Comment to the OP.

I think someone wrote that Wittgenstein didn't understand basically Turing's example (the Turing Machine) and Turing failed to describe it for him. The anticipated meeting of the two didn't create great advances. But it's telling that even for Gödel to understand that Turing's findings were actually similar to his did take a long time, so it's no wonder if Turing and Wittgenstein didn't understand each other.

I'm not sure when reading Murphy's article he understands this either. Using negative self-reference as Turing or Gödel did isn't similar to Liar paradox. Close, but they don't result in a paradox. That perhaps is the crucial point to understand.

It's not about a paradox here or there. Rather Wittgenstein's idea was to happily welcome logical contradictions in mathematics.

I agree it will never happen. It was just another absurd idea from Wittgenstein.

I've had many a discussion with contributors here who are convinced that ideas are essentially 'brain structures', and that the brain is shaped by evolutionary adaptation — Wayfarer

I think such a position is easy to rebut, in that it is a self-referencing negative statement structurally identical to the liar's paradox, or to "this sentence is false". The idea that ideas are essentially 'brain structures', if true, is itself a mere brain structure.

It's a point I used to make quite often here, at start, so I agree that this place (TPF) has more than its fair share of this kind of kindergarden version of materialism.

what we make of that math does seem to be a matter for philosophers (and everyone else, really.) — hanaH

Why yes, we are all amateur philosophers anyway, even those of us who are professional mathematicians. And there is no subject matter that philosophy cannot legitimately discuss. Mine was a joke.

The one we talked about:

[...] allowing contradictions in math is equivalent to dropping the law of the excluded middle from mathematical logic [...]
— Olivier5 — TonesInDeepFreeze

Wonderful. So now, how would you write down the proposal that we allow contradictions in mathematics, syntactically and semantically? What sort of axiom would that translate into, in your opinion?

So , in sum , if one understands context in a formal categorical sense, then the LEM is applicable in some contexts
and not in others. — Joshs

And then the interesting question (for me at least) becomes: in which types of contexts does it apply, and in which types of contexts does it not apply.

But if one equates context with absolute situational and perspectival contingency , then the LEM can no longer find the minimal categorical identity over time in the idea of context necessary for it to contribute anything useful.

My system crashed again. In other words, please?

how would you write down the proposal that we allow contradictions in mathematics, syntactically and semantically? What sort of axiom would that translate into — Olivier5

That is not a good question, because is has a trivial answer: Add any contradiction as an axiom.

A better question is: What systems preserve important parts of classical logic but not EFQ (explosion)?

https://plato.stanford.edu/entries/logic-paraconsistent

Also, contradictions, in the formal sense such as with inconsistent systems, are syntactic, not semantic.

Add any contradiction as an axiom. — TonesInDeepFreeze

You mean, one by one?

You mean, one by one? — Olivier5

You only need one. When you have one, you get them all. I have mentioned several times already the principle of explosion. Indeed, it is at the very heart of the Turing role in the matter of this thread, as mentioned in the originally linked article.

So anything of the type P & -P?

So anything of the type P & -P? — Olivier5

Correct.

And that would not contradict the LEM?

As I've said, it is inconsistent with everything.

And again, the point I made to you about LEM is not about what contradicts it, but rather that merely taking LEM out of a system does not make the system inconsistent.

It's pretty clear that you are not familiar with even the basic concepts in formal logic. To understand the topic raised by the reference to Turing in this thread, you need that basic background. I suggest starting out by reading a textbook in symbolic logic. I can offer a suggestion if you like.

I think I got the point you're making, which is -- correct me if I'm wrong -- that the LEM is different from the Law of Non Contradiction in that the "or" in LEM is inclusive and can accommodate (P & -P) which only the LNC excludes. Now, when I was taught formal logic, the 'or' of the LEM was described as exclusive. That is, either P or non P. And it's not my poor memory because it is still presented as such in the French wikipedia article about it, with such typical example as 'a door is open or closed.'

It seems therefore that as I was taught it, some decades ago in another country than yours, the LEM included the LNC by way of an exclusive or.

In any case, and this clarification being made, I agree with you that the 'inclusive or' version of the LEM is not the relevant logical law in relation to Wittgenstein's proposal to allow contradictions in mathematics. The LNC is.

the LEM is different from the Law of Non Contradiction in that the "or" in LEM is inclusive and can accommodate (P & -P) which only the LNC excludes. — Olivier5

Classically, LEM and LNC are equivalent, since all logical truths are equivalent. That is, for any logical truths P and Q, we have P <-> Q as a theorem.

But LEM and LNC are different in these senses: (1) They are literally different formulas and (2) If we remove certain classical assumptions, then they are not equivalent. Most saliently, in intuitionistic logic, we cannot derive LEM from LNC.

/

No, the literal formulation of LEM in ordinary symbolic is NOT exclusive or. Literally, LEM is:

P v ~P, where 'v' stands for inclusive disjunction.

either P or non P — Olivier5

Yes, inclusive 'either or'.

/

Also, Wikipedia is sometimes okay to get a quick and rough summary of a topic, but one must be on guard against misinformation and poorly formulated expositions there. However, in this instance it is correct. I took the French article and translated it to English (as I don't know French) and do not find a claim that the 'or' in LEM is exclusive. Indeed, the article says, contrary to you, exactly what I have been saying: It is from LNC that we get that it is not the case that both P and ~P are true, while it is from LEM that we get that it is not the case that neither P nor ~P are true.

/

If you want to understood such topics in logic, I strongly suggest studying an introductory textbook in a systematic and thorough way rather than relying on a cobbled together misunderstanding from bits and pieces mis-gleaned here and there.

I've lapped all this stuff years ago. The LEM was presented as I said, with an exclusive or. Honest mistake.

Then the presentation you were subjected to was egregiously errant. I would be on guard about anything else that was offered to you in that presentation.

It was just different, but logically equivalent in a system where both laws are in force, so it matters not. In fact I still think it is more elegant with an exclusive or because you only needs one law instead of two.

It matters because the 'v' ('or') connective should never have been conflated with exclusive-or.

Also, your notion that exclusive-or has an advantage of elegance is ill-conceived, as I could explain also.

No need, you're not a very eloquent writer. It would take you ages.

What's an event? Does it involve objects? 'Transformation' implies some thing that is transformed, maintaining its identity in some sense. As another poster has mentioned, this kind of point threatens to 'deconstruct itself,' which is not necessarily a bad thing. — hanaH

I think the key issue here isn’t materiality or objectness but relative stability. The question is, how does a position like Heidegger’s, which claims to deconstruct the self-identical object , achieve stability of meaning? The answer is that an event doesn’t occur into a vacuum , but into an exquisitely organized referential totality. That is precisely what an event is, a way that this totality of relevance changes itself moment to moment. So there is a tremendously intricate and intimate overall coherence from one event to the next. Each event is a subtle variation on an ongoing theme, and it’s very appearance shifts the sense of this theme without rending its pragmatic consistency. This relation between a referential background and new events allows one to say that the world can continue to be the same differently, as an ongoing style.

This is the paradox of this kind of model. On the one hand , it is more radically and immediately transformational than models
positing empirical objects. On the other hand it avoids
the arbitrariness of objective causality. There is a radical
belongingness of one moment to the next that is missing from causal approaches. We tend to assume that it is only by nailing down objects as self-identical that we achieve the possibility of order and stability in our models of the world. And this was true in comparison with pre-modern thinking. But it isn’t self-identically that gives us the order we crave , it’s the extent to which such assumptions facilitate our ability to discern real-time among such entities. The assumption
of self-identicality actually limits the possibilities of relationality that we can find in the world.

you're not a very eloquent writer — Olivier5

My posts aren't models of eloquence, but they are, for the most part, articulate and more precise about technical matters than normally found in a casual context such as this forum.

After a series of clearly correct explanations by me, and over a course of your stubbornness, finally those explanations provided you with understanding why you were incorrect to begin with, and also provided you with context of other notions that are important to the topic. But the very fact that I provided you with ample explanations leads you to gripe that they were not eloquent enough for you and too ample.

It would take you ages. — Olivier5

Yes, an explanation why your notion of an advantage of a certain elegance is wrong headed would be longer than a couple of lines, because your notion is wrong headed in so many different ways.

We have every reason to believe that because it is the best explanation for a shared world; in fact it is the only explanation apart from some form of idealism; some notion that all minds are somehow conjoined or that there is a universal mind we all partake in.. — Janus

Here are alternatives from those who reject idealism. Maybe you can agree with the gist of their arguments.

From Evan Thompson:

“Many philosophers have argued that there seems to be a gap between the objective, naturalistic facts of the world and the subjective facts of conscious experience. The hard problem is the conceptual and metaphysical problem of how to bridge this apparent gap. There are many critical things that can be said about the hard problem (see Thompson&Varela, forthcoming), but what I wish to point out here is that it depends for its very formulation on the premise that the embodied mind as a natural entity exists ‘out there' independently of how we configure or constitute it as an object of knowledge through our reciprocal empathic understanding of one other as experiencing subjects. One way of formulating the hard problem is to ask: if we had a complete, canonical, objective, physicalist account of the natural world, including all the physical facts of the brain and the organism, would it conceptually or logically entail the subjective facts of consciousness? If this account would not entail these facts, then consciousness must be an additional, non-natural property of the world.

One problem with this whole way of setting up the issue, however, is that it presupposes we can make sense of the very notion of a single, canonical, physicalist description of the world, which is highly doubtful, and that in arriving (or at any rate approaching) such a description, we are attaining a viewpoint that does not in any way presuppose our own cognition and lived experience. In other words, the hard problem seems to depend for its very formulation on the philosophical position known as transcendental or metaphysical realism. From the phenomenological perspective explored here, however — but also from the perspective of pragmatism à la Charles Saunders Peirce, William James, and John Dewey, as well as its contemporary inheritors such as Hilary Putnam (1999) — this transcendental or metaphysical realist position is the paradigm of a nonsensical or incoherent metaphysical viewpoint, for (among other problems) it fails to acknowledge its own reflexive dependence on the intersubjectivity and reciprocal empathy of the human life-world.

Another way to make this point, one which is phenomenological, but also resonates with William James's thought (see Taylor, 1996), is to assert the primacy of the personalistic perspective over the naturalistic perspective. By this I mean that our relating to the world, including when we do science, always takes place within a matrix whose fundamental structure is I-You-It (this is reflected in linguistic communication: I am speaking to You about It) (Patocka, 1998, pp. 9–10). The hard problem gives epistemological and ontological precedence to the impersonal, seeing it as the foundation, but this puts an excessive emphasis on the third-person in the primordial structure of I–You–It in human understanding. What this extreme emphasis fails to take into account is that the mind as a scientific object has to be constituted as such from the personalistic perspective in the empathic co-determination of self and other. The upshot of this line of thought with respect to the hard problem is that this problem should not be made the foundational problem for consciousness studies. The problem cannot be ‘How do we go from mind-independent nature to subjectivity and consciousness?' because, to use the language of yet another philosophical tradition, that of Madhyamika Buddhism (Wallace, this volume), natural objects and properties are not intrinsically identifiable (svalaksana); they are identifiable only in relation to the ‘conceptual imputations' of intersubjective experience.” (Empathy and Consciousness)

From Dan Zahavi and Hilary Putnam:

Knowledge is taken to consist in a faithful mirroring of a mind-independent reality. It is taken to be of a reality which exists independently of that knowledge, and indeed independently of any thought and experience (Williams 2005, 48). If we want to know true reality, we should aim at describing the way the world is, not just independently of its being believed to be that way, but independently of all the ways in which it happens to present itself to us human beings. An absolute conception would be a dehumanized conception, a conception from which all traces of ourselves had been removed. Nothing would remain that would indicate whose conception it is, how those who form or possess that conception experience the world, and when or where they find themselves in it. It would be as impersonal, impartial, and objective a picture of the world as we could possibly achieve (Stroud 2000, 30). How are we supposed to reach this conception?

Metaphysical realism assumes that everyday experience combines subjective and objective features and that we can reach an objective picture of what the world is really like by stripping away the subjective. It consequently argues that there is a clear distinction to be drawn between the properties things have “in themselves” and the properties which are “projected by us”. Whereas the world of appearance, the world as it is for us in daily life, combines subjective and objective features, science captures the objective world, the world as it is in itself. But to think that science can provide us with an absolute description of reality, that is, a description from a view from nowhere; to think that science is the only road to metaphysical truth, and that science simply mirrors the way in which Nature classifies itself, is – according to Putnam – illusory. It is an illusion to think that the notions of “object” or “reality” or “world” have any sense outside of and independently of our conceptual schemes (Putnam 1992, 120). Putnam is not denying that there are “external facts”; he even thinks that we can say what they are; but as he writes, “what we cannot say – because it makes no sense – is what the facts are independent of all conceptual choices” (Putnam 1987, 33).

We cannot hold all our current beliefs about the world up against the world and somehow measure the degree of correspondence between the two. It is, in other words, nonsensical to suggest that we should try to peel our perceptions and beliefs off the world, as it were, in order to compare them in some direct way with what they are about (Stroud 2000, 27). This is not to say that our conceptual schemes create the world, but as Putnam writes, they don't just mirror it either (Putnam 1978, 1). Ultimately, what we call “reality” is so deeply suffused with mind- and language-dependent structures that it is altogether impossible to make a neat distinction between those parts of our beliefs that reflect the world “in itself” and those parts of our beliefs that simply express “our conceptual contribution.” The very idea that our cognition should be nothing but a re-presentation of something mind-independent consequently has to be abandoned (Putnam 1990, 28, 1981, 54, 1987, 77)

too ample. — TonesInDeepFreeze

Too slow, rather. You could have said a long time ago: "you must mean the LNC, because the LEM does not actually rule out contradiction."

You could have said a long time ago: "you must mean the LNC, because the LEM does not actually rule out contradiction." — Olivier5

I don't know what you mean by 'a long time ago' relative to the duration of our exchanges. But many posts ago, I wrote:

the 'but not both' clause for exclusive or is demanded by the law of non-contradiction: ~(P & ~P). — TonesInDeepFreeze

You're blaming me for the fact that you didn't bother to read what I posted.

It matters because the 'v' ('or') connective should never have been conflated with exclusive-or.

Also, your notion that exclusive-or has an advantage of elegance is ill-conceived, as I could explain also. — TonesInDeepFreeze

Since P and ~P are mutually exclusive, what difference does it make whether the disjunction is inclusive or exclusive?

Since P and ~P are mutually exclusive, what difference does it make whether the disjunction is inclusive or exclusive? — T Clark

P and ~P are mutually exclusive in classical logic, but not necessarily in other logics, especially paraconsistent logics.

To answer your question:

(1) It is important to not be confused as to which connective is actually used.

(2) With inclusive-or, we can take out LEM to get intuitionistic logic, and also, by not subsuming LEM within an LEM/LNC combo, we can take out LNC to get paraconsistent logics.

(3) The notions of LNC and LEM go back to antiquity, and have been critical in the discussion of logic through the centuries, so to suddenly say that LEM now means something different would be quite confusing.

[Edit: I should not have written, "We can take out LNC to get paraconsistent logics."]

We started this discussion two or three days ago though.

We started this discussion two or three days ago though. — Olivier5

Oh, no, I didn't deliver your desired angle on the subject soon enough, even though what I did say was correct at every point while you persisted otherwise!

I gather that you have no idea what a nudnick you are being.

I adduced, entirely gratis, the point about LNC and LEM, because it was at that stage that I sensed it might bear upon your confusion. It's not my job to immediately anticipate what is confusing you about the subject and then to immediately warn you about those confusions. I gave you clear corrections, which is itself gratis, and then at other stages added more explanation also gratis. For that matter, there is even more about the subject I could add now, but, again, it's not my job to try to figure out why you are mixed up so that I choose just the right points customized for you.