On an alleged counterexample to Leibniz’s Law – Part 1

In a recent post I put forward my own preferred version of “Leibniz’s Law,” or more accurately, the Indiscernibility of Identicals. It’s a bit complicated, so as to get around what are some apparent counterexamples to the simpler principle which is commonly held.

Aside for non-philosophers: philosophers are usually after universal principles, truths which hold in all cases, rather than mere non-universal generalizations, i.e. rough rules of thumb which have exceptions. (An example of the latter: Boys love trucks.) Thus, when a philosophers makes a (universal) claim, other philosophers come along and try to show that it is false with “counterexamples” – real, or even merely possible, examples which show the principle to be false (as it doesn’t apply to them). For example, if someone says that all Texans love tacos, a counterexample to this would be a person who is from Texas and doesn’t like them. Just one counterexample is enough to show a universal claim to be false. When provided with a counterexample, of course, one will often refine, as it were, the original claim (e.g. All native Texans love tacos, or All Texans who appreciate Tex-Mex food love tacos) and the game goes on. This is all in the interest of discovering together what is true and what is false. (In my example, of course, those “refinements” would admit of easy counterexamples too.)

So my principle said, to paraphrase, that for any x and y, x just is (=) y, only if they don’t ever intrinsically differ. (I put this in terms of one having a “mode” at a time if and only if the other also has that mode at that time. Others would call these “intrinsic properties.”)

Here our friend, philosopher and blogger Brandon offered a counterexample, in comment #35 on that post.

if there is any entity that necessarily knows itself completely, its being both a subject of self-knowledge and an object of self-knowledge would seem like an intrinsic property. Now, if its complete self-knowledge is genuine, itself as known by itself just is itself as knowing itself. But itself as object can’t have all intrinsic modes in common with itself as subject, because the intrinsic properties of objecthood and subjecthood themselves are different: objecthood and subjecthood are intensionally different and this is essential to what they are. Thus it would seem that itself as subject and itself as object are intensionally different, that this intensional difference is intrinsic. So it seems at first glance that we have itself as subject just being itself as object, and yet itself as subject being distinct as to intrinsic modes from itself as object. I assume you’ve considered cases like this, so the question is, why isn’t this a counterexample?

Brandon is describing a case where, in his view, x = y and yet it is false that one intrinsically is a way if and only if the other is too. In other words, this is supposed to be an example of it being true that x = y and yet x and y differ. In subsequent exchange (comment 47) Brandon accepts my paraphrase of this in terms of God-as-subject and God-as-object. He’s assuming those are numerically identical yet they differ. How so?

That God as subject is subject and God as object is object.

In other words, the first is subject of knowledge but not object of knowledge, and the second is object but not subject.

I’ve granted earlier in the discussion that being a subject of some knowledge (e.g. knowing that pizza usually has cheese) is a “mode” or an intrinsic property of a person. So if there is any actual or possible case in which something simultaneously has and lacks this mode, then my principle is false.

I’m not sure he’s given us this. But let’s see what else he says:

After all, it’s at least enough to distinguish them that we can put God, under the intension of ‘subject’, into x, and God, under the intension of ‘object’, into y, and keep the two distinct all the way through. If this kind of intensional distinction is or can be a distinction in intrinsic modes of subjects as opposed to those of objects, then the consequent equivalence is broken without breaking the antecedent identity. If extensionally identical values of variables can under any circumstances have intensionally distinct intrinsic modes, the conditional doesn’t hold for those.

Whew – the philosophy lingo is coming hard and heavy here! Let me try to translate or paraphrase:

First sentence: we have concepts of being known, and of knowing. And we can think of God in either way – as being known (by himself) or as knowing himself. When we think of God in the first way, let’s call that x, and when we think of him in the second way, call that y.

Second sentence: this x and y differ, and if this can be a difference of mode/intrinsic property, then Dale’s principle is false. (It would be true that x = y, but false that they don’t differ – so the whole thing, that x = y only if they don’t differ, would be false).

Third sentence: the x and y refer to the same thing (are “extensionally identical”) yet x differs from y. Booya!

Brandon continues:

All identity statements have to assume that some intensional distinctions don’t matter. In x=y, we obviously are intensionally treating x and y differently in some sense — they get different letters to indicate that they are different variables and they have different locations in the equation (to the left and the right of the equality sign, for instance). We simply assume that these can be ignored to make sense of the statement as an identity statement; this allows us to focus on purely extensional matters. It’s when we get into the sorts of intensions that are typically handled by things like modal operators that things get tricky. It’s precisely this that causes problems for the standard version of the Indiscernibility of Identicals — it fails in certain kinds of plausible temporal logics, epistemic logics, etc. (because it fails to take the quirks of the relevant intensions into account), which is equivalent to saying that you can propose temporal, epistemic, etc. scenarios that are plausible counterexamples.

I would say yes, an identity sentence treats “x” and “y” as different terms. But this doesn’t assume any difference whatever in that to which those terms refer. But a sentence like “x = y” is not asserting the terms to be one, but rather the things. I don’t think any differences are being ignored; all agree that we can refer to things using different words. About these other alleged counterexamples – let’s just deal with this one first.

I’ll pause here to make sure I’m getting all this right; I’ll respond in my next post.

About Dale

Dale Tuggy is a Professor of Philosophy at SUNY Fredonia, where he teaches courses in analytic theology, philosophy of religion, religious studies, and the history of philosophy.

3 Responses to On an alleged counterexample to Leibniz’s Law – Part 1

  1. Kenny says:

    It seems to me that Brandon’s observations could easily be treated as an argument against direct realism, i.e. as a reason to think that talk about ‘the object itself existing in the understanding’ (as the Aristotelians would say) is incoherent. The thing that exists in the understanding must, simply in virtue of existing in the understanding, differ (intrinsically) from the thing that exists in the world, and so they are non-identical.

  2. Brandon says:

    So if there is any actual or possible case in which something simultaneously has and lacks this mode, then my principle is false.

    This would indeed be a way it could be falsified, but it is not actually required for falsifying the principle, at least as you’ve stated it so far; all that is required is that list of intrinsic modes it has when it is the value of one variable is not wholly intersubstitutable with the list of intrinsic modes it has when it is the value of another. This is why I keep hitting the topic of intensional equivalence-breaking: intensions can take one and the same thing with a given set of features, characteristics, or properties, and divvy up those properties according to different intensions. So, for instance, the man who is Mark Twain can be the same man as the one who is Samuel Clemens, but if John doesn’t know this, then this creates a context in which there are intensional descriptions under which one cannot attribute the same thing to Mark Twain and to Samuel Clemens, despite their being the same person. It’s not that the man in question simultaneously both has and does not have something, outright; it’s that he can have it if we consider it only under one intensional description and lack it if we consider it only under another. So any context in which those intensional descriptions are important becomes a potential source of counterexamples.

    I think, from things you keep saying, that you keep seeing this as confusing terms and things, or some such, but this is not the issue. If there were no problem with intensional contexts and descriptions, your whole project here would be otiose: it is only if intensional descriptions, such as those involved in tenses, affect what modes/properties/whatever are available when a thing is ‘plugged in’ to a given variable that there are any problem cases for the original Leibniz’s Law at all.

    x=y -> (Fx < -> Fy),

    in other words, is only problematic in the cases you’re trying to work around if in those cases it really does matter (for whether F can apply to something) whether you are plugging something into x or plugging it into y. Since, ex hypothesi, we are plugging the same thing into x and y, that means that x and y must be taking the same value in different ways (i.e., they are intensionally different). The original only needs to be reformulated if intensional descriptions, like temporal or epistemic modalities, already can make a difference; if they don’t, your reformulated principle is unnecessary.

    Also, as I think I said, the subject-object case wasn’t proposed as a counterexample but as a clarifying non-counterexample: I’m trying to get a grip on how you understand your principle, which I still don’t grasp because I don’t understand how the W predicate is working, by seeing exactly why this example isn’t a counterexample to it.

    I would say yes, an identity sentence treats “x” and “y” as different terms. But this doesn’t assume any difference whatever in that to which those terms refer. But a sentence like “x = y” is not asserting the terms to be one, but rather the things.

    I agree: In other words, again, we are simply assuming that the referents are not themselves qualified by any intensions of the referring terms; that is, we are assuming that we can ignore any intensional differences due to the letters used, the location of variables in the formula, etc. The question, though, is how far this is going to go. There are contexts where it is not so easy to ignore intensional differences in this way; as I note above, that’s why (and it is the only reason why) we get the modal problems with change, personal identity, etc., that we get with the standard principle of Indiscernibility of Identicals.


    It certainly could be turned into an argument against direct realism; but I don’t think it’s a very plausible one, because one could equally deny one of the other assumptions about identity, which, in fact, I think people do already — nobody really thinks that real identity depends on there being no reasoning-relevant intensional differences whatsoever.

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