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?

130 thoughts on “A=A

  1. eigenstate: 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.

    I agree — that’s why I called it “the logical concept of the object.” The logical concept of the object, or if you prefer, formal concept of the object, is the sense of object in which we talk of any object in any possible world in which there are objects of any kind. Hence nothing spatio-temporal, perceptible, material, causal, etc. plays any role at all in the logical concept of the object.

    In the face of this rationalistic metaphysics — which has been revived in the 20th century as “analytic metaphysics” and which also, in that guise, serves as an interpretative framework for the revival of Scholastic metaphysics by Feser and others — it fell to Kant to point out the fatal flaw: that we have not one but two basic cognitive capacities, a capacity for judging in accordance with rules and a capacity for being affected by objects.

    Since these capacities are individually necessary and jointly sufficient for objectively valid assertions (assertions which could be true or false), the operations of the former — understanding alone — are never sufficient for knowledge. As Rob Hanna puts it:

    According to Kant, both the origins and limits of human cognition or Erkenntnis are determined by the nature of our specifically human sensibility or Sinnlichkeit (CPR
    B1, A19-49/B33-73). In particular, there is an inherent cognitive-
    semantic constraint on all fully or “thickly” meaningful cognition: a cognition is “objectively valid,” i.e., fully or “thickly” meaningful, if and only if it it presupposes actual or possible externally-triggered sensory intuitions or Anschauungen of empirical objects (CPR A238-242/B298-300, A289/B345), presented within the global, framing structures of egocentrically-centered, orientable (i.e., it contains intrinsic enantiomorphic directions determined by a subject embedded in the space or time) phenomenal space and time.

    For more, see Kant, The Copernican Devolution, and Real Metaphysics by Hanna.

    Somewhat more contentiously, I think that Kant’s insights here about the impossibility of grounding metaphysics in reason alone — contra Aristotle, Aquinas, Spinoza, Leibniz, Lewis, Armstrong, Kripke, Chalmers, et al. — are not only basically correct but also quite separable from his overly empiricist conception of sensibility (as noticed and corrected by James, Bergson, and Merleau-Ponty), his overly intellectualistic conception of the understanding (as noticed and corrected by Wittgenstein, Heidegger, and Gadamer), and his commitment to the ideality of space and time (as noticed and corrected by all naturalists since Kant, but esp. Sellars).

  2. So the consensus answer seems to be my second option?

    And that the reason it’s worth devising such a system of logic is so that we can talk coherently about objects?

  3. Assume A=A, that means A is only itself not anything else. Well that leads to the problem one can’t use this claim to make truly general statements about the real world except “something is itself, not anything else”. Trivially useless.

    To make abstractions, we actually have to ignore A=A in the physical world, that is to say

    Property( 4 quarters) = A
    Property (100 pennies) = A

    and
    A=A
    thus
    Property( 4 quarters) = Property (100 pennies)

    You don’t even need to appeal to fuzzy logic, just appeal to the fact the Property operator is a total contrivance and there is no constraint that the Property operator is infallible, in fact, it might be said it is meaningless to say it is infallible or not, because it depends on what the Observer’s goal is. A banker would agree the above Property transformation is correct, a physicist would balk at the it, and a philosopher would say he’d have to think about it.

    This leads to the question of:
    1+1 = 2
    What we are really saying is:

    Property(1+1) = Property(2)

    in the natural numbers. But in the Modulo 2 world
    1+1 =0 that is

    Property(1+1) = Property(0)

    in the modulo-2 world.

    One can build polynomials with either natural numbers or modulo-2 numbers, and as I pointed out earlier, modulo-2 numbers are vital for building computer and communication error correction systems that utilize polynomials in the modulo-2 world. Is one system more real than another?

    A=A is foundational to classical logic, and for formal logic it is a given otherwise there would be instability in the inferences. This is illustrated by the fact that building computers which make the formal logic mechanical, if we allowed A= not A, the computational machinery would break down. Hence, it is not operationally sensible to allow statements like A= not A in a formal system.

    But, assuming we extrapolate A= not A to the real physical world vs. restricting it to the formal world, one gets a relatively useless trivial conclusion, “something is itself, not anything else”, which says nothing about the ability to make generalizations about reality. And the hope of the UD “rationalists” is to prove they can make sweeping generalizations about all things significant by the rules of right reason. All they really have in there hand are trivialities that they only pretend are theologically profound.

  4. Elizabeth: So the consensus answer seems to be my second option?

    Right.

    Logic applies to logical objects. Ordinary things are not logical objects. For sure, it is often useful to idealize, and treat them as if they are logical objects. But when we do that, we are working with a logical model, not with reality.

  5. A=A is infallibly, necessarily true

    If one says the above is an axiom, then it is accepted by faith. I accept it as an axiom. Formal logic works well with that assumption, but it’s used to build machines that try to approximate formal logic, like say, computers. It’s used to make science possible, but at it’s core is the saying “something is itself, not anything else”, and as pointed out above, we have to concoct Property operators to actually make A=A, useful. It thus then takes us out of the realm of trivial infallibility.

    If one says the above is a proposition, then since all proposition are either true or false, then the above claim is dependent on something else being infallibly true. You then get this:

    “A=A is infallibly, necessarily true
    because
    “A=A is infallibly, necessarily true and…..”
    because
    “A=A is infallibly, necessarily true and ….”

    It is the vicious circle paradox which Russell and Whitehead mentioned specifically in Principia Mathematica. Gödel showed, no non-trivial system will escape the vicious circle. Russell hoped he could build mathematics upon a foundation which could avoid the vicious circle. Gödel dashed Russell’s hope.

    You can assert it, you can’t prove it, you can only accept it on faith.

    PS
    The problem in the UD culture, people with actual background in relevant fields who disagree with the propaganda line are persona non-grata. It doesn’t matter if you ever held the title of scientist, you have to say, “ID is science”. It doesn’t matter if you’ve studied graduate level thermodynamics, you have to say, “2nd law shows entropy must decrease to increase biological complexity”. It doesn’t matter if you’ve had to apply information theory in you job, you have to say, “information theory proves ID, look at my non-existent CSI calculations”. It doesn’t matter if you actually have more depth in training in formal logic, you have to say, “principles of right reason show that God exists and that therefore ID is true and morality is self-evident”.

  6. Neil Rickert: Right.
    Logic applies to logical objects.Ordinary things are not logical objects.For sure, it is often useful to idealize, and treat them as if they are logical objects.But when we do that, we are working with a logical model, not with reality.

    Map, territory.

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

    What would be more than just mildly interesting is for you to make the case that all logic is formal logic and that no logic is important for rational discourse. If that is what you are implying why not just come out and say it?

  8. Neil Rickert: Right.

    Logic applies to logical objects.Ordinary things are not logical objects.For sure, it is often useful to idealize, and treat them as if they are logical objects.But when we do that, we are working with a logical model, not with reality.

    And so saying that it is “infallibly, necessarily true” is a nonsense. It is true when we declare it as an axiom in some formal system, right?

    I guess the other sense in which I could, less hyperbolically, agree that it is “necessarily” true, is the sense in which it follows directly from the parsing of the world into objects. An object is something that cannot not be itself.

    But again, that’s definitional – it’s not an infallibly true statement about the world.

    I suspect that parsing the world into objects isn’t the only way of parsing the world, anyway. It’s the way we evolved to do it, for good reasons. But Nature’s joints aren’t that jointy, it turns out. There are both spatial and temporal cusps, but few (any?) clean divisions/

  9. As Einstein wrote to a man who had lost his son:

    A human being is a part of the whole, called by us “Universe”, a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the rest — a kind of optical delusion of his consciousness. The striving to free oneself from this delusion is the one issue of true religion. Not to nourish the delusion but to try to overcome it is the way to reach the attainable measure of peace of mind.

  10. “It doesn’t matter if you ever held the title of scientist, you have to say, “ID is science”.” – stcordova

    It’s not just UD that requires this, it’s almost the entire IDM! Can you, Salvador, name a single Discovery Institute Fellow that doesn’t absolutely insist that “ID is science”? I don’t think so.

    Mike Gene disagrees. nullasalus disagrees. VJTorley by capitalisation disagrees.

    Yet it seems like you are one of those who insists that ‘ID is science’, Salvador. If not, then why not start a thread explaining why you don’t think so?

  11. Elizabeth: And so saying that it is “infallibly, necessarily true” is a nonsense.

    Again, “infallibly” doesn’t mean the same thing as “necessarily.” One is a psychological predicate, the other is a logical term.

  12. walto: Again, “infallibly” doesn’t mean the same thing as “necessarily.”One is a psychological predicate, the other is a logical term.

    Is that equivalent to saying that the former is an epistemic term and the latter is a metaphysical term?

    Just wondering how you would carve up the difference between epistemology and psychology, and between metaphysics and logic.

  13. Elizabeth: And so saying that it is “infallibly, necessarily true” is a nonsense. It is true when we declare it as an axiom in some formal system, right?

    That’s about right.

    I guess the other sense in which I could, less hyperbolically, agree that it is “necessarily” true, is the sense in which it follows directly from the parsing of the world into objects.

    I don’t much like that way of talking. We divide (not parse) the world into things that we treat as if objects. But what makes something an objects, is how we divide the world. That’s why we have problems with questions such as “If I replace the radiator cap on my car, is my car still the same object?” We divide the world into what we call objects, but we do it in a way that leads to logical paradoxes. It works for us pragmatically, as long as we are sensible enough to not push the logic to its limits.

  14. Neil Rickert,

    I think I’m more inclined towards realism about objects than you are. I find this curious because you and I both appreciate Gibson’s work on affordances, and I am a realist about affordances — affordances are real features of the organism-environment relationship But perhaps you are more inclined to think of affordances as “projections” from the organism onto the environment?

    I think of objects in ways similar to affordances: we should be realists about objects because acting on the basis of representations of objects is, generally speaking, practically effective. “Objects” are that which pushes back against us, thwarts us, offers resistance to our actions. A jellyfish in a liquid world (to use C. I. Lewis’ example) would have no conception of objects. (Lewis is not quite right about the details of that claim, but one can appreciate his point!)

    If Nature had no joints at which to be carved, then any criteria for successful action would be equally arbitrary as all other criteria. But that’s just not the case, and science (among other things) wouldn’t be possible if it were the case. So while we should always be on guard against the assumption that any theory correctly describes the structure of reality, we really cannot do away with the assumption that reality does indeed have an intelligible structure, and indeed one that is knowable by us because our cognitive capacities are a part of that structure and informed by its history.

  15. Kantian Naturalist: Is that equivalent to saying that the former is an epistemic term and the latter is a metaphysical term?

    Just wondering how you would carve up the difference between epistemology and psychology, and between metaphysics and logic.

    I guess it’s either psychological or epistemic (or both) depending on how you define it. And ‘necessary’ is metaphysical or logical or both. But the conflation on this thread is pretty rampant (though not omnipresent!)

  16. Neil Rickert: We divide the world into what we call objects, but we do it in a way that leads to logical paradoxes. It works for us pragmatically, as long as we are sensible enough to not push the logic to its limits.

    That seems right to me. I would build on that point by saying that the logical concept of an object leads to paradoxes because, when we construct a formal language, we stipulate a criterion of consistency that we do not need (or want) in our practical copings with the world as mediated by a natural language.

  17. Neil Rickert: I don’t much like that way of talking. We divide (not parse) the world into things that we treat as if objects. But what makes something an objects, is how we divide the world. That’s why we have problems with questions such as “If I replace the radiator cap on my car, is my car still the same object?” We divide the world into what we call objects, but we do it in a way that leads to logical paradoxes. It works for us pragmatically, as long as we are sensible enough to not push the logic to its limits.

    Well, I won’t quibble over verbs! I’m coming at it as a perceptual scientist, you as a mathematician. I think we end up in the same place.

    Our brains have evolved to construct a model of the world (if you don’t like parse the world) in terms of objects with properties, and also events that result from interactions between objects. You could even consider an “event” an extremely transient object, or an object as an extremely long-lasting event. But both have fuzzy edges – I parse – modelled – my father as an object, even though he was not the same substance when he died as when I first knew him, yet was the same substance, substantially, when I still knew him an hour before his death (and he made a funny crack about the upcoming election) as he was an hour afterwards (still warm, but dead, and no longer my father).

    He himself was very taken with the idea of people as waves – specifically the idea of ocean waves, having an identity over time, even though the substance changes, and the energy on going, even after the wave crashes on the shore, and relinquishes its identity.

    I don’t think there’s anything special about “object-oriented” modelling of the world, but I do see that it depends for its coherence on the LoI.

    But the LoI is not a truth about the world, and certainly not an “infallible” truth about the world. It’s simply implicit in the notion of an object (as Mung, I think, said).

  18. walto: I guess it’s either psychological or epistemic (or both) depending on how you define it. And ‘necessary’ is metaphysical or logical or both. But the conflation on this thread is pretty rampant (though not omnipresent!)

    I was afraid you’d say that.

    I think I follow a Kantian critique of empiricism in distinguishing between epistemology (as normative) and psychology (as descriptive), and a Kantian critique of rationalism in distinguishing between logic (as ranging across all possible worlds) and metaphysics (as ranging across all possible worlds that are epistemically available to beings whose cognitive capacities can be formally specified as the conjoint functioning of sensibility and understanding).

    That’s why the merely logical concept of an object (as defined by the three laws of reasoning*) cannot be milked for any metaphysical significance — to do metaphysics, you also need to do the epistemology of metaphysics. That is, all descriptive claims about objects, whether observed or posited, must be justified in terms of the process whereby one can have epistemic access to the object. And that’s what distinguishes ontological commitment from mere fantasy.

    * However, we now know that there are alternative formal systems — “non-classical logics” — that reject the law of non-contradiction. In those systems one is reasoning, but not about objects. For the same reason, the logic developed by Nagarjuna is a logic for reasoning in an ontology without objects, since Nagarjuna, as a Buddhist, taught the co-dependent origination of all things. There are no substances, in the Aristotelian sense, in a Buddhist understanding of reality. Interestingly, this does not prevent Buddhists from understanding and practicing Western science.

  19. Elizabeth: But the LoI is not a truth about the world, and certainly not an “infallible” truth about the world. It’s simply implicit in the notion of an object (as Mung, I think, said).

    I would say that the laws of identity, non-contradiction, and the excluded middle are, in fact, pretty much what Aristotle thought they were: they are necessary pragmatic presuppositions of any discourse that purports to be about objects.

    And our understanding of these laws consists in appreciating that they are metalinguistic: they allow us to say what we are already doing — that is, doing in language — when we have any discourse that purports to be about objects.

    It follows, on my view, that any being that didn’t already know how to engage in discourse about objects would not be able to understand why it should accept the validity of the laws of reasoning.

    Elizabeth: Our brains have evolved to construct a model of the world (if you don’t like parse the world) in terms of objects with properties, and also events that result from interactions between objects.

    Granted, you are a perceptual neuroscientist and I’m just a philosopher, but I have some vague qualms about this precise way of putting it. I think that Merleau-Ponty was right when he said, in Phenomenology of Perception, that we don’t perceive objects; at any rate, what we perceive doesn’t have the same structure as what we talk about when we talk about objects.

    In order to have a discourse about objects, we have to be able to keep track of differing embodied perspectives on objects — we have to understand that you can be entitled to claims about an object that I’m not entitled to, because you can see a side of the object that I can’t see. We keep track of the compatibility and incompatibility of our claims — each of us keeping track of both yours and mine — by specifying an object of shared attention and attributing different properties to the object.

    However, if we follow Merleau-Ponty in describing perception itself (insofar as it is possible to abstract from all discourse and yet also be describing anything at all!), we find the figure-ground structure of Gestalt psychology, which is not the thing-property structure of Aristotelian onto-logic. The sensible features of the figure disclose themselves over the temporality of the experience as an integrated multiplicity (the orange chicken at the Chinese restaurant is an all-at-onceness of sweet and sour and orange and juicy and . . . ). And one’s own lived embodiment is, as M-P notes, the ‘third term’ in the figure-ground structure – pushing the ground back and pulling the figure forwards relative to our potential, motivated bodily movements.

    What we might be interested in doing, in neurophenomenological terms, is explaining the dynamics of brain processes that can be correlated with the figure-ground-movement dynamical structure of perception — which is not quite the same as the thing-property-relation structure of discourse.

  20. Lizzie,

    …but I do see that it depends for its coherence on the LoI.

    But the LoI is not a truth about the world, and certainly not an “infallible” truth about the world.

    My brain wants to see the capital I as a lower-case ell, so that you appear to be saying something profound about the Internet acronym.

    In the beginning was the LOL, and the LOL was with God, and the LOL was God.

  21. Kantian Naturalist: walto: I guess it’s either psychological or epistemic (or both) depending on how you define it. And ‘necessary’ is metaphysical or logical or both. But the conflation on this thread is pretty rampant (though not omnipresent!)

    I was afraid you’d say that.

    I think I follow a Kantian critique of empiricism in distinguishing between epistemology (as normative) and psychology (as descriptive), and a Kantian critique of rationalism in distinguishing between logic (as ranging across all possible worlds) and metaphysics (as ranging across all possible worlds that are epistemically available to beings whose cognitive capacities can be formally specified as the conjoint functioning of sensibility and understanding).

    That’s why the merely logical concept of an object (as defined by the three laws of reasoning*) cannot be milked for any metaphysical significance — to do metaphysics, you also need to do the epistemology of metaphysics. That is, all descriptive claims about objects, whether observed or posited, must be justified in terms of the process whereby one can have epistemic access to the object. And that’s what distinguishes ontological commitment from mere fantasy.

    * However, we now know that there are alternative formal systems — “non-classical logics” — that reject the law of non-contradiction. In those systems one is reasoning, but not about objects. For the same reason, the logic developed by Nagarjuna is a logic for reasoning in an ontology without objects, since Nagarjuna, as a Buddhist, taught the co-dependent origination of all things. There are no substances, in the Aristotelian sense, in a Buddhist understanding of reality. Interestingly, this does not prevent Buddhists from understanding and practicing Western science.

    I’m not sure I follow all that, KN. Are you saying that what I’ve called “conflation” of the concepts here is a good thing? That if something is necessary it must also be infallible (and/or vice versa)?

  22. keiths: My brain wants to see the capital I as a lower-case ell,

    Finally! The promised follow-up to your “Arnie” post!! I knew you’d come through!

  23. Elizabeth: Well, I won’t quibble over verbs! I’m coming at it as a perceptual scientist, you as a mathematician. I think we end up in the same place.

    I’ll start with a different kind of quibble. I think of myself as something like a perceptual scientist, though perhaps “perceptual philosopher” would be a better term. I’ve been studying the principles of perception since around 1990, though the study is not empirical. I set out to understand human learning, and quickly concluded that I had to start with perception.

    My theoretical ideas of perception are informed by my mathematical knowledge, but I do not have a mathematical theory of perception. My ideas are much more tied to what is pragmatically possible and to how perceptual abilities could have evolved.

    Our brains have evolved to construct a model of the world (if you don’t like parse the world) in terms of objects with properties, and also events that result from interactions between objects.

    I’m not a big fan of “model” here either. I just drank some coffee, and it didn’t seem that I was drinking a model.

    In any case, here’s my issue with “parse”. As I see it, “parsing” is the term for an entirely mechanistic syntactic process. The way that we divide up the world is, in my view, semantic rather than syntactic. That’s why I prefer “divide”.

    He himself was very taken with the idea of people as waves – specifically the idea of ocean waves, having an identity over time, even though the substance changes, and the energy on going, even after the wave crashes on the shore, and relinquishes its identity.

    There’s something reasonable about that. I have a pending response to KN in this thread, though I may start a new topic for that. I’ll perhaps be commenting on the idea there (or at least the idea of phenomenology). I tend to think of a person as a process, with the body as an implementation detail of that process. So the continuity of the process is what matters, and the lack of continuity of the material atoms in the body isn’t as important.

  24. keiths: My brain wants to see the capital I as a lower-case ell, so that you appear to be saying something profound about the Internet acronym.

    Yes, that was my first impression.

    In the beginning was the LOL, and the LOL was with God, and the LOL was God.

    Excellent.

  25. walto: I’m not sure I follow all that, KN. Are you saying that what I’ve called “conflation” of the concepts here is a good thing? That if something is necessary it must also be infallible (and/or vice versa)?

    Not at all. I started off that post with a bit on Kant about how to avoid the conflation between epistemology and psychology, and also the conflation between logic and metaphysics. But I think the conflation you’re worried about here — between infallibility and necessity — can be avoided on the (very) roughly Kantian account I’ve adopted.

    It is true that if one has grasped a necessary truth, then one cannot be mistaken about it. (That’s an analytic truth, as well.) But it is not true that if one understands oneself to have grasped a necessary truth, then one has indeed grasped a necessary truth. For there can be classes of possible worlds in which the putative necessary truth doesn’t hold, as Riemann discovered when he rejected the parallel postulate and as paraconsistency logicians like Restall and Priest discovered when they relaxed the constraints on the law of non-contradiction.

    I think that you and I are in agreement that we can be mistaken about what we take to be logically (also metaphysically and physically) necessary. Put otherwise, it is always possible that we are mistaken about what we take to be logically necessary. That’s just the human condition as rational animals, finite and inescapably bound to time and contingency.

  26. So the sum total of the thread is that

    A-ish is sorta like A-ish.

    For formal logic conceptual systems, A=A is a necessary axiom from a practical standpoint. Stuff won’t compute without that assumption, and admitting something otherwise will lead to computational instability.

    As far as infallible? For trivial stuff, yeah. Not so clear how it might handle capricious conceptual and physical objects. We can still assume A=A from a practical standpoint, but why add the qualifier it is “infallible”?

    Capricious conceptual objects: Self referential statements like, “this statement is false”. Gödel showed it made a wreck of Russell’s logicism. It is now suspected mathematics has some transcendence outside of logic, hence formal logic cannot capture all truth.

    Capricious conceptual objects: Schrodinger’s cat or some quantum system. Classically cannot both be dead and alive, you cannot be here and there at the same time. Schrodinger’s cat sort of wrecks all that at the bottom of reality.

    A=A is necessary from a practical standpoint in doing formal logic. A=A infallible for all reality? Why go there?

  27. I asked earlier Barry, or someone on his behalf, could identify an error that “materialists” make through “denying” that A=A.

    I’d be interested in what Barry might make of the rare but real cases where a person kills another while sleeping.

    If A=A has moral implications (as Barry appears to suggest) does the man A while asleep = the man A who wakes later?

    How could the Law of Identity, in the view of anyone who thinks it has moral consequences, help us decide on the identity of the killer of that man’s wife?

  28. Kantian Naturalist: It is true that if one has grasped a necessary truth, then one cannot be mistaken about it. (That’s an analytic truth, as well.)

    Yes, but I don’t think that’s a function of it (or anybody) being “infallible.” See below.

    You say (and I DO agree) that

    It is not true that if one understands oneself to have grasped a necessary truth, then one has indeed grasped a necessary truth….

    I think that you and I are in agreement that we can be mistaken about what we take to be logically (also metaphysically and physically) necessary. Put otherwise, it is always possible that we are mistaken about what we take to be logically necessary. That’s just the human condition as rational animals, finite and inescapably bound to time and contingency.

    Yes, exactly. Your other remark SEEMS to conflict with that, because it concedes that there are some propositions P which are such that, for any person S

    (1) Necessarily, if S believes P, S is right.

    That’s because P is true in every possible world. The thing is, there is no P which is such that S can infallibly know that P is one of the propositions that makes (1) true. That’s what I take infallibility to be about.

    That’s one of the things I think Fifth is confused about with many of his remarks on presupposition. I don’t know if this is consistent with what you write about Kant above, but IMO, there’s no real conduit between necessity and infallibility.

  29. How could the Law of Identity, in the view of anyone who thinks it has moral consequences, help us decide on the identity of the killer of that man’s wife?

    You don’t even need to pick such an extreme case. Consider statutory rape laws in the USA. If an 18.2 year-old girl has relations with 17.999999 year old boy, its a felony crime. So let’s say this happened because they started having relations when both were “17” and just continued. In the eyes of the law, they don’t treat the elder as the same class of person. It was legal when they were both “17”.

    If they are married (as in having some sort of exchange of vows recognized by the state), it’s not a felony crime. Is it the same person going from 17 to 18? Well, yeah. So why treat her different? Because she’s not the exact same person she used to be? But then if she’s not the same person she used to be, then why punish her now that she’s not the exact same person?

    For A=A to be applied to the real world of physical objects, one has to cheat and view “dissimilar things as the same” in one respect and “the same when they are dissimilar” in another respect. In the above example, we treat the girl as the same person in terms of punishment, but dissimilar because of the girl being 18 vs. 17. So in the area of ethics and law, the assumption of A=A isn’t strictly applied. From my vantage point, right and wrong don’t seem self-evident, we can only do our best according to our conscience and best understanding.

  30. walto: The thing is, there is no P which is such that S can infallibly know that P is one of the propositions that makes (1) true.

    I agree with that!

  31. Thanks for your posts on this thread btw Sal. They’ve been interesting and helpful.

    You are more than welcome. And thank you for giving me a chance to air opinions here that are not welcome to be aired at UD.

  32. Elizabeth,

    He himself was very taken with the idea of people as waves – specifically the idea of ocean waves, having an identity over time, even though the substance changes, and the energy on going, even after the wave crashes on the shore, and relinquishes its identity.

    I don’t think there’s anything special about “object-oriented” modelling of the world, but I do see that it depends for its coherence on the LoI.

    Your father’s view reminds me of a programmer I worked with a number of years ago. He was a big fan of functional programming and said that the data in objects was just an indication of where the developer got tired of modeling behavior.

  33. Kantian Naturalist: Is that equivalent to saying that the former is an epistemic term and the latter is a metaphysical term?

    Can you tell us why this matters, when “term” is a term of logic and “no logic is important for rational discourse”?

  34. I found this gem at Wikipedia:

    https://en.wikipedia.org/wiki/Logical_truth

    Logical truths (including tautologies) are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and no situation could arise which would cause us to reject a logical truth. However, it is not universally agreed that there are any statements which are necessarily true.

    A logical truth is considered by some philosophers to be a statement which is true in all possible worlds. This is contrasted with facts (which may also be referred to as contingent claims or synthetic claims) which are true in this world, as it has historically unfolded, but which is not true in at least one possible world, as it might have unfolded. The proposition “If p and q, then p” and the proposition “All married people are married” are logical truths because they are true due to their inherent structure and not because of any facts of the world. Later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations.

    The existence of logical truths has been put forward by rationalist philosophers as an objection to empiricism because they hold that it is impossible to account for our knowledge of logical truths on empiricist grounds. Empiricists commonly respond to this objection by arguing that logical truths (which they usually deem to be mere tautologies), are analytic and thus do not purport to describe the world.

    A=A can be stated alternatively as the XNOR operator. See:

    https://en.wikipedia.org/wiki/Logical_biconditional

    One thing to point out:

    Logical truths, being analytic statements, do not contain any information about any matters of fact. Other than logical truths, there is also a second class of analytic statements, typified by “No bachelor is married.” The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate. “No bachelor is married.” can be turned into “No unmarried man is married.” by substituting ‘unmarried man’ for its synonym ‘bachelor.’

    ….
    Quine rejects that logical truths are necessary truths. Instead he posits that the truth-value of any statement can be changed, including logical truths, given a re-evaluation of the truth-values of every other statement in one’s complete theory.

    I think Quine was referring to other formal logics that can get an equivalent result. Meaning, you can implement the formal system differently, hence certain statements are not necessary. For example, below for Gödel Logic, there is not just true or false statements, there are false, and true1, true2, true3…. 🙂

    https://en.wikipedia.org/wiki/Quantum_logic

    In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account. This research area and its name originated in a 1936 paper[1] by Garrett Birkhoff and John von Neumann, who were attempting to reconcile the apparent inconsistency of classical logic with the facts concerning the measurement of complementary variables in quantum mechanics, such as position and momentum.

    Quantum logic can be formulated either as a modified version of propositional logic or as a noncommutative and non-associative many-valued (MV) logic.[2][3][4][5][6]

    and

    https://en.wikipedia.org/wiki/Many-valued_logic

    Known applications of many-valued logic can be roughly classified into two groups.[8] The first group uses many-valued logic domain to solve binary problems more efficiently. For example, a well-known approach to represent a multiple-output Boolean function is to treat its output part as a single many-valued variable and convert it to a single-output characteristic function. Other applications of many-valued logic include design of Programmable Logic Arrays (PLAs) with input decoders, optimization of finite state machines, testing, and verification.

    The second group targets the design of electronic circuits which employ more than two discrete levels of signals, such as many-valued memories, arithmetic circuits, Field Programmable Gate Arrays (FPGA) etc. Many-valued circuits have a number of theoretical advantages over standard binary circuits.

  35. A=A is infallibly, necessarily true

    It is not necessarily true in many varieties of MV logics, such as illustrated here:
    https://en.wikipedia.org/wiki/Many-valued_logic#Examples

    Check out the Biconditional Operator (the same as the equal sign). Note, it doesn’t return “true” for all possible values of A = A.

    For example, under Kleene logic

    The proposition A=A does not always return true!

    A=A is infallibly, necessarily true

    Maybe so in binary logica (true, false) but not in MV logic.

  36. See:
    https://plus.maths.org/content/not-carrot

    Paraconsistent mathematics is a type of mathematics in which contradictions may be true. In such a system it is perfectly possible for a statement A and its negation not A to both be true. How can this be, and be coherent? What does it all mean? And why should we think mathematics might actually be paraconsistent? We’ll look at the last question first starting with a quick trip into mathematical history.
    ….
    Essentially, the idea is that if assuming something is true leads to an “absurd” state of affairs, a contradiction, then it was incorrect to make that assumption.

    This seems to work well enough in everyday situations. However, if contradictions can exist, say if Russell’s set both is and is not a member of itself, then we can deduce anything. We merely have to assume its negation, and then prove ourselves “wrong”. Thus contradiction trivialises any classical theory in which an inconsistency arises. Naive set theory, for example, is classically disinteresting, because it not only proves that 1+1=2, but also that 1+1=7. All because of Russell’s paradox. So to the classical mathematician, finding a contradiction is not just unacceptable, it is utterly destructive. There is no classical distinction between inconsistency (the occurrence of a contradiction) and incoherence (a system which proves anything you like).

    By removing RAA (or altering it as we see below), and making a few other tweaks to classical logic, we can create a logic and mathematical system where contradictions are both possible and sensible.

    and

    Classicists knew they were inconsistent

    There are further motivations for paraconsistency beyond those mentioned above. One such motivation is historical: at various times mathematicians worked with theories that they knew at the time to be inconsistent, but were still able to draw meaningful and useful conclusions. Set theory is one such area. The early calculus, as proposed by Isaac Newton, was another; its original formulation required that a quantity be small but non-zero at one stage of a calculation, but then to be equal to zero at a later stage. Despite the inconsistencies, mathematicians still adopted these theories and worked with them, drawing useful and sensible conclusions despite the presence of contradictions.

  37. stcordova: It is not necessarily true in many varieties of MV logics, such as illustrated here:
    https://en.wikipedia.org/wiki/Many-valued_logic#Examples

    Check out the Biconditional Operator (the same as the equal sign).Note, it doesn’t return “true” for all possible values of A = A.

    For example, under Kleene logic

    The proposition A=A does not always return true!

    Maybe so in binary logica (true, false) but not in MV logic.

    What are you saying? That Law of Identity is false? Or that it isn’t necessarily true? Or that Law of Identity is false in the context that BA meant it?

    Want to see Law of Identity at work?

    Let’s take your statement: “For example, under Kleene logic … the proposition A=A does not always return true!” This is true if by “Kleene logic” you mean precisely Kleene logic and not something fuzzy so that it could include classical binary logic. I.e. your statement has truth value if we presuppose Law of Identity!

  38. Erik,

    There is no law of identity in Kleene logic since there are no tautologies.

    See:

    https://en.wikipedia.org/wiki/Logical_equality

    Under Classical Logic, A is a logical object, it can be True or False:

    (A=A) is true, Symbolically,

    (A=A) = True

    Under Kleene Logic, A can be True, False, Indeterminate

    (AA) is not true when A=Indeterminate, but rather Indeterminate. Symbollically:
    (I=I) = I

    there are no tautologies in Kleene logic, hence no law of identity.

    Additionally, in paraconsistent logic

    (A and not-A) = True for some A

  39. stcordova: There is no law of identity in Kleene logic since there are no tautologies.

    Does this somehow change the fact that anything you say (provided that it’s supposed to make sense) presupposes law of identity? When you say “Kleene logic” are we supposed to take it as “Kleene logic” or not? If we are supposed to take “Kleene logic” as “Kleene logic”, then we are presupposing law of identity.

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