Philosophically, if we have two purported objects X and Y, and for any property P, if X has P then Y has P and if Y has P then X has P, then X and Y are in fact the same object. Otherwise, X and Y are different objects. So different behaviors indicate that the researcher is dealing with different objects.

RE: For when the same conditions apply, the object under test behaves in the same way, and if under the same conditions behaves differently, it is not the same object.

There is an ambiguity here in the notion of sameness. Suppose I have a new plastic bottle and hit it with a hammer; it is flexible and does not crack or shatter. But plastic becomes brittle when it ages; if I hit the plastic bottle again under similar conditions years later, it will crack or shatter. It is the same bottle -- there is spatioptemporal continuity: it wasn't replaced with another when no one was looking -- yet it has undergone the changes that made it brittle. 

We must distinguish between numerical and qualitative identity. Consider:

"Mary and Fred drive the same car" is ambiguous between 1 and 2:

1. There is only one car and Mary and Fred take turns driving it.

2.  Mary and Fred each drive a car of the same type (i.e. make, model, color, etc.).

• 1 claims numerical identity.

• 2 claims qualitative identity.

Back to the plastic bottle: What could “It is not the same bottle it was before” mean? (Note the two occurrences of “it”, which presumably have the same referent or implicit antecedent.) The statement must merely mean that something about the one and only bottle under consideration is qualitatively different.

RE: if we have two purported objects X and Y, and for any property P, if X has P then Y has P and if Y has P then X has P, then X and Y are in fact the same object.

Commonsensical as it sounds, this principle, Leibniz's Principle of the Identity of Indiscernibles, is actually more controversial than its converse, the Indiscernibility of Identicals. Some have argued that it is violated at the quantum level. There are also some philosophical paradoxes (e.g. those arising out of Derek Parfit's personal-identity fission cases) that might be ameliorated by denying or qualifying this principle.

"Philosophically, if we have two purported objects X and Y, and for any property P, if X has P then Y has P and if Y has P then X has P, then X and Y are in fact the same object. Otherwise, X and Y are different objects."

Interesting.

I see this differently.

If both X and Y have property P, then P cannot belong to the same set as X and Y, but belongs by definition to the reference frame (the superset) defining (containing) X and Y, because a property common  to all objects of a set cannot itself be part to the set.

It can only mean that X and Y have at least P as a common property. They may possibly also have other properties in common, that is, other properties defining their common reference frame.

For them to really be identical, all of their properties must be common.

For example, the electron and the positron can be identified as having the following common reference frame:

Elementary, charged, massive, electromagnetic, having equal masses, scatterable against other same sign elementary particles, and no doubt some more...

@André Michaud

You are correct, André; the semi-English is carelessly worded and subject to a different interpretation. (My brain made me see what I expected to see, instead of what was written.) What is intended is this, which does say unambiguously that all of "their" properties must be in common:

 (∀x)(∀y)(∀F)(((Fx → Fy) & (Fy → Fx)) → x = y)

 which is redundant, since this will do:

 (∀x)(∀y)(∀F)((Fx → Fy) → x = y)

Karl raised the issue of identity, but rather on the basis of human identity. The bottle sample is anthropomorphize. As for human identity, we also have a problem with the Matuzalem.

I mean it

Vx iff Fx in cond Z then x = x iff Gx in cond Y

The function has the same property under conditions Z, and the same conditions Y under conditions Y behave differently than under conditions Z, although it is the same object. For this reason, we can estrapolate how an object behaves under the same conditions if we know its properties under conditions Z.

Vx ^ Vy in codn Y, iff x ^ y is Fx ^ Gy in cond Y then x is different then y.

To formalize law of extrapolation we need to add viariable "cond" what mean condition. 

I am not talking about identity as identity, but substance identity, for example, sulfuric acid under conditions Y behaves Z and condition B behaves K. I call it identificates, ie the same substance.

Why we could extrapolate. We could extrapolate just in this case, how we know behawior of identificates in conditions what we knew. Basis of extrapolation is knewed conditions and behawior of identificates in this conditions.

We must distinguish personal identities and substantial identities. 

Vx iff Fx in cond Z then x = x iff Gx in cond Y

Vx ^ Vy in codn Y, iff x ^ y is Fx ^ Gy in cond Y then x is different then y

Vx Iff in cond Y, Fx and in cond Z is Bx but x=subsx.

How we could identify identity ? From what interfere substantial identity diferent then empirical ?

I add

cond - condition

=subs - substancial identity

x=x iff in cond Y or in cond Z is Fx or Gx but is different then z becaouse z is in this same cond Kz 

When man reaches any of his limits, everything he can do is to resort to God or to Philosophy. God does not offer answers, because we are too insignificant in order to be able to reflect them. However, Philosophy is forced to offer us answers, although the answers are infinite, not yet identified, or impossible. This is also valid for the medicine as science, that must resort to God or Philosophy, when they reach the limit of their possibilities. God does not offer answers or we cannot reflect them, but the philosophical extrapolation is an answer. This is one of the reasons why the science or nonscience, whatever their nature, should always resort to Philosophy. The Unconventional/Philosophy tries to make these extrapolations known for anyone who wants to get familiar with them or who intends to surpass his own limits. This is one of the many reasons why you should not exclude Philosophy or the philosopher from your researches. The philosopher must replace the unknown from beyond any limit with extrapolations of the conventions we are. We could not understand this unknown in the absence of these conventions.

There are two ways of logical deduction: one that starts from the whole to the parts, or from the general to the particular, as is the case of the Unconventional/Philosophy and another one that starts from the parts to the whole, or from the particular to the general, as it happens in the case of sciences and nonsciences. By deduction and extrapolation, we can sustain that any whole assumes the properties of the parts, that it multiplies, under the form of a bigger whole, whereas the parts maintain all the properties of the whole, on the basis of which it divides. Sometimes the division of the whole is to be found beyond the capacity of reflection specific to any entity/universe, some other times, multiplication takes place beyond any capacity of reflection specific to the entities/universe. This does not mean that the parts lose from the properties of the whole, but only that the properties of the whole are no longer visible in the case of the parts, because of our limits and not because the limits of division or multiplication. A piece of metal divided into bigger parts remains the same metal. The atoms of that metal have the same properties as the piece of metal they were part of. It is ourselves that cannot reflect, know, relate all these properties to a system of reference, as they are too small for us and our devices. Each atom has a spirit, as long as the simultaneity of the atoms in the human body generates, creates ideas. We cannot create something from something that does not exist. However, we can reflect the existence that is to be found beyond our limits. This is the reason why, in our opinion, the extrapolation of matter/energy is still matter/energy, even if the extrapolation tends to the smallest or biggest unlimited or if we ourselves or others can reflect, compare, relate this thing to a system of reference or not. Any divided or multiplied unlimited notion is still a small or big unlimited, (notion) and we are part of this unlimited. Furthermore, let us not forget that there is no identity between the parts of the whole. There is no identity between the entities/universe of any kind as a whole, because of the internal and external space/time dimension, (transformation), which are always different.

The ether is only a convention

”It is a convention of our inability to understand the phenomena beyond the virtual world of our ideas.” Mankind must understand that we cannot make logical extrapolations by means of elements that do not exist. We can make extrapolations only with known sequences, if we wish to observe the human logic. The virtual cannot be extrapolated in the absence of a real matter, energy or matter/energy support, as we call it in the unconventional. The only virtual world is the world of the ideas specific to the entities/universe or of their spirit, which is defined in the unconventional as a simultaneity of ideas. Let us not omit the fact that no idea can be transferred from one entity/universe to another, in the absence of the intermediary support matter/energy. The ideas are the only virtual representation of reality, whatever the nature of this reality. The ether does not exist, it is only a convention of our inability to understand the phenomena beyond the virtual world of our ideas. The ether, (as reality) represents the extrapolation, beyond ourselves or the devices that help us, (matter/energy conventional entities/universe). From nanoscience to astrophysics, nobody has discovered other forms of existence, except for matter/energy simultaneously. There is no matter in the absence of energy, such as there is no energy in the absence of matter. Our limits, or the limits of the devices that help us do not allow us the reflection of the matter support for the energy or the reflection of the energy that sustains that particular matter. There is no entity/universe that can reflect reality and its existence beyond a certain bigger or smaller limit. Starting from this interpretation and from the fact that the division of a material form or energy, (whatever the degree of division, matter or energy) is of the same matter or matter, we can make a logical extrapolation. There is no ether, such as there are no virtual particles or God in the sense of the actual interpretations, if we wish to overcome our own limitations. All these elements, (the ether, the virtual particles, or God, the entities/universe) are matter/energy simultaneously, that lie beyond our capacity or the capacity of any entity/universe of reflection or comprehension. ”The virtual particles” are a nonsense, (such as light is a nonsense as a wave-particle, but a non-matter particle), if we take a close look at the classical conventions of the dictionary, according to which these are,

”A very small part from a substance, from a matter”

Matter cannot be both virtual and real at the same time. The ether cannot be virtual, as long as it gives shape to matter. We can only extrapolate the matter/energy that exists and that we can reflect in the world we live in. An entity/universe exists or it does not exist. If we were to consider the ether to be another form of existence beyond the matter/energy, we would go to ridiculousness. That would mean that the ether does not belong to this world, either because it cannot be reflected, or because it does not exist. The ether exists beyond the space/time dimension around us.

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Every proton is the same, becaouse in conditions y his behawior is z, because got the same properties what other protons, if his behawior in cnditions y is

other, then it is not proton.