The law of identity, they call it.
It is one of the three classical laws of thought and states, apparently simply enough, "each thing is the same with itself and different from another." Particularly, it urges us to realize that A is not 'not A,' for every anything A.
In some ways it is a big deal and it is important. Notably, it's important in regard to linguistic points. When we use the term firefighter, we mean something specific--which Ray Bradbury used to good effect in Fahrenheit 451. Particularly, if we talk about firefighters in a particular context, it's considered falling afoul of the law of identity to suddenly switch to another meaning for the term. Of course, this kind of equivocation is very common, particularly on terms like "God." There's nothing that bridges the gap, for instance, between the philosopher's abstraction known as "God" and the somewhat more concrete variant of that notion embraced by Christians or Muslims--a point where apologists are too eager to equivocate.
The law of identity seems to get abused heavily in this way, in fact, and it seems to come up a lot when someone with a little philosophical savvy wants to prove the existence of God or, more recently, that Platonism is legitimate. I don't really understand what motivates people to believe that the law of identity somehow counts in philosophical favor for the existence of God, to be honest. I'd welcome a cogent explanation, but I mostly expect that these people just want to be able to say something that sounds superficially smart, using that as a talisman meme to ward off attention from the fact that we don't have any credible evidence that God exists.
Anyway, abuses aside, the law of identity seems entirely straightforward, but is it? I'm not so sure.
There is an easy objection to it by bringing time into the matter. A relatively famous maxim states that "no man jumps into the same river twice, for it is not the same river, and he is not the same man." We have a sense of sameness of object, particularly of person, but it's hardly clear that when A is me or you or somebody that there's any real permanence there. For brevity, I'll skip the almost banal discussion of how quickly our tissues renew, our patterns of thought change, etc., with the sole exception of the tricky issue of whether or not Jane is still the same Jane after she develops dementia.
As the maxim indicates, other complex cases, like rivers for example, are obviously tricky on the law of identity as well--for the same reason. Given any cross-section of any river, the particular collection of molecules (be those part of the water, particulates, bacteria, wood, fish, etc.) flowing through that cross section at any given instant in time is entirely unique. The course of the river changes with time, the depth and current of the river change with every rainfall, even miles away, and so on. Is the Mississippi River the Mississippi River? What is even meant by that?
There are good ways around the problem, for most intents and purposes--clearly we know what is not the Mississippi River, for example. We could define the mighty Mississippi or our friend Jane by a set of properties, Jane's dementia making those properties a bit more difficult to pin down exactly, and those properties seem to make the matter less ambiguous. This problem is a bit too big to wriggle out of, though, just by using "properties." Why? Because almost everything changes almost all the time, and we don't have any real reason to believe that past or future states really exist. It may not mean much to say that something, any something A, is ever A again, at least in the exactly-the-same sense implied by equals.
We might agree to call a particular body of water the Mississippi River, for instance, but the Mississippi River we're talking about at this moment already isn't that Mississippi River and never will be again, nor was it ever that Mississippi River ever before. Jane is still kind of Jane when she develops dementia, but she's already really "not herself." Further, what is she when she dies? A=A implies a certain sense of transcendent eternity--which we literally do not have. So if A=A, then A=A right now, and that's as much as we can say about it, but right now is already over.
This problem of identity being fundamentally ephemeral, even down to the quantum scale, it appears, is a serious issue of incoherence for the law of identity if we want to accept that this law is true (or even better, obviously true). I mean, in a sense it is true (even obvious), but in another sense, it isn't at all true. In fact, in that second sense, it is clearly false. Given this, I'm not entirely convinced that the law of identity is what lovers of classical thought want it to be: a universal truth, often tied up in mathematics since it is usually taken as a fundamental axiom in mathematical systems. Near the end of this little essay, I hope to point out under what conditions it is that this "law" could be considered a universal truth, but I want to clarify this "in one sense true, in one sense false" business first.
Let me allude to Taoist thought to illustrate what I mean. In Taoist thought, there are two fundamental concepts: yin and yang. These concepts are stunningly hard to define on their own because they are polar opposites to one another. Literally, yin refers to the shady side of a hill, and yang refers to the sunny side. Relatively speaking, yin is dark, cold, receptive, earthy, feminine, solid, sinking, etc., while yang is bright, hot, projecting, airy, masculine, rising, etc. On the famous taijitu, what most people call a yin-yang symbol, yin is the black parts, and yang is the white.
Now consider a pool of water, perhaps just off the Mississippi, one where
Jane used to dip her toes on hot summer days, though she hardly
remembers that now. Is that pool of water yin or is it yang?
It's tempting to say that it is yin, but that is incorrect, as the earth that forms the basin it rests in is more yin, rendering the pool relatively yang in comparison. It's less tempting to say that the pool is yang, and as it turns out, that too is incorrect, as it is far less yang than the sunlight glinting off it. The pool may appear quite still, yin, but it is still presenting movement within it and on its surface, yang. Both answers, yin and yang, are incorrect--and yet both are, in a measure, correct.
What this Taoist construction presents is a third truth value to our familiar Boolean values of true and false. Here, the statement "the pool is yin" is neither strictly true nor strictly false; it's somewhere in between. An argument could be made that the statement is false because the pool is not "pure yin," but "pure yin" doesn't really exist. There's no such thing, and so such a comparison is vacuous and inadmissible. Faced with a third truth value in that system, we could call it "neither" or "both," I suppose, but that's immaterial.
Now, let's revisit the question of whether or not Jane is the same person today as she was yesterday, before or after dementia or death, and if the Mississippi River is still the same Mississippi now as it was five minutes ago, or as it will be in five minutes time. The answer, I think, is that there's something wrong here with the word "same," it applying on two different levels, giving the sense of "yes" and "no" simultaneously being correct. This is just like saying that the pool is yin: yes, but no.
The problem hopefully uncovered, before nailing this down, let's consider something a bit different, a "universal" idea. Consider the number three and the property that is "threeness." Three trees, three pencils, three-letter words, three meals a day--all of these exhibit fundamental threeness. So, if we take A=3, is it accurate to say that 3=3? Well, of course it is. Will it be 3 later? Yes. Was it 3 before? Yes. Would it be 3 even if there were no minds to name it so? In a sense, yes--though this is murky territory. So here, the law of identity works just fine, just like it seems to for Jane or the Mississippi River if we transcend the "mundane" details (in quotes because, actually, they matter quite a lot).
What gives? Why is 3 different than Jane here--but not totally different from Jane? Specifically, why does 3=3 timelessly but Jane=Jane have serious and obvious issues with it?
Timelessness. That answer sounds like a cop-out, though, so we have to ask where timelessness occurs. The answer: in abstractions. Anywhere else? I don't think so. Three is an abstract notion that describes a certain property, and that abstract notion endures beyond the details of what exhibits it. The person of Jane, legally defined or done better by, is also an abstract notion in the minds of everyone who knows her, including herself. That gives us a sense of continuity to Jane, the myriad states that have been Jane, are Jane, and will be Jane, be she a fetus in the womb, toddler with a water sprinkler, firefighter, or dementia-crippled shadow of her former self (or is it better to identify her as an entirely different person in each of these cases, each taking the label "Jane" with a sense of evolving continuity from one state to the other?).
"Jane," referring to the "same" person, means different things at literally every moment in time, so Jane does not equal Jane literally except when utterly frozen in time. Our abstraction of Jane, mostly composed of memories and guesses, however, is timeless. That Jane=Jane, even as Jane changes throughout her life and eventually after the end of it. The same is true of the Mississippi River, even after the land changes, and the great river is no more. The abstraction of the river is a timeless thing, and it "exists" as do all abstractions in the minds of anyone able to think about them. Certainly, as the details change, the relevant abstraction can be updated continually to encompass the changes, and by considering the problem from this meta position, it makes sense to apply the same term.
Abstractions, yet again, are the special things, in that they are timeless--because they are not real. Here, they are the things that satisfy the law of identity--because they are timeless--because they are not real.
Maybe, then, this is why the apologists and the Platonists are so interested in it? Maybe it is that they want to equate abstractions with reality. Sure, we only experience reality through our abstractions of it, but this is a question that is likely to have more to do with the nature of consciousness than with whether or not abstractions have concrete reality. Religious experience, at least under the Abrahamic faiths, requires a certain degree of Platonism, a belief that ideas have a sort of concrete reality to them. Thus, this law is important to them, perhaps. It's a centerpiece of classical thought about abstract ideas.
This ties right back into the usual use of the law in the first place: avoiding equivocation. When we talk about a firefighter, we're appealing to a certain abstraction--the word not being the thing--and it is disingenuous to imply one abstraction while meaning another.
Does A=A? Sure, as long as A is abstract.