Wednesday, September 24, 2008

Supposing the Impossible

There is often thought to be something special about the Cartesian conclusion 'I exist'. It is, as Descartes says, "necessarily true whenever it is put forward by me or conceived in my mind." But there are any number of transparently necessary truths, e.g. in logic and mathematics. So why is this one so special? Descartes thinks we can doubt the others, after all:
[J]ust as I consider that others sometimes go astray in cases where they think they have the most perfect knowledge, how do I know that God has not brought it about that I too go wrong every time I add two and three or count the sides of a square...?

How do you know that God has not likewise brought it about that you go wrong every time you reflect on the necessary implications of your own thinking or doubting? It is admittedly nonsensical to seriously entertain one's own non-existence, but it is no less nonsensical to seriously entertain that the second successor of 3 is not 5.

One objection (a friend pointed out to me) is that we often suppose impossible things to be the case, at least for sake of argument. I take it the purpose of this practice is to highlight some important partial truth. But we can do this just as well in case of the cogito. ("Suppose, for sake of argument, that I don't exist. So my parents only have four children!")

So I think the apparent difference is merely psychological. In case of any incoherent supposition, the thought experiment eventually breaks down, if pushed to the point where the contradiction becomes explicit and unavoidable. But some contradictions are better hidden than others. It seems easier to suppose that there are finitely many prime numbers, for example. We can create a rough mental model which seems (prima facie) to accommodate this possibility, and it will take a bit of pushing before it explodes on us. Denying one's own existence, on the other hand, leads to much more obvious and immediate difficulties. We can't even pretend to make sense of this, the way we can when entertaining the denial of some more complicated mathematical thesis. But it's worth noting that even the latter is mere pretense. There's isn't really anything there for us to grasp -- the deep incoherence renders the scenario ultimately unintelligible -- we are merely playing along for the sake of argument; drawing inferences and highlighting partial truths. You can't, strictly speaking, suppose gobbledygook. So logical necessities are on no less firm footing than Descartes' cogito.


  1. A can of worms: Might you get somewhere by distinguishing logical from semantic truths? Perhaps the claim "There is no such entity stating this sentence" is literally incoherent, whereas the claim "2+2=5" is simply necessarily false. The test being that I can make sense of the second assertion, but the first does seem to be a straightforward misuse of the words involved.

  2. I'm not sure I follow your test. Neither statement corresponds to a sensible or coherent state of affairs when taken as a whole. Both invoke 'locally' meaningful terms and relations, and so there's some looser sense in which I can understand or 'make sense of' what is alleged by both. (I can draw inferences from them, etc.) So in what remaining sense is only one of the two comprehensible? And - if we assume mathematics is synthetic - on which side of the boundary would analytic falsehoods, e.g. 'some vixen aren't foxes', fall?

  3. I think you've missed the boat, Richard. Descartes' test on the foundation is nothing but a psychological test: what we find we cannot doubt. So, we can doubt the truths of mathematics and logic, because we can look at any necessary formal truth and say "Gee -- is that really true?" (If you don't think that's possible, try teaching intro logic to undergrads some day -- some aren't even sure about the law of excluded middle.) However, we can't doubt our own existence as a thinking thing (or, at least, the existence of a thought), because the moment of doubt is identical to a thought. Doubt requires thought because doubt is thought. Hence why doubt leads us to the existence of a thought.

  4. I think ADHR is right that it's all psychological for Descartes, because, of course, the problem of skepticism is itself a psychological one. Think of it this way: it's entirely possible for us to make basic arithmetic mistakes, in a moment of lapse. So we can think mathematical and logical truths and be mistaken. But I can never think "I think; I exist" and be mistaken. This is the sort of anti-skeptical point Descartes is trying to find for his foundation; and then, given this, he can ask, "OK, but when I affirm this truth about which I am never mistaken, what, precisely, am I affirming?" And from then on we have the step-by-step building of the Cartesian system.

  5. I'm unconvinced. One might ask 'Gee -- is that really true?' about anything whatsoever, including one's own existence. Granted, it'd be dopey to do so, as the whole "doubt leads us to the existence of a thought" argument shows; but it's equally dopey to doubt whether 2 and 2 make 4.

    I see two claims that might be made here:

    (1) As a matter of contingent psychological fact, humans aren't capable of sincerely questioning their own existence, the way they can question just about anything else. I'm not sure whether that's actually true -- as noted, undergrads seem capable of doubting pretty much anything, or at least adopting a convincing pose as if of doubt -- but it wouldn't seem to be of any philosophical interest regardless.

    (2) More plausibly, the claim might be that one cannot reasonably doubt one's own existence, that anyone who manages to do so thereby exhibits egregious irrationality. That seems true enough, and plenty interesting to epistemologists, but (again) it's also true of simple logical truths and other easily provable propositions.

    Brandon - you're applying a double standard: assessing the cogito in its specific form, and logical claims in the abstract. Really they are all on a par. I can never think "I exist" and be mistaken, but equally, I can never think "2+3=5" and be mistaken. I can be mistaken about arithmetic operations in general, but equally, I can be mistaken when reflecting on the implications of my own thinking (in general). In both cases, the specific argument is necessarily infallible, while the general process of reasoning (abstracting away from the particular details of this instance of reasoning) is fallible. I see no grounds for distinction here.

  6. Still doesn't work, I'm afraid.

    Per Descartes, doubt doesn't lead us to anything; doubt is the existence of thought. That's the point he's trying to make. If you ask "is that true?", then you're thinking the doubt the question expresses. If you ask "is the law of excluded middle true?", then you're thinking a thought of doubt. If you're hypothesizing the existence of an evil demon or that one is dreaming, then you're thinking a thought of doubt.

    The distinction between what's reasonable to doubt and what simply can be doubted is not viable at this opening stage of the Meditations. The evil demon or genius is a grossly unreasonable doubt, but it's a doubt, and thus Descartes says you have to take it seriously. But from that, he thinks he can pull self-existence: he thinks that the existence of doubt is the same as the existence of himself.

    The point is you can't doubt doubt -- doubt is there when doubt is there. (Basic self-identity.) And that's all Descartes says: doubt is there.

    I think what you may be objecting to is the way Descartes elides quickly from "doubt exists" to "I exist as a thinking thing" -- that much can be questioned (and Russell and Nietzsche, to name two, did). But the claim you're making, that necessary truths are as unquestionable as self-existence on Descartes' picture, isn't true.

  7. I think the reason many people doubt obvious mathematical truths like 1+1=2 (and many people sincerely do) is because they don't understand what the concepts involved actually mean. There are a lot of math dyslexics out there. I've had people - bright people - tell me that math is an empirical science. I chalk it up to an imperfection in their brains and/or plain ignorance. On the other hand, I don't think I've ever encountered anyone doubting the affirmation "I exist", although I don't doubt there are a couple people out there who do.

    So, understanding the proposition "1+1=2" apparently requires slightly more cognition than understanding "I exist", and that's probably why it's less transparent to most people -- but they're on the same footing logically in the sense that, in each case, if you can actually understand the proposition uttered, you _automatically_ know it's true. This, by the way, is of course not true for non-trivial necessary statements, unless, perhaps, if you have perfect reasoning ability (although I'm not sure it's true even then).

  8. It's worth noting that it's not actually nuts to think that math is an empirical science. That was Mill's view.

    That said, though, whether math is empirical and whether its claims are necessary may be two separate questions. (e.g., If mathematical terms designate rigidly.)

  9. Of course it's nuts to think that math is an empirical science. It doesn't matter what Mill or any other philosopher thought about it. Mathematicians never use induction to establish theorems, and theorems are never refuted by experience (it's not even open to possibility), so on what basis could one classify mathematics with the empirical sciences? Certainly not because of shared methodology, as I just showed; but then that's how the empirical sciences are usually _defined_ -- by their utilization of the scientific method.

    Mathematics is necessary a priori, and I'm not sure what you might mean when you wonder whether mathematical terms may 'designate rigidly'. Mathematical terms are not names.

  10. Anyway, back on topic...

    "you can't doubt doubt -- doubt is there when doubt is there."

    Again, I'm on board with the conclusion that 'doubt exists'. That's obviously true. But it doesn't follow that "you can't doubt doubt"; it merely follows (once again) that you'd be obviously mistaken if you did so.

    Put it this way: doubt is a propositional attitude. As such, it can take any proposition whatsoever as its content, including the proposition that doubt exists. Just like you can believe that you have no beliefs. You can have the belief, well enough. You'd just be wrong, is all. (Self-defeating, even.)

    Maybe the thought is that self-defeating claims are on worse epistemic footing than claims that are incoherent on other grounds? But I'm not sure why that would be so.

  11. Well, pardon the digression, but this actually illustrates perfectly how some otherwise intelligent people come to doubt mathematical platitudes. Because of very superficial similarities, one decides to assimilate mathematics with the empirical sciences (especially physics), and then carry the characterizing methodology of these sciences back to mathematics, where it is grossly misapplied. All of a sudden, it starts to sound sensible to say "well, I've never been in a situation where 1+1 was not equal to 2, but that doesn't mean it might not happen in the future." This kind of reasoning (which I've heard from several people) is obviously not possible with the statement "I exist", where the truth of the statement stares you in the face -- or is 'self-affirming', as ADHR says.

    I'm not saying the above reasoning is why Descartes decided to (briefly) doubt mathematical trivialities in the Meditations, because it's obviously not. But it does illustrate my previous point that, trivial as they are, mathematically obvious statements like 1+1=2 do require a little more mental processing than a statement like "I exist", and may be doubted by intelligent people given the proper background beliefs. Again, the same is not true for "I exist" or "doubt exists", and therefore the two kinds of statements are on a different psychological basis, although they obviously have the same epistemic footing if you just care to look.

  12. "Neither statement corresponds to a sensible or coherent state of affairs when taken as a whole. [...] on which side of the boundary would analytic falsehoods, e.g. 'some vixen aren't foxes', fall?"

    On the first point, my thought was that "2+2=5" corresponds to a necessarily false proposition, whereas that "there is no entity stating this sentence" corresponds to no proposition at all. As I say, this is a can of worms and I certainly don't claim to be able to fill out the details: but perhaps someone else can, and I'd guess that someone has tried before if you look around.

    On the latter, I'm not sure, though it's not impossible for the boundary between logical and semantic truths to be vague.

  13. Hi, Richard,

    I don't think there's any double standard here. Precisely what is significant about the cogito is that it doesn't matter whether I take it as 'specific' or 'general' -- it's a singular proposition, so they're the same. (A single instance considered generally and specifically is still that same single instance.) This is actually dealt with fairly clearly by Descartes: the sort of initial doubt he considers for mathematical propositions is explicitly a "slight and metaphysical doubt" about the specific case arising from consideration of the general case. No such slight and metaphysical doubt can arise from the cogito because there is no distinction between the specific case and the general case here: there are both me, thinking.

  14. Lvb,

    Just to argue Mill's side for a moment.... It doesn't matter what mathematicians do unless you prejudge the issue against the empirical account of mathematical judgements. Similarly, it doesn't matter whether we would or would not take future evidence as disconfirmation of a mathematical claim -- that could just as easily be a psychological quirk of humans as some deep epistemic insight. (Although, FWIW, that was Ayer's argument against Mill's view.) The question is what can be supported by argument: what is it that characterizes a necessary truth, and does that characterization fit mathematics or not? Which is where rigid designators could do some work.

    I don't really see on what basis you say mathematical terms are not names. Is this a fictionalist position you're defending? But, take a mathematical term: "one". If "one" doesn't name an abstract object (whether this is a real Platonic whatsit or a presumed object per Quine -- or was that Putnam? -- or what have you; let's suspend the ontological questions as much as possible), what does it do? How does it mean anything? Is it a purely logical term?

    Anyway, dragging this back to Descartes, I think he's got a point in that it's less obvious that mathematics is a domain of necessary truths -- as evidenced by the fact that Mill, a very smart guy, denied it -- than that doubt exists. So, mathematics is more dubious than the cogito.


    Brandon covered it (I think), but let me repeat: if you doubt that doubt exists then doubt exists. You can't get out of that.

  15. ADHR, there's no need to repeat what I've granted all along. The disagreement lies elsewhere.

  16. I know Richard hates when the discussion strays, but I hope he'll bear with this,

    Of course it matters what mathematicians do. Empirical science is defined by its methodology; if mathematics partakes in none of that methodology, the burden of proof is on _you_ to show how on earth it still makes sense to call it 'empirical'.

    And no, it's not a psychological quirk of mathematicians that they refuse to acknowledge that theorems may be refuted in light of new evidence. It's _logically impossible_ to refute a theorem with evidence.

    And fine, you can say that mathematical terms name abstract objects if you feel like it. It still won't make sense to question whether they are 'rigid designators' or not: abstract objects are not actually _in_ the world, so their designation naturally won't change across different worlds.

    Richard, ADHR, you guys are talking past each other. ADHR said in his first post already that "Descartes' test on the foundation is nothing but a psychological test". There is no ontological OR epistemic disagreement between the two of you as far as I can see (unless ADHR actually agrees with Mill). Like ADHR and I have both been stressing, it is just a psychological fact that mathematical platitudes are more dubious than the cogito. The case of Mill (and other misguided philosophers) shows this.

  17. The presumption that empirical science is defined by what its practitioners do isn't enough to get the claim that mathematics consists of necessary truths off the ground. (And it is a presumption. Empirical science can also be defined by the standards the practitioners should live up to. It's surely a mistake to quickly equate sociology of science with philosophy of science.) There's a gap between what mathematicians report doing and what mathematicians actual do. They may say they don't allow evidence to undermine their mathematical claims, but we don't need to take this seriously. Indeed, there is reason to think we shouldn't -- the empirical fact that other mathematicians disagree can sometimes serve to undermine a belief in a mathematical claim, which looks like at least a close cousin to empirical refutation.

    I don't want to push the point too far, as I do think Mill is quite wrong, but I think it's too quick to conclude that his view is just silly or crazy. It's a substantive and non-trivial matter to show that mathematics consists of necessary truths, hence why (for example) Ayer takes a big chunk of a chapter (the fourth, if I remember right) of Language, Truth and Logic to do just that.

    I've just picked up, though, that you're talking about theorems, while I was talking about mathematical claims. Denying a theorem is logically impossible, clearly. But I don't think anyone -- even Mill -- would say theorems are empirically confirmed. If that's the point, then, yes, it would be crazy to claim these follow through some sort of empirical method. (Although, if logic were argued to be empirically confirmed, then it's still possible -- that looks pretty radical, though.) The issue for Mill is things like standard arithmetic or Euclidean geometric claims -- 1 plus 1 equals 2, for example. Apologies if I was talking past you.

    Re: the rigid designator issue, as I understand the notion of a "possible world", it can be cashed out in a non-ontological, purely logical way. Given that, abstract objects do end up in the world (there are true sentences involving them, after all). If possible world is ontologically loaded, then of course abstract objects don't wind up in the world -- they're abstract, after all -- but that makes it mysterious why we would take abstract objects to exist at all.


    Lvb is right. Descartes clearly (to my eye, at any rate) opens the Meditations talking about doubt as some brute psychological state. He tries to build epistemology out of those pieces. So, the disagreement -- if there is one -- has to be on whether we can undermine the claim that doubt exists. And Descartes, rightly, says we can't. We can't doubt that doubt exists, for doing so entails the existence of doubt. So, that's the foundation, and we build up from there.

    I suspect what's being lost is that epistemology doesn't exist in Descartes' system until he's got the foundation laid. The basic structure of the argument -- radical doubt; I am, I exist; God exists; God is no deceiver; my impressions come from outside my mind -- is intended to create (or, speaking more precisely, explicate and analyze) epistemic categories.

    I'm not sure whether you and I are ontologically or epistemically disagreeing, because there doesn't seem to be an ontological or epistemic point at stake. What's at stake is the psychology of doubt.

  18. For the most part, there really isn't that much to mathematical truth. Mathematical claims are true or false within a framework of axioms. The peano axioms correspond to our normal arithmetic based on the basic operation of counting. In this framework, 2+2=4 is a theorem. Of course, you can mess around with the axioms of any mathematical system if you want, as long as the result is consistent. We all know there are consistent alternatives to Euclidian geometry. And you can even make 2+2=5 if you feel like it, but the numbers 2 and 5 in this case obviously won't be the same entities as the 2 and 5 described by the peano axioms, although they might share several properties in common. Mathematics is more concerned with structural relations rather than the particular entities involved anyway.

    It's true that there are controversies in the foundations of mathematics (but not about such issues as whether 2+2=4). Is the Continuum Hypothesis 'really' true? Is the Axiom of Choice valid? Do transfinite numbers really exist? But these controversies are really philosophical problems, they don't belong to mathematics proper (a lot of mathematicians don't even think the questions make sense), and in any case there is certainly no conceivable way in which they could be settled empirically (I'm not convinced there is any way to settle them at all). Of course, the question of which mathematical systems best describe our actual world IS an empirical question.

  19. "there doesn't seem to be an ontological or epistemic point at stake."

    The whole point of my post was to address the question whether Cartesian conclusions ("I exist", or "doubt exists") are on firmer epistemic footing than logical necessities and the like.

    But anyway, I also disagree on the merely psychological point: it's perfectly possible to doubt doubt, i.e. for 'doubt exists' to be the propositional content of an attitude of doubt. (Since when are there metaphysical restrictions on the possible contents of a propositional attitude!?) It's just that one is clearly mistaken to do so. I point this out, and you respond with an argument that shows such doubt to be stupid, and I repeat "yes, that's my whole point."

  20. Heh, I appear to have several 'whole points' :-)

  21. Lvb,

    Two questions. (1) How does that presentation of mathematics -- theorems derived from an (possibly arbitrary, although possibly empirically satisfactory) set of axioms -- show that mathematical claims are necessary? It looks like it makes them analytic, but the connection between analyticity and necessity is obscure. The connection between the analyticity and the a prioricity (is that even the right word? I've never been sure of the noun) is also not clear to me.

    (2) I was under the vague impression (I may be misremembering) that, unlike the first-order sentential calculus, arithmetic was (per Gödel) either incomplete or inconsistent. Which means that the idea that mathematics is a process of deriving theorems from axioms is at best a partial representation of what mathematics is. Right?


    Okay, let's see. First, you're saying that what's psychologically undoubtable is not obviously on firmer epistemic footing than logical truths, mathematical axioms, and so on. I disagree, on Descartes' behalf, because he's using doubt to analyze what "firmer epistemic footing" is. You're not entitled to build in an understanding of that concept, as Descartes would reject any such analysis on the grounds that it can be doubted. This may be bad methodology, but it's his methodology.

    Second, you're saying that "doubt exists" can be the propositional content of an attitude of doubt. Sure. But if it's the content of an attitude of doubt, then the attitude of doubt exists -- which is the very content of the attitude. So, in doubting doubt exists, doubt exists (necessarily). But this entails that you cannot doubt that doubt exists without contradiction.

  22. "So, in doubting doubt exists, doubt exists (necessarily). But this entails that you cannot doubt that doubt exists without contradiction."

    Here's the situation:
    1) I doubt whether doubt exists
    2) So doubt exists
    3) So my doubting this was certainly in error.

    1) I doubt whether 2+2=4
    2) Regardless, 2+2=4 (by obvious logical necessity)
    3) So my doubting this was certainly in error.

    In both cases: (i) The attitude of doubt is psychologically possible; and (ii) the attitude of doubt is certainly in error, and as unreasonable (or epistemically unjustified) as any doubt can be.

  23. 1) Well then I don't know what you put in the word "necessary". There is no way you can have a mathematical claim be true in one situation and false in another, unless you change the axiomatic system in which you state the claim.

    2) True, which is why I wrote "for the most part" in my previous post. For the most part, true = provable. But some statements are undecidable - so you can state them within a framework of axioms, without being able to derive them from those axioms. What one should do with such statements is a big controversy in the philosophy of mathematics. But one thing is certain: mathematicians would never _accept_ a statement as true without a clear proof, ultimately tracing its validity back to a set of consistent axioms. And all such true statements will be necessarily true, at least in the only way I can understand "necessarily true".

  24. "The whole point of my post was to address the question whether Cartesian conclusions ("I exist", or "doubt exists") are on firmer epistemic footing than logical necessities and the like."

    Well this just depends on what you mean by 'epistemic footing', doesn't it? Obviously all logical necessities are on the firmest epistemic footing possible in the sense that when you know them, there's no way you can be wrong. Of course, you _don't_ know all logical necessities, and some of them are psychologically more inaccessible than others (as you mentioned yourself in your first post). Thus Fermat's last theorem was psychologically completely inaccessible up until 16 years ago, and its 'epistemic footing' was therefore (in the psychological sense) relatively weak, even though it was of course logically impossible to doubt its truth and not be in error.

    Obviusly 2+2=4 is much less psychologically inaccessible than Fermat's last theorem, but what ADHR and I have been arguing (with empirical evidence) is that it's still slightly psychologically inaccessible when compared to "I exist", and in that sense is on a weaker epistemic footing, especially when you are embarking on a project like Descartes did -- starting with the assumption that you know nothing and working your way up. Then it makes sense to move through the rank of increasingly psychologically accessible statements.

  25. Richard,

    I'd also add that in the first argument, (3) doesn't follow. What follows is: "(1) was impossible". This modified conclusion doesn't follow in the second argument. And this, I take it, is Descartes' point.


    Necessary = true in all possible worlds. As soon as you hedge by adding "unless you change the axiomatic system", then we're not talking necessity any more. For there is a possible world where the claim is not true -- one where the axioms are different.

    So, I suspect what mathematicians look for is a demonstration that any claim is analytic (that is, in effect, just restates the axioms). But it's non-trivial to add to that the claim that mathematics consists of necessary and a priori truths.

  26. ADHR - "I'd also add that in the first argument, (3) doesn't follow. What follows is: "(1) was impossible"."

    Both sentences are so obviously false I don't know what to say to you.

  27. Oh, honestly. Read the thread, Richard. Dogmatically digging in your heels is irrational.

  28. Yes, I realise you said those obviously false things upthread too. I explained why they were false, and then you repeated them. I'm not sure what else I can do but throw up my hands in exasperation.

    One last try.

    You say that "in doubting doubt exists, doubt exists (necessarily)". From this it follows that the doubt in (1), i.e. with content "that doubt exists", is certainly mistaken; there's no possibility of the doubt here being vindicated. That's my (3). It's just an obvious entailment.

    You, on the other hand, claim that it follows that the state of doubt in (1) [taking as its propositional content "that doubt exists"] is itself "impossible". Not just necessarily mistaken when held, as I say, but a propositional attitude that cannot possibly be held in the first place. But this doesn't follow at all. It's a total non-sequitur. People may be mistaken, and they may even hold attitudes that are necessarily mistaken. That's not impossible, it happens every day.

  29. But ADHR, axioms are not world-dependent. As I said above, certain sets of axioms are better suited to describe various aspects of our world than others, but the majority of the mathematical systems studied by mathematicians bear no relation to anything resembling the real world. Furthermore, there's no restriction on what sets of axioms one may study, as long as the sets are consistent: math as a whole encompasses all conceivable sets of axioms. So individuals in different possible worlds are in the end all studying the same mathematics, although they may (as a contingent fact) be studying different parts of it.

    "So, I suspect what mathematicians look for is a demonstration that any claim is analytic (that is, in effect, just restates the axioms)."

    Yes well this is why Goethe had such a low opinion of mathematics. I'm inclined, though, to say that whereas trivial demonstrations merely restate the axioms in an analytic way, non-trivial demonstrations combine axioms in novel ways which lead to non-expected (I'm tempted to say "emergent") results, which cannot be predicted from a merely analytic investigation of the axioms.

  30. "If you don't think that's possible, try teaching intro logic to undergrads some day -- some aren't even sure about the law of excluded middle." - ADHR

    Well, it's not just undergrads. Some non-classical logicians aren't sure about the law of excluded middle.

  31. (I suspect he meant the law of non-contradiction, though I guess even that has been questioned by the odd dialetheist. Mind you, they say there isn't any view so crazy that some philosopher hasn't defended it!)

  32. Don't most dialetheists limit where the logic applies though?

  33. "Don't most dialetheists limit where the logic applies though?"

    Most probably do, yeah. I think Priest just insists that we need good justification for thinking we have actually found a dialethia (that is, it's not confined a priori to certain domains, but nor is it a catch-all solution to dilemmas, as some dialetheists seem to think in argument...)

    In any case, it still amounts to their rejecting LNC as an exceptionless law.

    I know this is way off the original topic of the thread*, but it's interesting that LNC and excluded middle are actually logically equivalent, by DeMorgan's and double negation. (DeMorgan's only works, though, if excluded middle holds in your system anyway - that is, if your system is complete.)

    * although maybe not, as whether or not we can conceive of true contradictions, and the epistemic/metaphysical implications (or lack thereof) of this, is an interesting issue.

  34. All I can get from this is that there are those that share the OPINION that "2+2=4" is more psychologically inaccessible when compared to "I exist." As opposed to the FACT that they are both equivalent in the face of logic. Otherwise, were is the metric to determine this? What MATH are you using to determine this?


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