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Indian philosophy has a logical category of non-difference (related somewhat to abheda) in which something is different from something else but identical to it as well. An example is that of the the drop of ocean water to the ocean.

So the concept of abheda is relevant to describe the nature of something.

A drop of water is neither identical to the ocean nor is it different from the ocean. Can we say this is a logical category of non-difference?

This is a separate category from identity and difference. Does it or can it exist in Western philosophies?

A drop of ocean is in one sense the same as the ocean by being of the same quality and in another sense it is different by quantity. So there is difference and non-difference. Does this category of non-difference violate principles of traditional logic by violating non-contradiction? The drop of ocean is the ocean in one sense and not the ocean in another sense.

So can we explain a drop of water being different from the ocean and yet non-different to the ocean in logical terms?

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    Difference in one sense and sameness in another sense does not violate non-contradiction. You can have predicates P and Q such that P(a) = P(b) but Q(a) ≠ Q(b). So a and b are the same in the sense of P and different in the sense of Q. It is only difference and sameness in the same sense that violates it. – Conifold Oct 26 at 7:51
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    Fascinating topic. If you have a better link than WP, have at it. I thought something was better than nothing. – J D Oct 26 at 9:34
  • @JD Yes the WP is ok but is too broad, it mixes a lot of concepts and gets into the nature of the absolute truth and the living entities without getting into abheda as a logical category. I think I left out of the question a designation of abheda as referring to the nature of something. – Amala Oct 26 at 10:07
  • I narrowed the link down in the edit, and I also came across anekantavada. I like the English translation one-sidedness. Good luck! – J D Oct 26 at 10:16
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Short Answer

As far as I'm aware, no. In the West, logic is strongly rooted in the Law of the Excluded Middle, Law of Identity, and the Law of Non-Contradiction. It is possible to have a system of logic where one can have a statement (i.e. S :='P=Q') and accept that it AND it's negation (S* := 'P≠Q') are both true. This is called a dialetheia, but strictly speaking this isn't the same as a relation between identity and the negation of identity, although it seems they're inter-related. Also related are multi-valued logics and concepts like paraconsistent logics which deal with the explosion principle and fuzzy logic provides a quantitative formalism.

Long Answer

From your example:

A drop of water is the ocean. (W=O, They are the same.)
A drop of water is not the ocean. (W≠O, They are different.)

Note that sameness and difference are negations, and so the first statement is really the negation of the second. To a Westerner, this idea seems more prominent in the East than the West, not only in Indian philosophy but also East Asian philosophies like Taoism and Zen, where the latter routinely uses such contradiction in koans. Looking around, I can't find a formal logic equivalent.

The non-difference relation seems to be dealt with in informal logic by explaining the contradiction or characterizing one of the propositions as a 'literal truth' and the other as a figurative one such as a 'metaphor'. To wit:

A drop of water is the ocean, and yet not.

Were this a line of poetry, one might analyze the text by explaining that a drop of water is similar to the ocean because both are wet, thus an analogy is at play. Yet, it is a literal truth that a drop of water is not the ocean. To call it the ocean is just a metaphor.

In order to deal with the nature of the contradiction, let's look at the SEP:

A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true. If falsity is assumed to be the truth of negation, a dialetheia is a sentence which is both true and false. Such a sentence is, or has, what is called a truth value glut, in distinction to a gap, a sentence that is neither true nor false. (We shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as one’s favourite truth-bearer: this would make little difference in the context.)

Lastly, with fuzzy logic, it is possible to have fuzzy membership so that relationships can have a three-valued membership based on a set of membership functions like min(x), 1-x, max(x). According to WP, there are fuzzy first-order logics that use general and existential quantification. You probably can find more information about these logics on one of the SE Math sites.

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  • Thank you, I think this is very detailed answer. Thank for the link to dialetheism. It seems to have the closest parallels in its acceptance of contradictions but my even shorter answer is that I think this logical category does not exist because of the LNC. – Amala Oct 26 at 10:29
  • One more avenue for you to try exploring is paraconsistent logics given whatever your ambitions might be. – J D Oct 26 at 10:31
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    In Euclidian geometry when two triangles have the same proportions but different sizes they are not congruent, as the linked article asserts:"In geometry, two figures or objects are congruent if they have the same shape and size". There is no "non-difference" or dialetheias there, it is a theory built over classical logic. I do not think equivalence relations generally much resemble either, the former is about overlaying metaphysical "perspectives" and the latter about doubling down on identity, not diluting it. – Conifold Oct 26 at 11:01
  • @Conifold Thanks. I'll reflect on your guidance. – J D Oct 26 at 11:09
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The areas of Western analytic philosophy that would give an account of your drop of water/ocean example are mereology, the study of parthood, and more specifically material constitution.

Just as you have said, the drop of water is not identical to the ocean (because there are predicates which are true about one but not the other, like 'you can swim in ...' or '... could fit in a glass'). But if the ocean didn't exist then the drop of water would not exist, and the drop of water is clearly part of the ocean. Moreover, the drop of water is spatially coincident with the ocean; it takes up a subset of the space which the ocean takes up.

Trying to ascertain exactly the nature of the relationship between the drop of water and the ocean is definitely within the remit of mereology, and it's a very tough problem! If we accept that there are two objects which can take up the same space, we have to account for why other distinct objects (me and a wall) can't take up the same space. If we say that the ocean is a specific combination of drops of water then does the ocean stop existing if I take a drop out? All of these are very difficult problems and active areas of research in modern analytic philosophy.

I'm not sure what you're looking for specifically when you say you want to describe the relationship in 'logical terms', but both of the SEP articles linked to above include a good amount of formal logic in the way they specify different notions of parthood and coincidence, and they certainly address the exact situation you're looking at. Logic doesn't give us answers, but it enables us to be extremely specific about what we mean and to carefully investigate the consequences of a particular definitional choice, without worrying that we are confusing ourselves in any given natural language.

If you're reading further into this, it's useful to be aware that much of the Western discussion of these concepts uses a statue and the lump of clay constituting the statue as an example.

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    @Amala I understand that you've already accepted J D's answer, but just tagging you here in case this answer is also useful! – dbmag9 Oct 27 at 10:58
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    And rightly so. The more I think about it, the more the question of Non-Difference goes to mereological notions of shared identity. Well met! – J D Oct 27 at 17:53
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The "water in the ocean" example is closely related to the Sorites paradox or the "paradox of the heap". How many grains of sand do you need for a heap of sand?

A significant part of the confusion is caused by the mix of the abstract ("sand", "water") and the concrete ("this grain of sand", "this heap of sand", "this drop of water", "this ocean"). An example of the concept is not the concept itself.

To a large degree, this was already sorted out in classical Greek (Aristotelian) logic.

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