That’s fair: in re-reading, I realize that you were actually commenting on my comment to someone else. Apologies: I thought you were the original person I was replying to.
From a web search: Says electrons lack an identity and so x = x doesn’t apply to them. There’s a Schrodinger logic in which \neg x = x.
I’m not sure what you mean here: could you elaborate? An infinite regress of justifications entails no member has the ability to provide inherent justification; and thusly none of them would be capable to have it since nothing would be providing it to them. This is analogous to an infinite chain of ovens without a power source: no oven would be on. If someone tells you that each oven gets its electricity from the previous ad infinitum, then they simply are not understanding that an oven cannot innately provide electricity; so if even one oven is on then there’s an external power source to this infinite chain of ovens.
I don’t think @Banno nor @Jay are right here: ‘x=x’ is a valid proposition. ‘x’ represents anything quantifiable; and, as such, it is truth-apt. Either any given ‘x’ is identical to itself or it is not: we don’t need ‘U(x)(x=x)’ for this.
CC: @Scorpion
The issue is that @Banno is denying that they are affirming ‘x=x’ without any justification to sidestep my objection that they are simply biting the bullet as a response to this Trilemma (by affirming the ‘dogma’ option as valid). If Banno can assume ‘x=x’, then I can assume ‘x!=x’; and there’s no way to prove each other wrong or right.
You’re right about the missing grounding of propositions that is the essence of infinite regress. I was wrong.
\neg x = x is just another stipulation, like x = x is. Google for logics that reject x = x.
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It was stipulated by @Banno as an attempted solution to the Trilemma: I think, if I’m being charitable, Banno is trying to convey the concept of self-evident beliefs (i.e., a belief that is grasped as true by merely understanding what is contained intrinsically in its own concept). Hence why they say things like “equality is just what ‘=’ does”: it’s analytic a priori I suppose.
The problem is that (1) some self-evident truths are not fundamental (e.g., ‘1 + 1 = 2’ is self-evident but is not fundamental like ‘a=a’) and (2) fundamental beliefs are justified from within themselves (as opposed to Banno’s view where they are assumptions). E.g., we are justified in believing ‘1 + 1 = 2’ is true because we can further justify it with underlying beliefs (like “1 = 1”); but ‘x = x’, which is also involved in justifying ‘1 + 1 = 2’ is not further justified in this manner: it is justified through its inevitable deployment by reason (even when arguing against it).
You could equally say that x=x because x=x, and so the justification is circular. This has the same validity as “dogma”. What this shows is a limit built in to the trilemma. The Trilemma doesn’t address constitutive rules.
So Asking “what justifies x=x?” is a false move within the formal system; that x=x is constitutive of the formal system. Much like asking what justifies keeping the bishop on the red squares during a chess game is to misunderstand what constitutes the game.
It’s quite certain that the bishop stays on the red squares. Supposing anything else is to stop playing chess, or at least to change the rules. So it simply isn’t true that we do not have any certainty at all. It seems odd to say that the rules of chess are “baseless” - they are how chess works. And yes, @Scorpion seems to understand that, given their last post. If we instead stipulate ~(x=x) we are not changing how things are, so much as the things we can say.
That the bishop stays on the same colour is hardly self-evident - after all, it could have been otherwise. Nor is it quite right to call it an assumption in chess; it is a part of what chess is.
Indeed, this:
points in the same direction as @Jay and I. x=x does need clarification, something more to make it a proposition; either an instantiation or a quantification.
See especially from about 4:00
In hindsight, it might have saved confusion had we worked with a=a.
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Right, and it is part of the Wittgenstenian move to say, “Ah, but that’s not a proposition, so that doesn’t count.” This sort of answer employs a technical sense of “proposition” which evades the very point at issue.
Note too @Banno’s recurrent insistence that we must be playing a game, like chess. Or look at his anachronistic portrayal of Leibniz’ law, pretending as if Leibniz was playing Banno’s “game” of modern predicate logic. Leibniz’ law is anything but Banno’s symbolic game; it is formulated as a metaphysical principle.
Part of the sense of the OP is the assumption that justification is more than game-playing, and that moving a chess piece is a fundamentally different sort of act than justifying a proposition or affirmation.
Oh, Leon.
The difference between free and bound variables is common to all of modern logic; it’s not to do with Wittgenstein, it’s what happens in first-order logic, coming into play as soon as we introduce variables.
Now Leibniz never wrote ∀x ∀y (x = y → (φ(x) ↔ φ(y)). We can agree to that. While that’s the presentation usualy found in first order logic, it’s not the whole of Leibniz’ Law. We might add a variant(∀F)(Fx ↔ Fy) → x=y. Notice the quantification over the predicate F that makes this second order.
Now the first is a tautology, treatable as a stipulation in the way I have been suggesting.
The second is closer to Leibniz metaphysical claim, saying that there can’t be two things that are indiscernible. But we can read this as a bit of metaphysics about the actual world, or as a specification about the domain, for example that it never contains two indiscernible iron balls. We could equally specify a domain containing two indiscernible spheres of iron and so not including Leibniz’ Law. In either case it is not a tautology.
What this means is that in the first case, the first order example, it is a tautaology. But not in the second - if F is restricted to qualitative predicates and leave out God.
That’s the confusion here, set out as clearly as I can make it - that Leibniz did not have the logical apparatus available to make this distinction, so he thought a metaphysical proposal was a tautology.
“Justification is more than game-playing” is no more than a demand that we acquiesce to Foundationalism, that justification bottoms out in a direct grasp of how things are, not another move answerable only to further moves. Questioning this is precisely what generates the Trilemma. If Leon wants to insist justification is categorically unlike a chess move, he owes an account of what a terminal justifying act looks like that isn’t itself either (a) another inferential move, subject to Agrippa, or (b) a bare causal impact from the world, which isn’t yet a justification at all…
Added: this problem with Leibniz was first pointed out by Bertrand Russell, who first used modern logical apparatus in order to understand Leibniz’ confusion.
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As far as I can tell, Banno’s on the right track. x = x is a stipulation; it’s also an axiom. We can assume the axiom \neg x = x sometimes. Google AI points to quantum/particle logic where identity doesn’t exist e.g. every electron is like any other electron as a situation where x = x may not apply.
I don’t know if x = x is self-evident. Consider the infinite regress problem here.
AI says x = x is a defense against equivocation.
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It doesn’t particularly matter whether a “logic” “rejects” x=x. Agrippa’s Trilemma has to do with justification, whether “logical” or otherwise. A stipulation does not justify. In that respect it is much like the “dogma” option of the OP.
@Jay wants to say that he has no idea what x=x means if the variables are not bound via first-order predicate logic. Remember, though, that @Banno is the one who introduced the term “x=x” into the thread, and he did not do so within the context of first-order predicate logic. In order to understand what a term means we must look at context. No term is intrinsically meaningful or unmeaningful, and first-order modern logic is not some kind of supreme arbiter of these issues.
There are philosophers who substantively oppose the law of identity. Deleuze and Heraclitus before him are two that come to mind. Their approach would bear on justification, and not merely on “logic” (in the sense of detached formalisms). Even if the “stipulation” was held by everyone, this would not count as justification; and it turns out that the “stipulation” is not in fact held by everyone. This is what I was getting at in the paragraph beginning, “How should we respond to Wittgenstein here?”
And it says a lot more besides. I tried to quote some of it, to show the consensus among logicians, but forgot that this would violate our guidelines, and so the post was removed. But the point I would have made was that the status of “x=x” is not controversial, proposition-wise. It isn’t one, unless you’re using shorthand and dropping the quantifiers. Gotta bind that variable!
You are engaged in errors at every turn.
- Not everything is first-order predicate logic.
- To speak of Leibniz’ law in terms of such a thing is anachronisic.
- AI does not represent a “consensus among logicians,” and it is particularly inept when we are speaking about specialized meta-logical questions.
- The response about “equivocation” is a case in point—a complete failure to apply the context we are working with.
Note too that, in response to Jay’s appeal to Sider, @Banno literally denies that quantification can occur over a “domain containing everything,” which is precisely what Leibniz’ law would represent if someone wanted to try to adapt it to first-order predicate logic.
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Part of the crux here is the fact that grammar does not avoid metaphysics, and contemporary philosophers have a really hard time with this. It reminds me of Srap wrestling with the fact that his favored anti-metaphysical thesis would end up denying the difference between nouns and verbs in sentences (i.e. the difference between being and doing). Whenever someone claims to be engaged in something instead of metaphysics they are confused.
Identity claims of all kinds take for granted the metaphysical concept that the Medievals called the transcendentality of unum (or oneness). This is incidentally what Deleuze attempts to undo.
Similarly, Banno’s emphasis on the innocence of “1=1” depends on metaphysical assumptions related to realism and universals:
(Note here that whoever would want to deny such a recognizable sameness in and across several singulars would have to deny that he is able to recognize the same words or the same letters in various sentences; so such a person would not be able to read, write, or even to speak, or understand human speech. But then we shouldn’t really worry about such a person in a philosophical debate.)
Identity relations are not metaphysically innocent, and any formal logic that presupposes the validity of identity relations is not metaphysically neutral.
Oh, Banno. I never said it was. The Wittgenstenian point and the first-order logic point are obviously separate.
Again, part of the problem is that Leibniz is thinking of an unrestricted “domain,” and on your view modern logic cannot express such a domain. Thus:
Presumably you would agree that an unspecified domain is necessarily “metaphysical,” and therefore if you also agree that Leibniz is not restricting his domain then you yourself should agree that his claim is metaphysical.
Regarding your logic:
If you say that this is first-order and not second-order then φ apparently belongs to the meta-language. Either way, in your formulation φ is unbound (and I wonder why @Jay doesn’t care in your case, even though you are explicitly trying to write first-order logic). If you meant the quasi-sentence as an axiom formula, then a petitio principii is in play.
To the more general point, I agree that your first thesis is often understood tautologically. The problem comes when it is asked to do work in the real world. As soon as it is applied we have a judgment that the logic is applicable to metaphysical realities in one way or another, and that is where “Agrippa’s trilemma” would come to bear. Stipulation qua stipulation is not subject to justification.
I will also say that your posts in this thread are much more substantial than is usual, so I appreciate that.
In a simplified sense I think @Banno is saying that he can stipulate anything he likes without being subject to questions about justification. He could stipulate “1=2” just as he could stipulate “1=1”, and he would not need to justify anything. Perhaps in the first case the three letters (‘1’, ‘=’, and ‘2’) are stipulated to mean something different than they do in the second case (‘1’, ‘=’, and ‘1’; where we could further stipulate whether the “same” letter on the right and left side have the same meaning).
But if you press these issues you are going to run into nominalism and voluntarism. The OP is presupposing a paradigm of realism and intellectualism, where truths can be known. In the nominalist/voluntarist paradigm everything gets reduced to will instead of intellect, there are no truths in the classical sense, and all language is reduced to mere words (“nomina”) divorced from reality.
The thoroughgoing voluntarist sees a discussion about the OP as being the same as a discussion about chess:
- Eudemian: Why did you move your bishop?
- Voluntarist: Because I wanted to.
- Eudemian: Why did you move him diagonally?
- Voluntarist: Because chess requires it.
- Eudemian: Why are you playing chess?
- Voluntarist: Because I wanted to.
On this view there is no truth to be had; there is no knowledge to be had. In the end there are only the different things we in fact do, and the only reason we do them is because “we wanted to” (will/voluntas). It doesn’t make much of a difference whether the answer is, “I wanted to,” or, “We/society wanted to.” It has nothing to do with that goulish spectre of “Foundationalism,” but rather with the deflationist’s insistence that there is no normativity, not even as relates to truth and falsity, justification, etc.
@Banno is always a step away from this move, and he will take it when pressed. Still, his tradition flows from empiricism in general and logical positivism in particular, and that tradition is surely not confessedly nominalistic. Banno retains realist streaks in his philosophy. @Jay, on the other hand, is strongly drawn to the nominalist and voluntarist paradigms, albeit for largely moral reasons. In any case, both of them will try to deny the terms of your OP.
The law of identity x = x is a simple idea. It means a thing is itself. An example is its instantiation, Roger = Roger. The main advantage we gain is that of consistency. Otherwise we would have odd situations like, mountain = snow or bicycle = frog. 
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Oh, Leon. Your posts, directed at me in multiple threads and with such venom, are frankly beginning to look neurotic.
Logic doesn’t reject x=x. It’s a common shorthand for U(x)(x=x), a standard formula in first-order logic with “=”. Yes, I introduced the term, mistakenly presuming a sufficient grasp of a common logic in my readers for them to see it as at least a first-order formula. Most understood.
And yes, Agripa’s Trilemma is about justification, rational justification, and therefore it is about what can logically be justified. An illogical justification is no justification.
Stipulations can be used in justifications. That the rules stipulate that four aces beats a flush of any sort is justification for taking your money.
Yes, not everything is first-order logic, however it is a good starting point. I am happy for you to produce an alternative and show why it should be preferred. I hope it clear that retreating to Aristotelian syllogistic is inadequate to that task, since that is already subsumed by first order logic.
In First-Order logic Leibniz’ Law becomes reflexivity: U(x)(x=x), together with substitutive: for every predicate P x=y → (P(x) → P(y)). Together these set out how “=” is to be used. It is not a theorem of pure first-order logic, it is an additional logical principle governing equality. It is introduced axiomatically or equivalently by stipulating inference rules. Once introduced, one derives the familiar substitution results from it.
What it is not is some Rule of Logic. Nor is it self-evident. It is not true in every logical system.
I suppose I should be flattered that you are following me from thread to thread. In the other thread you mentioned, I did not say that quantification cannot occur in a domain containing everything. Rather I pointed out that Cantor’s theorem would render such a domain forever incomplete. Whatever totality one specifies, Cantor shows that there is a larger one, so the notion of a completed domain of absolutely everything is mathematically unstable.
Of course there are metaphysical implications for many of our utterances. x=x rules out x≠x. We can choose the one that best suits our needs. We choose a logic that fits in with the metaphysics we would adopt. Now since Leibniz’ Law is not a Rule of Logic in the grand sense supposed, it is open to choose logics and their metaphysical interpretations that do not use it. Again, Max Black does exactly this in rejecting the Identity of Indiscernibles with his two balls of iron.
Leibniz could not have been thinking in terms of an unrestricted domain of quantification, since the modern model-theoretic notion of a domain did not exist in his time. Rather, he took logic itself to be universal in scope, to apply to everything that is. The later distinction between a formal language, its domain of interpretation, and the semantics relating the two belongs to Frege, Tarski, and twentieth-century model theory, not to Leibniz. Again, Leibniz lacked the logical machinery to seperate out reflexivity and substitution. We have that machinery and can make the distinction.
The difficulties surrounding unrestricted domains also postdate Leibniz, arising only with the development of modern set theory, model theory, and the associated paradoxes. Leibniz had no conception of a domain of quantification in the modern sense, so he could hardly have anticipated objections based on Cantor’s theorem, Russell’s paradox, or the impossibility of a universal set.
∀x ∀y (x = y → (φ(x) ↔ φ(y))) is indeed in the metalanguage, and again the universal quantification is presumed, the shorthand taken as granted. We could have written instead U(φ)(∀x ∀y (x = y → (φ(x) ↔ φ(y)))). Just as x=x is shorthand for U(x)(x=x).
Fun as this is, addressing your errors is time-consuming. Cheers.
Just so. Doing so would be to change the way “1” and “2” work, or to change the use of “=”. So I won’t. And I would not recommend it.
That is the justification; how what one stipulates fits in with everything else one says. And working that through is what logic is for.
Again, I am flattered at the sheer exertion you put in to misunderstanding me. Cheers.