How is indissoluble not the same as soluble even though the prefixes would essentially cancel each other out?

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Thanks for asking!

The relationship here isn't quite what you might expect. In fact, 'soluble' has the opposite definition of 'indissoluble,' while 'dissoluble' and 'soluble' have almost identical definitions!

soluble: able to be dissolved, especially in water --> comes from Latin 'solubilis,' from 'solvere,' and is related (via Old French) to English 'solve." (related to "solution")

Dissoluble: able to be dissolved, destroyed, or loosened --> comes from Latin 'dissolubilis,' from the verb 'dissolvere,' is related to English "dissolve." (related to English "dissolute/dissolution.")

indissoluble: unable to be destroyed --> from the prefix in- + dissolubilis (above)

So, the question is: what is the function of "dis" as found in "dissoluble/indissoluble?"

In the Latin from which these words are derived, (and actually the Proto-Indo-European from which Latin is derived,) "dis-" means "apart," "in different directions." In the same, "solvere," means "to loosen." So, the also-Latin form, "dissolvere" means something like "to break apart, to loosen apart." From this we get both "soluble" and "dissoluble," which have almost identical meanings, and admittedly they seem pretty redundant. That kind of thing is an accident of language development, and it happens all the time. The redundancy is evident in the fact that the definition of "soluble" includes the word "dissolve," which is directly related to "dissoluble."

Today, it is my impression that "soluble" is usually used in a purely chemical sense--some substance is soluble if it can be dissolved in water. In the time since the innovation of these words, the word "solve" in English (from solvere, related to soluble) has come to mean "explain," or "answer," (since about the 16th century actually) so its relationship with "soluble" has been obscured, and we have innovated a new noun form of "solve," which is "solvable." In the meantime, "soluble" has the new opposite form "insoluble." An insoluble substance cannot be dissolved in water.

By contrast, it is my impression that "dissoluble" can be used in wider applications, in reference to relationships, contracts, etc. "Indissoluble" is therefore the opposite of "dissoluble," where some referent (a friendship, a relationship, a contract, etc.) is dissoluble if it is vulnerable to being destroyed, but indissoluble if it is lasting and (near) invulnerable to destructive forces.

So we have two seemingly redundant forms: 'soluble'~'dissoluble,' which have strictly near-identical definitions but which generally serve different, more specific purposes in common Modern English usage. These two forms have their own opposites: 'soluble' has the opposite 'insoluble,' while 'dissoluble' has the opposite 'indissoluble'.

In fact, the prefixes "in-" and "dis-" do not cancel each other out, as they are serving different purposes. 'Disoluble' is not the opposite of 'soluble,' but its near-synonym. I hope that clears things up for you!