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Suppose , I have two objects A and B. A has a set of properties P and B has a set of properties Q. It is possible that when I combine A and B the combination has a property c that is neither in P nor in Q (i.e c ∉ P∪Q ). Examples are everywhere so i am not feeling the need to provide any. Now my question is : what exactly is property and where do new properties come from?

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    See Properties: "Properties (also called ‘attributes,’ ‘qualities,’ ‘features,’ ‘characteristics,’ ‘types’) are those entities that can be predicated of things or, in other words, attributed to them. Moreover, properties are entities that things are said to bear, possess or exemplify." Commented Jan 25, 2018 at 11:46
  • pvspade.com/Logic/index.html You might also be interested in Prof. Spades "Warp and Woof of Metaphysics"
    – Gordon
    Commented Jan 25, 2018 at 15:17
  • (On the above page, Things to download, Medieval Philosophy, then you will see this PDF.)
    – Gordon
    Commented Jan 25, 2018 at 15:20
  • Are you interested in a philosophical notion (which is loose and vague as actually used) or some precise formalization? In formal theories properties are identified with one-place predicates. Predicates (not necessarily one-place) are built from some pre-specified basic predicates by using logical connectives and quantifiers. What you mean by "combining" objects is unclear, why should we expect that the "combination" would even have the properties of the originals? Say, green and red objects are both mono-color, but their combination isn't.
    – Conifold
    Commented Jan 25, 2018 at 21:36
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    Hydrogen's not wet and oxygen's not wet but water is wet. Is that an example of what you mean? Surely the whole often has properties not possessed by any of the parts. A semicircle isn't round but if you put two of them together it's a circle, which is round. This by the way is why the concept of "emergence" is murky and explains far less than its proponents think it does.
    – user4894
    Commented Jan 25, 2018 at 23:36

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Your profile mentions an interest in math, which you seem to be pursuing in the formulation of your question. So let me suggest how physics approaches this same kind of question. Terminology-wise, your "object" is a "system", and its "properties" are the system's "observables".

Then the collection of all possible states a system can assume is modelled by (the unit vectors of) a Hilbert space H corresponding to that system. And then the collection of bounded Hermitian operators B(H) on H represent all the system's observables, and this B(H) comprises what's called a C*-Algebra.

Now, if you have two systems, A and B represented by H_A and H_B, then the composite system is represented by H_A x H_B, and the C*-Algebra of observables B(H_A x H_B) of that composite system is way more complicated than B(H_A) and B(H_B) alone (and way more complicated than their simple set-theoretic union).

Now, regarding the comments, a property like "wetness" wouldn't really be a formal observable in the above sense (at least I'm pretty sure -- make that almost absolutely positive -- not). But if you're looking for a rigorous mathematical treatment of your question, it couldn't hurt (except for the mental time and effort) to google some of the above stuff and start with that kind of approach, which has been carefully developed by many people during the last ~75 years (I'm dating it since Irving Segal's 1947 seminal paper).

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