Are Indiscernibles necessarily identical? I believe they are and this paper will attempt to argue for its case. The paper will first elaborate Leibniz’s law of the Principle of Identity of Indiscernibles (PII), qualitative and numerical properties and Max Black’s dialogue, which consists of Arguments for PII and against PII. It will also include brief sections on the Bundle Theory of Universals and Michael Della Rocca’s 20-spheres analogy. Following which, I will present a revised version of the Argument of Meaningful Verification and state its reasons for effectiveness in defending the Principle of Identical Indiscernibles. Lastly, I will conclude why indiscernibles are necessarily identical.
Leibniz’s Law: The Principle of Identical Indiscernibles (PII)
“There are never in nature two beings which are perfectly alike and in which it would not be possible to find a difference that is internal or founded upon an intrinsic denomination.” (Monadology, 1991: 62)
Leibniz's Law is a bi-conditional that claims the following: Necessarily, for anything, x, and anything, y, x is identical to y if and only if for any property x has, y has, and for any property y has, x has. Because this is a bi-conditional, it is comprised of two conditional statements (i) and (ii):
Indiscernibility of Identicals
(i) If x is identical to y, then for any property x has, y has and for any property y has, x has.
Part (i) refers to the Indiscernibility of Identicals because of the claim that self-identical objects must be indiscernible themselves. If “two” objects are identical, then they are not different in any way, which follows the PII.
Identity of Indiscernibles
(ii) If for any property x has, y has, and for any property y has, x has, then x is identical to y.
Part (ii) is the converse of part (i) – the Identity of Indiscernibility, where it has the claim that indiscernible objects must be identical. Simply put, if two objects are not different in any possible manner, they are identical. This is a weak version of the principle.
In formulating this principle, there are several distinctions regarding properties which are used to define any two objects similarity and differences. PII can also be defined in a few different ways, some of which include only intrinsic properties, others of which include both intrinsic and extrinsic properties. PII can also be defined so that it is, if true, necessarily true (true in all possible worlds), or again so that it is, if true, contingently true.
Qualitative vs Numerical Qualities
With regards to this debate over the validity of the argument, it is important to ascertain properties that allows us to “distinguish” between two identical objects. These are known as “distinguishing properties” and the two main examples are qualitative and numerical properties.
Qualitative properties are properties that equates to being a thing of a certain kind. A qualitative property does not need to apply to one thing, it is neither a property of being a particular object nor a property of being related to a particular object. These properties usually cannot be measured numerically, it is a measurement of quality. Examples of it include hardness or temperature.
Numerical properties are the property of being able to distinguish objects through counting. The most common example would be the example of twins. Identical twins are themselves qualitatively identical but not numerically identical simply because there are two of them. In the case of Superman, since Clark Kent is Superman and Superman is Clark Kent, they are numerically identical because they are intrinsically the same person.
From knowing these types of properties, we can write the strong version of the principle,
PII: Necessarily, if X and Y share all the same qualitative properties, then x and y are numerically identical (or x = y)
Pertaining to the PII is a questionable objection concerning that these properties and their definitions. One of the most well-known analysis would be Max Black’s dialogue.
Max Black: Black’s Dialogue
Argument from Identity and Difference
In Max Black’s story, there are two characters A and B, A argues for the PII while B argues that PII is false. A’s first argument is the argument from the property of identity.
1. If X has the property of being identical to X, then X is X.
2. Y lacks this property that X has, if not Y would be X.
3. Therefore, X has a property in which Y lacks, the property of being identical.
4. Hence, PII is true.
This is a weak claim by A which is immediately refuted by B questioning the legitimacy of the property of identity, arguing that it is a useless tautology that goes in a circular manner of saying nothing. Character A then changes his argument from the property of identity to the property of being different which in a way makes it trivially true but at the same time questionable to the same objection of legitimacy. However, B retorts these properties as mere “haecceities”, or properties that trivially allows objects to be “distinguished”, but it remains a circular argument in proving the PII.
Argument from Meaningful Verification
A then proposes another argument using relational properties. Character A argues that for PII to be false, it must be verified through experience for it to be “meaningful”. Anything that is un-verifiable in principle would be meaningless. X and Y are numerically distinct when we have relational properties that allows us to distinguish that the two objects are not identical in nature. An example of a relational property would be its location. Here Character A incorporates the property of spatial location into the identity of Indiscernibles where spatial points exist, and the points differ in relation to them. These points in space are real, existing things, which means that if X and Y are at a distance from each other, they share a relational property which means them distinguishable in nature. B counters this line of reasoning by arguing that the solution requires there to be such things as space-time points, which are exactly alike, but which are distinct. So, this response is committed to the existence of things, which violates the identity of indiscernibles. Empty regions of space have no features all by themselves, yet they are supposed to be distinct.
The Two Spheres Counterexample
With that, character B brings forth his counterexample of the two spheres, in which even with the inclusion of relational properties, PII would still be false because it is logically possible for there to be two spheres with the same relational and monadic features. The two spheres B considers are made of iron and are positioned two miles apart, they are alike in all their intrinsic properties. Even if they are located at separate places, their relational properties remain the same.
1. If there is a universe with two spheres with the same relational properties, then PII is false
2. A universe with two such spheres is logically possible.
3. Therefore, PII is false.
Bi-Located Spheres (Objection to Black’s Two Spheres)
The problem with Black’s Two Sphere example is a tricky case, but one can claim that instead of two spheres, there is but one alone, stipulating that the single sphere is a certain distance from itself, since in the example, space-time is curved. The problem is this appears to be begging the question by re-describing the case. It seems perfectly conceivable for two spheres to exist in a universe where space is not curved. John Hawthorne, a bundle theorist, proposes that where one sphere is in two places at the same time, a bi-located particular that exists in that exact location. He argues, that because universals can be multiply instantiated, objects are simply bundles of universals and hence it is possible that what is believed to be two spheres is actually a single group of universals instantiated in two locations (Hawthorne 1995) This property of being bi-located is also echoed by Varzi and Calosi (Calosi & Varzi, 2016) in their brief sequel to Black’s dialogue.
Hawthorne solution maybe an ingenious move, but what if something happened to the sphere at one of its locations, but not at the other? How does one then give an account for its bi-locative property? Hence, this plausible objection is strong enough to reject the claim of bi-locative property.
Rocca’s 20-Sphere Analogy (Objection to Black’s Two Spheres)
Michael Della Rocca believes that the PII should be accepted and that the possibility of overlapping objects in space is absurd. His argument lies in the inexplicability of the distinctness of the objects. Rocca believes that there cannot be indiscernible things in precisely the same place, at the same time, and he states it as a brute fact. His analogy goes to the extent of saying that instead of two spheres why not there be 20 spheres instead? How will we ever know the actual numerical value of the sphere(s)? With this claim, it effectively rules out the logically possibility of two such spheres in a universe. However, an opponent against PII would claim that unlike the twenty-sphere case, we would be able to tell that there is a multiplicity of objects, or as Rocca puts, “There cannot be two or more indiscernible things with all the same parts in precisely the same place at the same time.” (Rocca, 2005).
Character A also states an objection against B’s Two-Sphere analogy. One of them relates to the problem of Arbitrary Reference where A claims that it is still possible to pick out one of the spheres and distinguish it with a label. B counters by saying that this still does not identify the one you actually picked out in the first place and its duplicate. Hence, it is clear that B’s Two-sphere analogy is not yet cleared and refuted.
Objection to Character B’s Two-Sphere Argument
Before I put forth the revised version of the Argument from Meaningful Verification, we must first have an objection to character’s B Two-Sphere counterexample, aimed to object to the Argument from Meaningful Verification. The objection adopts a verificationist stance to object to Black’s Two-Sphere analogy,
1. What is logically possible should be verifiable or detectable in reality or in our universe.
2. Anything that is un-verifiable is not meaningful.
3. Since the two spheres proposed by Character B is not verifiable in reality, it is therefore not meaningful.
Here, this objection specifically objects to Premise 2 of Character B’s argument (A universe with two such spheres is logically possible) because it can be seen that such two spheres at the same spatiotemporal location at the same time is impossible in this universe (our reality). Hence, if Premise 2 is not true, then its conclusion must be false.
Argument from Meaningful Verification (Revised)
In the case of Max Black’s analogy, the character A in his story presented an Argument from Meaningful Verification which I believe is the strongest version in defense of the PII. The argument, revised, goes like this:
1. If PII is false, then we can never detect whether a and b are the same.
2. Whether a and b are the same should be detected by means of verification in reality (our universe).
3. We can detect whether a and b are the same.
4. Therefore, PII is true.
The method character A in Max Black’s story is using a form of verification to assert its meaningfulness. Character A argues that the only way we discover that two objects are distinct is by discovering a distinguishing feature. Failure to do so would only mean that we cannot tell them apart. That lack of property to distinguish them shows that a and b are not distinct, therefore PII must be true. However, he encounters the problem of the two iron spheres posed by character B. In order to defend the argument from verification then, is to refute premise 2 of the two-sphere counterexample. This is done with the addition of premise 2 above in the argument from meaningful verification. Here, the definition of whatever is logically possible should be verifiable in reality and not just by pure imagination alone. If I believe pigs can fly, does it entail that it is logically possible for pigs to fly in reality? The answer may be yes, given evolution or by attaching it with wings or putting it on an airplane, but it is fundamentally unsound to assume that pigs do or can fly. If we do accept that possibility, then almost anything I imagine would be possible, that I can miraculous make an identical counterpart of myself in reality. This, I believe, makes no coherent sense, which makes Black’s analogy of the two iron spheres an indeterminate possibility.
Character B might also argue that the two iron spheres have the same relational property, that there are “two rulers”, on either side of the ball to preserve symmetry of relational properties. However, why would you introduce an instrument of measurement into the picture in the first place if you do not believe in its distal relation? The fact that he requires a 3rd element of measurement to remove the problem of relational property is already conceding to the notion that there are already two numerical spheres in the first place, and that its relational distance, regardless of its similar distance, is already a distinguishing property that separates the two spheres.
Evaluation
Any means of disputing the PII is simply put, trying to violate the intuitive metaphysical principle of physics. By rejecting the notion of possibility in other universes in Black’s two-sphere case, it allows us to reject the notion of having logically possible two spheres having a relational property to each other. And this claim will hold true until the day it has been empirically proven that other universes exist outside the realm of our own.
This Argument from Meaningful Verification is the strongest version in defense of the PII because firstly, it appeals to the only method by which we, as human beings, can differentiate identicals and indiscernibles. Secondly, it utilizes the notion of a measuring instrument or device (a 3rd element) to differentiate two entities as distinct which logically, should be the only method to differentiate objects of the same properties. And lastly, even if we use our minds to think of two imaginary objects as indistinguishable, it does not undermine the argument because it is still unverifiable regardless of your resolve or imagination. Even if it is possible in my mind or imagination, it does not logically entail that it is possible in the real world. Hence, even if Black states that two such spheres can exist in the mind or any such universe except the real one, it definitely cannot exist at the same spatiotemporal location of our reality since no two things can occupy the same space at the same time.
Conclusion
If it is proven that Identicals can be verified in nature to be numerically distinct, such as in the case of Superman and Clark Kent, then it is safe to assume that the PII is false. However, as long we believe that Superman and Clark Kent are one in the same person, it remains futile to argue that identicals are discernable unless you opt for the option of difference in reference, which is a different matter altogether.