyad and monad, since by combining those we do not get a triad. Why? If we work just with the notion of composition, suppose we decompose it into the dyadic relation of two internal parts and the monadic relation of the whole unto itself. When we put them together, we just have a dyad and monad, but not the relation between them of parts-to-whole. It is also not made of two dyadic relations, namely, that between a part and a part and that between a part and a whole. Why? Because then we do not have the notion of composition. We have the relation of part 1 to part 2, the relation of part 1 to the whole, and the relation of part 2 to the whole. But that does not render us the notion of composition, which requires that the relation of parts to whole be a relation involving the togetherness of the parts in their combined relation to the whole. For, otherwise we do not have a whole but rather just a part of a whole. The only way parts can compose a whole is in their combining relations with one another. Thus the triadic relation in this example is indecomposable. To recap, we need three relational structures for the idea of ‘composition’: {1} the monadic self-relational structure of part 1 to itself and part 2 to itself, (which makes them components that can be relatable), {2} the dyadic relation of part 1 to part 2, and {3} the triadic relation of both parts taken together and then related to the whole. So the triadic, at least in this example, is indecomposable, because there is a third sort of relation that is not simply a dyadic one between part and whole but rather requires the additional operation of taking the parts together to relate them, as a grouping, with a third entity, the whole they compose. Now we need to understand why a tetradic relation is in fact decomposable into two dyadic relations. Without any examples, I am not sure how. Let me venture a wild guess. When studying emergentism and brain states, I recall the following idea. Consciousness is an emergent phenomenon that arises from the dynamics of particular brain states. However, that emergent phenomenon of consciousness can then have some influence on how the parts are operating (something like conscious volition with regard to what one is choosing to think or how one is doing one’s thinking). This idea I think is called “downward causation”. So here we have {1} the monadic self-relation of the biological parts (or operations) of our cognition; {2} the dyadic relation of these parts or operations interacting with one another; {3} together their operations generate a third term to which they are compositionally related, namely, the emergent phenomenon of consciousness; and{4} that emergent consciousness then can act again back upon its own constituent parts and operations, affecting their structures or behaviors. But what kind of relation is 4? Is it a new kind? Perhaps it is reducible to a dyadic, in that it is one thing related to the parts as another thing. Or perhaps it is a triadic, because it is one thing relating to constituent parts. Another way to think about it is the following. We needed to add an operation to get the triadic relation of conscious parts to relate to the emergent conscious whole. But to go back from whole to parts, we need no additional relational operation. We already have the parts taken together, as a left-over from the triadic relation. And we also have the parts taken by themselves, from the monadic relation. So there is no need for an additional relational “invention” to conceive the idea of downward causation. I am not sure. A less interesting example would be “A loves B who loves C who loved D”. Maybe this is tetradic, and it is clearly decomposable into three dyadic relations. Any more complicated such relations between parts, like A, B, C, and D each loves each of the others, still involves dyadic relations. At any rate, Peirce places a very high philosophical importance to the impossibility of tetradic and higher –adic forms being indecomposable. Let me quote.]


It is a priori impossible that there should be an indecomposable element which is what it is relatively to a second, a third, and a fourth. The obvious reason is that that which combines two will by repetition combine any number. Nothing could be simpler; nothing in philosophy is more important.





[There are three types of indecomposable parts of phanerons: 1) those that are what they are just by themselves, 2) those that are what they are in relation to something else, and 3) those that involve a combination that is then taken to be further related to something else.]


Peirce then summarizes his findings in this section. The phaneron’s indecomposable parts come in one of three varieties: {1} those that are simply positive totals [they are wholly what they are on their very own], {2} those that involve a dependence of one on another without them being regarded as together forming something that may take further relations, and {3} those that involve the combinations of parts taken together as a further relatable entity.

We find then a priori that there are three categories of undecomposable elements to be expected in the phaneron: those which are simply positive totals, those which involve dependence but not combination, those which involve combination.



Peirce then announces that he will examine the phaneron [in a more phenomenological way]: “Now let us turn to the phaneron and see what we find in fact” (p.147).




Peirce, C.S. Collected Papers of Charles Sanders Peirce, Vol 1: Principles of Philosophy.  In Collected Papers of Charles Sanders Peirce [Two Volumes in One], Vols. 1 and 2. Edited by Charles Hartshorne and Paul Weiss. Cambridge, Massachusetts: 1965 [1931].



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