A=A

and always = A and only a  TSZ “nihilist” .would deny it, says Barry Arrington.

A=A is infallibly, necessarily true

What does this claim even mean?  That something denoted by A is identical to something else also denoted by A?  Clearly not.

That if we devise a system of logic in which we declare that A always equals A , A must always equal A?  Well, duh.

That the only possible logic system is one in which A is always equal to A?  Well, no – fuzzy logic is a very useful logic system, and A is sometimes only approximately equal to A, or may equal A if it passes some threshold of probability of being A.

So what does he even mean?  Is his claim even coherent?

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130 thoughts on “A=A

  1. Well, it’s not a one off. Yes, I know he used it as a shibboleth for his last purge, but he must think it means something.

    I just can’t figure out what he thinks it means.

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  2. In the end, I think what it means is that “design of life” is self-evident, so is God, etc., and anyone who says otherwise simply wants to do evil–and is self-evidently wrong.

    He doesn’t have much else.

    Glen Davidson

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  3. It means that every object is necessarily identical with itself.

    Which is true, and in a certain sense of “self-evident”, self-evidently true.

    It is not, however, very interesting. It is a logically necessary truth, and logically necessary truths are no guide to how things are in the actual, contingent world.

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  4. That little dose of BA reminded me of what a horrible site that is. Everyone who disagrees is a liar, an idiot, insane, etc. Plus, there’s so much fucking pomposity flying around, hand in hand with all the insults. It’s just as awful as I can imagine any discussion to be–on the internet, or anywhere else.

    Re the merits, I think the whole conversation suffers from mixing up “infallibility” which is an ostensible property of a thinker or some thought or assertion of a thinker, with necessity, which is an ostensible property of some propositions.

    To talk about a proposition being “infallibly true” seems bound to cause confusion: it makes no sense to me, at any rate.

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  5. Kantian Naturalist:
    It means that every object is necessarily identical with itself.

    Which is true, and in a certain sense of “self-evident”, self-evidently true.

    It is not, however, very interesting. It is a logically necessary truth, and logically necessary truths are no guide to how things are in the actual, contingent world.

    Oh, right. Thanks.

    No, it isn’t very interesting. And actually not even very useful, because it begs the question as to what an object IS. And to call something an object, it pretty well has to be extended over time. And not everything extended over time remains the same thing! For instance, a conceptus can become two or more people….

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  6. This has been a long-running battle that started with the idea of objective morality and that it is self-evident that abortion at any stage is objectively wrong. He lost the argument when he couldn’t answer the hypothetical about whether to save a crying baby or a freezer full of blastocysts if you could only save one, without admitting that a breathing crying baby has more value than a blastocyst.

    He then devolved to the self-evidence of 2+2=4. His arguments, and his behaviour have degraded since then. I can’t imagine that Learned Hand will be permitted to comment for much longer.

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  7. What does this claim even mean?

    For BA this was the first link in a logical chain that would lead to the certain conclusion that abortion ( and the actions of Planed Parenthood) are evil.

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  8. Elizabeth,

    Thank you for raising this discussion.

    As someone who is an ID proponent, pro-life, and accepts the strong form of the law of non-contradiction, let me tell say a few things about UD culture. Here are the commandments, all of which I have broken:

    Thou shalt profess the 2nd law of thermodynamics as evidence of ID even though a living human has more entropy than a frozen dead bacterium

    Thou shalt profess and swear by CSI calculations even though Mathgrrl and I showed no one at UD is actually capable of doing them correctly

    Thou shalt profess that ID is science even though there haven’t been nor ever might be direct observations and experiments as seen in other scientific disciplines

    Thou shalt profess that killing babies is self-evidently immoral even though the Christian God of the OT commanded the killing of babies (probably by the sword) would hence show morality isn’t self-evident but must be revealed by the Law Giver

    Thou shalt never praise RDFish (who has deep understanding of formal logic) when he corrects StephenBs mistakes about “self-evidence”

    Thou shalt not only profess the law of non-contradiction, but thou must profess if using the words “ontological” or whatever other philosophical BS add-ons are demanded at the time

    Thou shalt comprehend the clear thinking clearly communicated by KairosFocus lest thou be brow beaten by FYI-FTRs and fishing reals

    The issue is not the axiomatic acceptance of the law of non-contradiction in formal logic and math. I accept that. It is about whether such things are self-evident. Are things self-evident? The answer is no, and it is easily demonstrated.

    Simply ask, “Is the notion of self-evidence itself self-evident?” If they have to explain the answer, then it isn’t self-evident! Thus if the notion of self-evidence isn’t self-evident, then statements like, “this is self-evident” cannot be self-evidently true as a matter of principle. QED.

    As an aside, consider this statement:

    C=C

    If a software developer saw that in a program, he’d likely think, “That’s a mistake! What was the guy thinking.”

    Or if an organic chemist saw it, he’d think. “Oops! Nonsense!” even though something like this would make sense:
    http://2.bp.blogspot.com/-d6Yh5J7hz8k/UQwx8J0IemI/AAAAAAAAADY/T59GVU88VIg/s1600/cc.png

    In the python computer language, this is both valid and functional

    C= not C

    One will complain, “you’re deliberately misunderstanding what I’m trying to say.” To which I’d respond, “No, I understand what you’re trying to say, but I’m driving the point home, if you have to ever explain what you mean, it isn’t self-evident. QED.”

    1+1 =2

    Is true but trivial.

    When the math gets non-trivial such as with conditionally convergent series, it’s not so straight forward. See:
    http://mathworld.wolfram.com/RiemannSeriesTheorem.html

    It shows the way you wish to perceive an object determines what it is! And such math has relevance to physics.

    Finally,

    http://www.phy.duke.edu/~rgb/Philosophy/axioms/axioms/node27.html

    This leads us in the most natural of ways to Gödel, who was the primary logician responsible for proving that logic is a tragically flawed tool even for the purpose of guiding abstract reasoning, let alone for fulfilling the rationalists’ dream of deducing the True Nature of Being from Reason Alone.

    StephenB is a “rationalist” with the rationalists dream of “deducing the True Nature of Being from Reason Alone”. That’s what this debate is really about, whether one can concoct a system of thought, call it “principles of right reason” and deduce the True Nature of Everything of significance.

    I say no to self-evidence. I say an element of faith as a little child and providential grace that gives eyes to see and ears to hear is what is really needed to know anything of any worth.

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  9. It can easily be demonstrated that A=A can be false. If the A on the left and on the right are actual physical objects, then they are not the same. They may be very similar but they are not identical. And what if the physical A on the left is two feet in front of you and the other one is a mile away? Then they are decidedly not equal.

    Don’t bother to get up, I will let myself out now.

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  10. Kantian Naturalist: It means that every object is necessarily identical with itself.

    But that is not true. Ever.

    You can’t step into the same river twice. Every object is constantly changing. Some change is trivial and can mostly be ignored, but non-trivial objects are not the same from one time to the next.

    The only things that meet Barry’s criteria are abstractions. Maps, not territories.

    ETA: I see everyone saying this in different words.

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  11. Did I miss something? Is this the “I Doubt My Own Identity Zone” now?

    Honored to be in such august company. Any day now TSZ will topple UD as the ID that is not ID Zone. Long may you reign as the Not Dogmatic Dogmatic Not Skeptical Skeptical Not Zone Zone.

    click not me

    Who said anything about objects?

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  12. Actually my first impulse was to start out saying how it was a good thing I didn’t come here for the wit. What a bunch of boring whiners.

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  13. But Elizabeth already pointed out that the law of identity is simply an axiom underlying a formal system, where equality exists by definition. Great for factoring equations, useless for empirical real-world application.

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  14. I laugh whenever Barry gets on his “principles of right reason” hobby horse.

    Here he is defending his LNC purge:

    Why is Barry Arrington Stifling Dissent at UD?

    If you visit some of our more vociferous opponents’ websites that is the question being asked. The answer, of course, is that I am not stifling rational argument on this site. In fact, just the opposite is true; my purpose has been to weed out those who refuse to engage in rational argument so that rational argument can be pursued by those who remain. Since, however, recent modifications to this site’s moderation policy have caused such a brouhaha, I feel compelled to lay out a formal defense.

    1. The Rules of Thought.

    The rules of thought are the first principles of right reason. Those rules are:

    The Law of Identity: An object is the same as itself.

    The Law of Non-contradiction: Contradictory statements cannot both at the same time be true.

    The Law of the Excluded Middle: For any proposition, either that proposition is true or its negation is true.

    And:

    Anyone who says a proposition can be simultaneously true and false demonstrates that his understanding is deeply disordered. It is pointless to argue with him, because he has rejected the basis of all argument, and arguing with him will only cause confusion and frustration, which is why Ibn Sina (Avicenna) famously wrote: “Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.”

    Now read what the hypocritical Barry has to say when the subject of God comes up:

    The sadistic maniac and the amiable bumbler schools are both wrong. God is powerful enough to combine apparent contradictions in his person. He is three, yet he is only one. He is both immanent and transcendent. He is sovereign, omniscient, omnipotent; yet despite the evil that exists in the universe he created, he is also omni-benevolent. It never ceases to amaze me that skeptics are surprised when they are unable to fit God into neat human categories. But if we could understand God completely, would we not be gods ourselves? I know I am no god, so I am unsurprised to find that I cannot comprehend God in his fullness or understand fully how such contradictions can be combined in him. Nevertheless, I am quite certain they are.

    Hypocrite, ban thyself.

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  15. Flint: But Elizabeth already pointed out that the law of identity is simply an axiom underlying a formal system, where equality exists by definition.

    Equality is not identity.

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  16. I don’t even have to judge him. Barry convicts himself. That’s the amusing thing about hypocrites.

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  17. A=A is infallibly, necessarily true

    A more accurate statement:

    “A=A” is trivially true

    Infallibility has connotations that aren’t justified, as if to claim “A=A” is some profound insight that empowers us to make unassailable claims about reality.

    To say that a has the property of being identical with a, is to say nothing more than “a is a,” which is a “roundabout way of saying nothing.” It is a “useless tautology.”

    http://homepages.wmich.edu/~baldner/black.htm

    Where there is greater insight is when there are two distinct objects that form an equivalence relation (I’m not using “equivalence relation” in the formal math sense, but just to drive home a point) such as:

    4 quarters = 100 pennies

    In such case, each side has different objects but are “equal” under a certain perspective.

    It starts to become non-trivial to state alternate descriptions of supposedly the same concept. It took Russell and Whitehead 362 pages to prove the following alternate statements were describing the same concept in their Principia Mathematica:

    1 + 1 = 2

    “1+1” is an alternate description of the same concept that can be described by “2”.

    See:
    http://humor.beecy.net/misc/principia/
    http://blog.plover.com/math/PM.html

    As an aside, in modulo-2 arithmetic which is used to create error correction polynomials for computer memory and disk drives.

    1 + 1 = 0

    or in Z11 arithmetic

    13 + 19 = 11

    http://www.millersville.edu/~bikenaga/abstract-algebra-1/modular-arithmetic/modular-arithmetic.html

    Assumption of non-contradiction in a formal system leads to very interesting claims in mathematics, but “A=A” is not one of them since that is trivial claim, a near useless tautology.

    What is really being promoted is the law of non-contradiction, whereby statements like “Bruce Jenner isn’t himself today” are not allowed.

    But since some insist on the law of non-contradiction, here is a contradictory claim: “the notion of self-evident truths is itself self-evident”. How does one define self-evident? Oops, the moment one has to start explaining what self-evident means implies the notion of self-evident truths isn’t itself self-evident. Hence the law of non-contradiction falsifies the notion of self-evident truths.

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  18. If the axiom is apparently self-evident, that only means, practically, that it is nearly indubitable; for things have been thought to be self-evident and have yet turned out to be false. And if the axiom itself is nearly indubitable, that merely adds to the inductive evidence derived from the fact that its consequences are nearly indubitable: it does not provide new evidence of a radically different kind. Infallibility is never attainable, and therefore some element of doubt should always attach to every axiom and to all its consequences. In formal logic, the element of doubt is less than in most sciences, but it is not absent, as appears from the fact that the paradoxes followed from premisses which were not previously known to require limitations.

    Russell and Whitehead, Principia Mathematica

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  19. Flint: But Elizabeth already pointed out that the law of identity is simply an axiom underlying a formal system, where equality exists by definition. Great for factoring equations, useless for empirical real-world application.

    The interesting question is whether, or to what degree, we can establish any ontology on the basis of logic alone.

    However, Arrington et al aren’t using the law of identity (together with the law of non-contradiction and the law of the excluded middle) in order to explore formal systems. They are using these laws to determine who is capable of having a rational conversation. The overall intent here is to show that one is not capable of having a rational conversation unless one is already willing to accept an Aristotelian view of the relations between thought, discourse, and reality.

    That is of the utmost importance to them because they need their opponents to concede that in order to compel us to accept that there must essences. And from essentialism it follows — they think — that abortion is objectively immoral and that same-sex marriage is conceptually incoherent.

    In other words, Arrington et al. think that the regressive sexual mores of the Republican Party circa 2015 are directly reflective of the fundamental nature of both thought and reality. It’s not surprising to me that someone who holds such a view would not be able to see anyone here as capable of rational discourse.

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  20. Mung: Equality is not identity.

    Right, and that was KN’s point. But identity at time 1 is not necessarily identity at time 2. So in what sense, in a world in which time is a dimension, does the law of identity hold?

    It clearly holds in a formal system in which it has been declared as an axiom. But that could be true of any axiom.

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  21. Mung: Sure. He’s talking about the law of identity.

    So what does the claim:

    A=A is infallibly, necessarily true

    Actually mean? I know he is talking about the law of identity” – I’m asking what he is saying about it it.

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  22. Mung, you wrote, at UD:

    I thought that Elizabeth was asking an honest question. She got a bunch of nonsense in return. The psychology of UD. The psychology of Barry. etc.

    She then asked for people who think A=A means something to weight in. At which point KN offered his wisdom.

    But then Elizabeth went on to ignore it. Which calls into question my original assumption, lol.

    Do they really not understand the law of identity? Or is it just that the truth doesn’t matter, esp. if it’s declared at UD.

    Why did you say:

    But then Elizabeth went on to ignore [KN’s answer]?

    Almost immediatly below KN’s post (one post intervening) I wrote:

    Elizabeth: Elizabeth on September 11, 2015 at 8:23 pm said:
    Edit

    Kantian Naturalist:
    It means that every object is necessarily identical with itself.

    Which is true, and in a certain sense of “self-evident”, self-evidently true.

    It is not, however, very interesting. It is a logically necessary truth, and logically necessary truths are no guide to how things are in the actual, contingent world.

    Oh, right. Thanks.

    No, it isn’t very interesting. And actually not even very useful, because it begs the question as to what an object IS. And to call something an object, it pretty well has to be extended over time. And not everything extended over time remains the same thing! For instance, a conceptus can become two or more people….

    Would you correct your error at UD please?

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  23. Elizabeth: So in what sense, in a world in which time is a dimension, does the law of identity hold?

    Identity could still hold synchronically even if it doesn’t hold diachronically — at any instant, treating “instant” here, as philosophers are wont to do, as an infinitesimally thin time-slice, an object is identical with itself.

    But I’m not sure if identity here means that the object never changes — it could simply mean that as any object changes (as all objects do), its self-relation also changes.

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  24. Barry’s response is:

    Over at The Skeptical Zone Elizabeth Liddle writes:

    Barry Arrington [says] A=A is infallibly, necessarily true. What does this claim even mean? . . . Is his claim even coherent?

    Really folks. You can’t make this up. She wrote that. If I made this stuff up someone would accuse me of lying. But she really wrote that.

    Fortunately, not everyone at The Skeptical Zone is a complete idiot. Kantian Naturalist replies:

    It means that every object is necessarily identical with itself. Which is true, and in a certain sense of “self-evident”, self-evidently true.

    Why do they kick against the goads with such vehemence? Not only do they make fools of themselves, they seem eager to do so. It really does beggar belief.

    I repeat my question to KN: does the Law not beg the question as to what an object is – and, if an object is extended in time, as things we normally call objects are, in what sense is it “necessarily and infallibly true”? Unless we hold that an object that changes is not the same object? [ETA: answered since I wrote that]

    In regard to Barry’s second point, it is odd to me that he should think this is such a bone of contention. The only bone of contention I see is Barry’s own use of the LoI as a shibboleth for entry to UD conversations, and as a stick to beat “materialists” with. So yes, I’m eager to demonstrate to Barry that his shibboleth is a nonsense.

    I challenge Barry to provide a single example of an error he thinks “materialists” have made because they have failed to recognise that A=A.

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  25. Kantian Naturalist: Identity could still hold synchronically even if it doesn’t hold diachronically — at any instant, treating “instant” here, as philosophers are wont to do, as an infinitesimally thin time-slice, an object is identical with itself.

    OK, but I still smell a tautology: that an unchanging object does not change.

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  26. Kantian Naturalist: But I’m not sure if identity here means that the object never changes — it could simply mean that as any object changes (as all objects do), its self-relation also changes.

    OK, well that would be interesting, but a long way from a “law of identity” – more a law about identity. How much does an object have to change before we can say it is a different object? Or how much can change while it remains the same? The Ship of Theseus?

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  27. Also, regarding “infinitessimal time-slices” – the neat thing about differential calculus is that it allows us to define rate of change even for an infinitessimal time-slice.

    So I’m still not sure this point holds.

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  28. Elizabeth:

    I challenge Barry to provide a single example of an error he thinks “materialists” have made because they have failed to recognise that A=A.

    Absolutely – the only time the law of identity comes into a debate is when arguing about what it means and why it is true or when comparing it to other “laws”. It is never plays a part in any debate about any subject other than itself.

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  29. For my part, the law of identity is a formal law — it tells us that in any possible world, every object will be identical to itself. (It is not necessary to add “at any given moment in time”, since this is a formal constraint that applies to all possible worlds, including timeless worlds.)

    It is an mildly interesting feature of formal logic. It is not important for ontology or for the conditions of rational discourse.

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  30. Kantian Naturalist,

    Wittgenstein must have written about the meaning of the laws of logic somewhere. What use are they? What form of life do they participate in? What are you doing when you assert “this apple is identical to itself”?

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  31. Mung at UD

    To say that a thing exists is to assert the law of identity. No wonder we here at UD take it to be fundamental to rationality. Foundational, even.

    Now THAT makes sense – the law of identity, conceived thus, does not beg the question as to “what is an object” but is no more (and no less) than to say that we CAN say that “things exist” – that there is meaningful sense in the concept of an object extended over time.

    However, such an object need not be identical with its former self, under that conception. It could even be the Ship of Theseus.

    However, I suggest that it is, still, not an “infallible law” but rather a declaration that we can consider “objects” as things that exist over time (and thus can be said to “cease to exist”), even though their physical properties and make-up my change. Thus Elizabeth today is the identical object to Elizabeth 63 years ago, even though Elizabeth today is not identical in either form or substance to the Elizabeth of 63 years ago.

    Agree, Mung? If so, then you would seem to disagree with KN.
    KN, am I wrong here?

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  32. I’ll return to this conversation tonight (EU time — I’ve been in Paris for the past few days) but will think about Mark’s very interesting question about what the later Wittgenstein would have made of all this.

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  33. To me, it has always seemed obvious that statements such as A=A apply to formal systems such as mathematics and logical models.

    It has seemed equally obvious that a natural language is not a logic system, and that the ordinary world is not logical.

    Note, that when I say that the ordinary world is not logical, I do not suggest that it is illogical (oops, there goes the law of excluded middle).

    For example, the way that we use “equal” and “identical” in ordinary life is not the same as the way that we use them in formal arguments. We talk about identical twins, but we are not suggesting that they are the same person. The founders of the US famously declared “all men are created equal”, but they were not suggesting that we are all the same person.

    As for laws of thought — there aren’t any. Thought is pretty much free flowing, not bound by laws.

    On where logic comes from: we tend to organize the world into a binary tree structure. We divide into plants and animals; we further divide animals into vertebrates and invertebrates, etc. Logic is the natural tool for traversing a binary tree. So the use of logic is associated with the way that we organize the world. It’s a useful tool. A craftsman doesn’t apply his tools blindly — he uses them where appropriate.

    On Wittgenstein: I don’t know what he would have said about logic. But I’m pretty sure that the later Wittgenstein would have agreed that a natural language is not a logic system.

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  34. Troubles start as soon as you forget the basics of logic. Delusionists like Barry are unable to distinguish between ontology (facts and things) and logic (statements, propositions, concepts). This is understandable actually, because this defect has plagued the entire so-called analytic philosophy all along. So, it’s also understandable when even KN gets stuck in trying to bring some clarity to a bunch of atheist puppets who are very much used to alternately deny and affirm logical principles as per random whim.

    The law of identity is a law of thought or a principle of logic. Laws of thought apply, first and foremost, to statements about things, not to things themselves. Things are whatever they are, but in order to analyse them rationally, we have to approach them in an orderly and organised manner, and this is where the principles come in. Things themselves may or may not be orderly, but whichever is the case, our approach must be orderly so as to verify it beyond reasonable doubt.

    Barry says, “I have no greater argument that A=A than the self-evident fact that A=A.” Actually, A=A is not a fact and it’s not self-evident either. A=A is a necessary presupposition in order to give a solid meaning to concepts like fact, evidence, truth/falsity, etc. A=A is a necessary law of thought, if we are to make sense of things. The other option is to simply refuse to make sense of things.

    To not make sense of things is a real option. Lots of people make no sense, so laws of thought are really not self-evident, and they are not necessary facts. They are necessary, if we are to make sense of things, but not everybody wants to make sense of things – and many nonsensical people live, as irrational as it may seem, long decent lives.

    Barry takes A=A to be an ontological statement, so he lost track of reasoning from the get-go and everything he derives from it is ever more hopelessly lost. It’s also a lost case to argue against Barry by saying something like “You can’t step into the same river twice. Every object is constantly changing.” This makes the mistake of accepting the false supposition that A=A is an ontological statement (which it isn’t) and makes it doubly worse by attempting to argue against an inevitable law of thought.

    Elisabeth says, “That the only possible logic system is one in which A is always equal to A? Well, no – fuzzy logic is a very useful logic system, and A is sometimes only approximately equal to A, or may equal A if it passes some threshold of probability of being A. So what does he even mean? Is his claim even coherent?”

    Well, if Elizabeth cared about logic, she would know what “coherent” means. It’s incoherent to reject a statement that is applicable within a formal system by pointing out that there is another system. It’s like rejecting the visual data obtained through your left eye because you also have a right eye. The fact that there are two eyes does not mean that one of them is invalid. They may very well be complementary, you know.

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  35. Formal systems applied to the real world actually put a simplifying approximation (and thus a distortion) and viewpoint on the real world that may not be actually there in the real world by making distinct objects indistinguishable (indiscernible).

    Consider this banking transaction:

    4 quarters = 100 pennies

    Well, when a financier is doing his banking analysis, it’s possible at some point he’ll be erasing the physical details and reducing it to

    1 = 1

    But now, forcing this formalism onto the banking transaction has actually lost and therefore distorted the specifics of the real world. It’s still a useful formalism. But real information about the objects is permanently lost in the process. If all I started with was:

    1=1

    I’m not going to be able to reconstruct back the details of the actual system being modeled (namely 4 quarters vs. 100 pennies). I certainly wouldn’t be able to talk about the weight of silver of the quarter nor the weight of copper in the pennies.

    The logical formalism is a triviality, it certainly doesn’t give you infallibility in understanding the real world.

    In physics, we also have to account for indistinguishablity in the atomic realm between objects we “know” are distinct. Example:

    a hydrogen atom = a hydrogen atom

    Well, is that really true?????

    a hydrogen atom on mars = a hydrogen atom on earth

    The are not the same object, but we will force them into this abstraction

    1 hydrogen atom = 1 hydrogen atom

    When our measuring apparatus can’t distinguish these objects, we treat them as identical because they are now indiscernible. This feature actually affects our calculations of the mixing entropy of a gas.
    https://en.wikipedia.org/wiki/Gibbs_paradox

    In statistical mechanics, a semi-classical derivation of the entropy that does not take into account the indistinguishability of particles, yields an expression for the entropy which is not extensive (is not proportional to the amount of substance in question). This leads to a paradox known as the Gibbs paradox, after Josiah Willard Gibbs. The paradox allows for the entropy of closed systems to decrease, violating the second law of thermodynamics. A related paradox is the “mixing paradox”. If one takes the perspective that the definition of entropy must be changed so as to ignore particle permutation, the paradox is averted.

    If we say “A=A” and we are really truly talking about the same object, it’s trivial, but if we say “A=A” and we got the left hand side by making one object indistinguishable from the object on the right hand side, we’ve actually distorted reality so we can apply a more simplistic and therefore tractable viewpoint of the world and hence “make sense” of a complex system in terms our feeble minds can understand.

    It might make some “rationalists” have a false sense of comfort that their minds have figured everything out. After all, they are holding on to “an infallible self-evident truth”.

    That’s why to claim:

    A=A, is infallibly true

    is better said to be

    A=A is only true in the real world if A refers to the same object, and therefore is trivial

    Infallible? Philsophical add-ons that don’t further understanding.

    Where in formal logic do we see “infallible” mentioned. See for yourself:
    https://en.wikipedia.org/wiki/First-order_logic
    https://en.wikipedia.org/wiki/Second-order_logic

    We have to assume axiomatically the law of identity to make the formal logical machinery have any hope of working.

    But applying the law of identity to the real world in a way that is useful means applying the formalism to objects that ARE NOT in the real sense the same object (thus not really A=A in the true sense) — such as a set of 4 quarters and a set of 100 pennies, or distinct atoms that we’re just not able to distinguish.

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  36. Good post, Sal. As your others on this thread have been.

    Thank you for the kind words. I did not post these thoughts at UD at the time since I didn’t want to get banned. But now I’m free to speak and put the sword to the folly being promoted.

    I assume “A=A” in a formal system, because it is pragmatic, and if is not ultimately true, I’m screwed anyway, so I have nothing to lose by making the assumption and every thing to gain by doing so. I don’t get into debates about it being “infallible”, it’s a necessary assumption.

    Let
    A = 1 hydrogen atom
    B = hydrogen atom on mars

    thus one element of what qualifies as a “not-B” is a hydrogen atom on Earth

    but from many physical theories we view

    hydrogen atom on Mars = hydrogen atom on Earth

    in a sense we are saying

    B = not B

    to make

    A = A

    🙂

    So the problem for the UD “rationalists” is this. A = A in the real world is only true for the same object. But this is so trivial so as to be totally useless even though they claim it is infallibly true.

    Where the notion of “A=A” becomes useful is applying to objects that are not strictly the same thing — like atoms on differently planets. In a manner of speaking, we are saying, “B = not B” in order to make “A=A”. But this invites the possibility of fallibility. Gödel found out kind of the same thing in mathematical logic. Non-trivial systems of any utility might remotely be flawed, one had to risk being wrong in order to capture a payoff of utility.

    So the UD “rationalists” are trapped. To remain in the fortress of infallibility, they are unable to make any generalizations about reality because every object is unique, and therefore no generalizations can be infallibly made, only assumed.

    Whereas to make useful generalizations (like say about atoms) one has to in a sense allow “B = not B” by only looking at select properties of atoms or other objects rather than the objects themselves, but to do so, one has to leave the fortress of trivial infallibility.

    Thus, the UD “rationalists” are then unable to extend their claims of infallibility to the real world because it violates the law of non-contradiction which they swear by, so they are stuck only asserting trivial stuff and can only pretend they are saying something profound when they are not really saying something profound.

    Live by the sword of non-contradiction, die by the sword of non-contradiction.

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  37. If any of you are interested in what Arrington is trying to say, you really should take a look at this: Laws of Thought. It’s much more clear than anything Arrington writes. The brief discussion of the law of identity is also quite nice.

    The statement “A=A”, it turns out — apparently first symbolically formulated by Leibniz, which I did not know! — expresses the view that “whatever is, is”. It conveys the thought that the fundamental structure of thought involves our ability to recognize that every object is itself and not some other object.

    Without this, we would not be able to recognize objects as having degrees of similarity or difference in perceptible, functional, or relational properties, and so we could not classify them, think about them, disagree about them, and so one. There would be no possible content to thought if we did not have the ability to recognize an object as being that specific kind of object and not some other kind.

    In that sense, “A=A” captures what we might call the logical concept of an object. One could reject it at the price of ceasing to have anything determinate to think about at all. (I am puzzled why Arrington calls that rejection “nihilism,” since the first teaching of many great mystical traditions is that there are no determinate boundaries between objects. “Thou Art That“.)

    But we would still need to consult actual experience in order to determine how the logical concept of an object is applied, and how it acquires any actual objective validity for us and not merely logical validity.

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  38. Kantian Naturalist: In that sense, “A=A” captures what we might call the logical concept of an object.

    That makes sense to me (as I said to Mung, who made what I think is a similar point).

    The statement rather than being “infallibly and necessarily true” is the definition of an object. And an object, thus defined is a useful concept, and possibly a necessary one for constructive thought (although sometimes a barrier, I would argue).

    It’s the notion that it is a truth claim I take issue with.

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  39. With

    4 quarter = 100 pennies

    and then the resulting “identity” which we project onto it

    1 =1

    This illustrates, there is no constraint on how we wish to force fit and declare objects identical. Nothing from physics says 4 lumps of silver should equal 100 lumps of copper. It is a purely human construction.

    If we can take such different objects and arbitrarily decide we want to look at them as identical, there is no guarantee invoking the trivial law of identity, A=A, will be infallibly applied anyway!

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  40. It’s the notion that it is a truth claim I take issue with.

    Why? It works for formal logic. It’s true enough for the domain where A=A is true. Perfectly reasonable to say an object is identical to itself.

    When we apply the formal system to the real world, we have to cheat. We treat objects that we know are distinct and different as being the same object. Hence we violate the law of non-contradiction (in a manner of speaking) by applying identity to objects that are not truly identical.

    So, assume “A=A” is true in theory, but recognize in practice, we violate it by choosing to view things the way we find convenient so that we can say things like “4 quarters = 100 pennies” — which by the way is a purely human contrivance.

    One might argue, “we’ll we’re comparing identical properties.” To which I respond, certainly not inherent properties in ALL cases. “4 quarters = 100 pennies” is true because we contrived a property that wasn’t inherently there, and most certainly not a “self-evident” property, namely the contrived property of having an economic value of 1 dollar.

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  41. Elizabeth: Now THAT makes sense – the law of identity, conceived thus, does not beg the question as to “what is an object” but is no more (and no less) than to say that we CAN say that “things exist” – that there is meaningful sense in the concept of an object extended over time.

    However, such an object need not be identical with its former self, under that conception. It could even be the Ship of Theseus.

    However, I suggest that it is, still, not an “infallible law” but rather a declaration that we can consider “objects” as things that exist over time (and thus can be said to “cease to exist”), even though their physical properties and make-up my change. Thus Elizabeth today is the identical object to Elizabeth 63 years ago, even though Elizabeth today is not identical in either form or substance to the Elizabeth of 63 years ago.

    Agree, Mung? If so, then you would seem to disagree with KN.
    KN, am I wrong here?

    I think this is wrong, in that inflates the LOI to something much bigger and ambitious than what it can bear in the place it holds as a fundamental principle of thinking. “A is what it is” does not implicate time or extension or any other property than identity. It’s just the minimalist anchor need to build negation and differentiation as “logical atoms” on top of it, building blocks for the kinds of propositions that might include “exist over time” or other qualifiers that rely on identity and differentiation.

    The LOI is not time bound. The LOI doesn’t address persistence. These are downstream concepts — far downstream, I’d say — from the axiom that is the LOI.

    I can agree with this: “Thus Elizabeth today is the identical object to Elizabeth 63 years ago, even though Elizabeth today is not identical in either form or substance to the Elizabeth of 63 years ago.” and at the same time dismiss this as irrelevant to the LOI. It depends on the LOI, as all statements do, but only insofar as all statements do. The LOI is not concerned with “identical” in the sense you’ve used it here, but rather with the fundamental concept that grounds differentiation (“not A”), which in turn gets deployed in forming your higher-level concepts of “identical” as it pertains to Elizabeth or any other real-world object.

    I note that when this topic comes up, when understanding and agreement is reached, someone will usually say “Well OK, but that’s just trivial, then”. I think that’s the case here. The LOI is much more fundamental, ubiquitous, and trivial than you are conceiving of it, here, I suspect.

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