Suppose I own a copy of Moby Dick and you own a copy of Moby Dick.
My copy is the same as your copy, meaning that they are both the same text.
My copy is not the same as your copy, meaning that they are not the same physical object.
Does this violate the law of identity?
This was answered already–those equations can be solved by setting all the variables to either zero or aleph-null.
If you don’t mean them to be interpreted numerically like that, then you need to explain how you mean them to be interpreted. Without having explained that, we have no way of knowing what the problem is supposed to be.
Let me offer a different way of thinking about it.
I have two finite mathematical objects (sets, graphs, groups, …) Let’s call them A and B. Is it possible that A is properly contained within B, and B is properly contained with A? It is not possible.
But what if I have two infinite mathematical objects, C and D. Is it possible that C is properly contained within D, and D is properly contained within C? Yes, it is.
Anyone who’s studied group theory, for instance, can easily give examples of infinite groups F and G such that F < G and G < F, while F and G are not the same group.
So if the Father, Son, and Holy Spirit are infinite, one can imagine that the Father contains all properties of the Son and Holy Spirit, the Son contains all properties of the Father and Holy Spirit, and the Holy Spirit contains all properties of the Father and Son, while the three are not the same. This is not a direct correspondence to mathematics, but rather is using mathematics as an analogy to understand a theological concept which the human mind cannot grasp in its fulness.
If Trinity is summarized as follows:
G = F + S + H
F = S = H
F = G
S = G
H = G
This means:
F = F + S + H
S = F + S + H
H = F + S + H
then aggregating 1), 2) and 3) would violate the Law of Identity.
The Law of Identity states that A is identical only with A (i.e. itself).
In light of the Law of Identity, 2) and 3) in the brief presentation of Trinity above make equivocal and contradictory statements.
define ‘god’ -
Christians (trinitarians anyway) define God as the combination of the three beings - all equal, all separate, all becoming (or being) the one thing called “GOD”.
You could similarly call man a “three pronged being” - body, mind, soul - and it wouldn’t be that far off the mark.
It is not a logical argument but one of faith - which does not require, by definition, any logic or reasoning.
You keep using ‘aggregate’ or a summation (plus ) - when it could be a multiplication issue -
1x1x1 still == 1
1 /1 still ==1
You’re also (like the OP) assuming “Identity” is something its not been defined as - at least as far as the Christian Trinity is concerned.
Mathematics is a language that I used to master relatively well before I went to university (i.e. long ago). I’m not against using math concepts, but I don’t see how this is useful in the case of Trinity.
For instance, point (1) states that G=F+S+H whereas point (2) states that F=S=H.
Whatever makes F, S and H different in point (1) should make it impossible for F, S and H to be the same in point (2). Accepting (1) and (2) at the same time leads to contradiction.
A sylogism cannot be equated to a series of mathematical operations. In a logical argument, premises cannot be multiplied. (N.B. The Law of Identity belongs to formal logic.)
Great points! And it’s amusing that two valid answers are infinity and zero … which pretty much sums up how different people assign the values. 
(BTW, I believe any infinity works, aleph null merely being the smallest.)
You’re assuming that justice has an objective definition, which is a very unusual position among philosophers and scholars. Ayn Rand would agree with you, but she at least goes to the trouble of explaining why. Religious people can say that God decides what is just; that’s still subjective to God.
When you say you feel something is just or not just, how do you know that’s correct, or is that just what you think? Sociobiologists will tell you that we have an evolved “moral sense” that emotionally informs us when something is fair or not. If they’re right, this moral sense is not based on some objective standard for right and wrong, it’s based on what works. We all use our moral senses as guides, but we should be careful not to assume it’s always the best judge.
Restatement doesn’t clarify.
If you said this:
F = G
S = G
H = G
F != S != H
then you’d have to be dealing with a situation where the equality operator is intransitive, or else you’d have an inconsistency. I’ll admit I have a problem with that; I can’t imagine any use for an intransitive equality operator. But is anyone actually saying that F != S != H? Perhaps.
In any case, I don’t need the Trinity to find the concept of God illogical. To me, it’s illogical from the get-go. But then, the existence of anything at all is a bit illogical. Everything would be so much simpler if there were nothing!
Sylogisms aren’t valid logic in any case. Since it doesn’t treat existence specially, you can prove all sorts of silly stuff. For more info, see Syllogism - Wikipedia .
Is that not what you are trying to do in your “summation” ?? You’re trying to construct a mathmatical model for “god” or the “trinity” - I’m simply showing that there is a mathmatical model that does resolve just fine for the trinity.
And what part of ‘God’ or “The Trinity” is a logical argument?
Had you actually started out with this as the OP, we’d have been further along with less snark.
The Trinity has been a point on which various Jews and Muslims have criticized Christianity as not monotheistic for a long time. Christians are aware of the criticism and it does not seem to have had any deleterious effect on their faith, but you are welcome to make your pleas as long as you remain civil in this thread.
Ice, water, steam.
The co-exist at the “triple point” of water.
Anyway, it’s religion. It means whatever the believer wants it to mean.
I wonder if you have applied prepositional logic, which is a branch of mathematical logic (about which I have no idea). I don’t think math is the key to the Trinity logical problem. And even if it were, the propositions and their implications should have noncontradictory equivalents in plain English or else logical paradoxes can be invoked by anyone in order to justify just about any flapdoodle.
It seems to me that the concept of the singleness of God’s person, who should be regarded as consisting of three distinct persons, violates the Law of Identity.
True. The question then is should we invoke logic anymore in learning what there is or there isn’t or we should we uncritically accept every statement issued by some authority?
I understand now that the way I was trying to formalize the ideas has made people think of a mathematical solution, but that was not my intention. I meant the equal sign to stand for “identical with.”
The statement above (i.e. “Ice, water, steam.”) is misleading. I know that in informal logic water is the same as liquid water, but if we want to be rigorous it should be stated as “Ice, liquid water, and steam.” Since ice is solid water and steam, gaseous water, there is no contradiction in holding them all as water since non-contradictory modifiers do not turn a single object into a plurality.
Well, presumably, the idea is that a wave is ‘not a particle’, and that thus, a photon, being both a particle and a wave, is therefore a particle and not a particle. But of course, that’s debatable from a quantum-mechanical point of view: a photon manifests wave-like properties only in the context of a specific experimental arrangement, and particle-like properties likewise. They’re complementary properties, and part of this complementarity relationship is precisely that both aspects are not simultaneously observable.
However, the view that quantum mechanics poses a challenge to the law of identity has certainly been expressed before, based on some other phenomena it gives rise to. It depends a little bit, however, on how exactly you understand identity: if, for instance, you understand it to include the identifiability of objects, their having some sort of individual existence that makes it possible to tell them apart, then it’s arguably violated in quantum field theory—two electrons are indistinguishable to the point that you can’t put a lable on each of them to say ‘that’s electron 1’ and ‘that’s electron 2’. This is in fact an important factor in the statistics ‘gases’ of electrons or photons follow, which are different from those of classical particles that you can imagine as being composed of tiny little balls that bounce off one another.
It’s not something I’m ultimately really qualified to comment upon, but it’s a view that has some adherents, I think, and has been defended in this book a couple of years back.
They are all unambiguously H[sub]2[/sub]O, however. They’re substantially the same, but different individually—different ‘modes of presentation’ of the same thing, perhaps. You can make sense of the trinity in the same way—the Father, Son, and the Holy Ghost are all God, and there is only one God, in the same sense there is only one H[sub]2[/sub]O, but they’re different ways the substance of God may present itself, just as there are different ways H[sub]2[/sub]O may present itself, as a liquid, as a solid, or as a gas.
I can see both the beauty and coherence of the analogy, but I still think this argument fails to logically support, for instance, the idea that the Father and the Son are distinct persons and yet make up the single person of God.
When steam leaves - its still ‘water’ although physically separate from the main body - it can even ‘stop being steam’ and rejoin the main body with no inherit change in either (other than state).