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The following sentence, which is part of a presentation of the doctrine of structural holism and is translated from the french, does not seem quite to make sense to me:

"each one [of the members of a system] is the system itself considered in one of its members."

Is this sentence logically correct or not? And if not, what is exactly the problem?

Is it possible to express this sentence in the language of formal logic?

What is exactly the meaning of "is" in this sentence?

Is there some sort of vicious circle here? or a contradiction?

Addition: After reading the first answer and comments, I realize I have to give some more information, in order to allow a better understanding of the possible or intended meaning of the sentence, as well as of the main words it contains.

A system, in the sense of structural holism, is an ordered pair or set, like a dyad, triad, etc. A system is a whole, and its parts, or members, are something different from elements or individuals put together. They are individuals taken under a description, which means they are considered in their reciprocal relationship and interdependance within the system, as fulfilling complementary roles, like giver/receiver/object given, or killer/victim.

The quoted sentence is taken (slightly modified) from a book by Vincent Descombes, The Institutions of Meaning, Harvard University Press 2014, and belongs to a passage commenting an example borrowed from Peirce, "Cain kills Abel" :

"What we add when speaking of a dyad rather than of two individuals is the idea that we are considering these individuals as the members of a dyadic system. Each of them is a dyadic unit; therefore each of them is the system itself considered within one of its members. Each of them is a dyadic unit because each is taken under a description: we are not speaking of Cain tout court; we are speaking of him insofar as he is a murderer."

  • can you clarify what you mean by "logically correct"? Without that, it's impossible to answer your question coherently. Surely, many positivists would have hated this idea, but is that what we mean by logically here? – virmaior Jun 17 '15 at 8:30
  • E.g., It is possible to express much of the meaning of the first sentence in formal logic. For any x, Mx -> Wx where M(x) = x is a member of the system and W(x) = x is the system itself considered in one of the members. It's going to take some more work to translate the two sentences for each of these functions but it won't be impossible. – virmaior Jun 17 '15 at 8:30
  • "Is" has typically two meanings that formal logic separates : in "Plato is a philosopher" the "is" must be translated as (set-theoretic) "belongs" (is a member of), i.e. "the individual Plato is member of the set of philosophers". The other is "identity", as in "Scott is the author of Waverley". – Mauro ALLEGRANZA Jun 17 '15 at 10:24
  • Depends on what you mean by "members" in each case. A steering system is a constitutional member of any car. However a steering system is not a representative member of the set of cars. Use both my word-senses in your quoted sentence, and you'll see that the sentence is true. – prash Jun 17 '15 at 10:49
  • @ virmaior: by "logically correct", I mean first of all the way the sentence itself is expressed, its logical syntax and internal coherence, independently of our judgment about the basic position of structural holism. It might be that we have here to do with a clumsy way of expressing an aspect of the true doctrine of structural holism. – istela Jun 17 '15 at 16:15
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Here's one proposed way of interpreting the sentence. Suppose that we have a partial function with two arguments that we call a "considered in" function, cons_in(X,a). Then suppose that we represent memberhood of a system by the relation m<<S, and assume a regular identity relation =. We can use a predicate calculus formulation like this: given a system S,

∀x. (x<<S) -> (x = cons_in(S,x))

Questions then arise as to what the "system memberhood" relation and the "considered in" function are supposed to be. These aren't typical mathematical terms, and don't in themselves appear to carry a lot of significance, but the idea that in principle no sense could be made of them seems a bit hasty.

Addition in response to comments: There is also another possibility, which might be also informative depending on how you want to read "considering in", which is that the two instances of "member" in the quote need not necessarily be the same. Let's suppose for instance that we take one such system to be the Natural Numbers. One form of "considering" a number might to take that number's predecessor and apply the successor function to it.

So here we would want to say that the sentence has a slightly different logical form. Given a system S,

∀x. (x<<S) -> (∃y. (y<<S) ^ (x = cons_in(S,y)))

Again, it's entirely logically sensible, though not of itself informative without further assessment of the two concepts above.

  • I have a problem with x appearing on both sides of the = sign. And this is, I think, where there might be a contradiction or vicious circle implied also in the english sentence. Am I right if I say that the sentence "Each member of a system is a particular way to consider the system" would be better, because it would avoid this problem? – istela Jun 17 '15 at 16:43
  • There isn't anything at all contradictory about having an x on either side of an identity relation! It's arguably an essential part of what it means to be an identity relation, since identity satisfies reflexivity - that is, x=x. If we think this is some sort of generative principle for what it takes for something to be an "x" then yes, there would be a problem, but again, this isn't so much a logical issue as much as a question about what we're analysing in the particular terms "system" and "considering". – Paul Ross Jun 18 '15 at 10:11
  • @istela One thing your rephrasing might do is that x is no longer an index in the "considered in" function, since we're just talking about "a particular way" rather than the system being "considered in one of its members". So instead we might use an extra existential quantifier, as in ∀x. (x<<S) -> (∃y. x = cons_as(S,y)), with y representing some choice variable from the "particular ways" in which the system might be considered. Again, perfectly logically sensible, though it does mean something different. – Paul Ross Jun 18 '15 at 10:24
  • @istela although now that I write that, I do see that it's also compatible with the original phrasing that x needn't necessarily be "the one member" in which the system is considered. For example, might 2 be the system of natural numbers "considered in" 1? That's again something that might be possible, depending on how we read "considering in" - I'll add that in as an edit to the answer. – Paul Ross Jun 18 '15 at 10:27
  • Ok, but (x = cons_in(S,x)) can only be true if the function cons_in(S) does not modify x at all. In other words, you can forget about cons_in(S). What remains is x = x. Or in English : each member of a system is a member of this system. But I don’t think this is what the English sentence is trying to say. Whatever "system", "member" and "consider" mean, I see a contradiction in this sentence because it says at the same time 1) that a member is a member and 2) that a member is more than (or something different from) a member. – istela Jul 23 '15 at 13:38
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I believe the sentence makes logical sense and that it is true when applied to a dyadic system, though it is not necessarily true when applied to a system of three or more elements.

According to your notes, when we consider Cain and Abel as individuals (tout court), we do not consider the relationships which exist between them. When we consider Cain and Abel as a dyadic system, we are considering these relationships. In particular, if a certain relationships exists between Cain and Abel, then when viewed dyadically, both Cain and Abel know of this relationship. Therefore, when we consider one element of the dyadic system, we have information about how that element relates to the other element. Furthermore, in a dyadic system, every such relationship is identifiable in this way. Therefore, "each of them is the system itself considered within one of its members".

In a system of more than two elements, this may not be true. This is because when we consider a particular element, we know only how that element relates to each other element, but not necessarily how the other elements may relate one another. For example, consider the triadic system { Peter, Paul, Mary }. When considering the element Peter, we would know how Peter relates to Paul and to Mary, but we would not know how Paul relates to Mary. So considering the element Peter is not equivalent to considering the whole system.

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