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.

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.

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.

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.

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.

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

Correct.

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.

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.

when I say we have an inconsistency I just mean that the intuition that .9999 does not equal 1 is inconsistent with any proof that the two are equal — Janus

That's not what mathematicians, including Turing, mean by 'inconsistency'. That's not (as far as I can tell) a sense of inconsistency at issue with the "bridge collapses" issue mentioned in connection with Turing, which is the sense of a system proving a formula and its negation, thus, by the principle of explosion, the system proving every formula whatsoever in the language of the theory.

Inconsistency, in a formal sense, is not a clash between a theorem and the fact that certain people have a different intuition.

The context in which .999... = 1 is classical mathematics as introduced ordinarily in Calculus 1 then more rigorously explicated in Real Analysis, especially made rigorously axiomatic by the set theoretic development of the real numbers. The infinite sum as the point of convergence of a sequence is what mathematicians MEAN by .999... Saying, in face of that context, that .999... does not equal 1 is not informed intuition but rather ignorance of the actual mathematics. Claiming, without mathematical basis, that .999... does not equal 1 is claptrap that needs to be remedied by study of the basics of the subject. (I am addressing here only what is ordinarily meant in mathematics, as I recognize that there are other approaches including finitistic views and computationalist views).

0.999999 can converge infinitely with 1 out ever reaching it — Janus

That's not the point.

'.999...'
is informal for

SUM[n = 1 to inf] 9/10^n

And SUM[n = 1 to inf] 9/10^n is the limit of a sequence. And that limit is 1.

/

By the way, when you wrote, "0.333333333 is no more equal to one third than 0.9999999 is equal to 1", I glanced over it too quickly and took you to mean that we can't assume .333... = 1/3 any more than we can assume .999... = 1". That is correct. But we can go on to prove both that .999... = 1 and that .333... = 1/3. We don't assume them, but we do prove them.

I'll leave my post as it is so that your following self-correction makes sense in context.

it is said that 0.9999999999999... equals 1, because .33333333333333... equals one third and three thirds equal one. — Janus

That is not a rigorous mathematical proof. However, there is a rigorous mathematical proof that .999... = 1.

'.999..' is an informal way of describing a certain infinite summation. And infinite summation is defined by convergence. And we prove that the sequence converges to 1.

I am wondering now whether I should have said I accept the following:

The consistency of certain systems (PA and the like) cannot be constructively proved by any means. — sime

The consistency of, for example, PA cannot be proven by finitistic means, but I don't know whether it can't be proven by constructive though non-finitistic means. First, to pin the question down exactly, we would need an exact mathematical definition of 'constructive'. Second, even if we say, for sake of argument, we mean a particular constructive set theory or other constructive theory, then we would have to prove that such theories do not prove the consistency of classical PA.

As to Gentzen's proof, if I am not incorrect, we can describe the situation in two ways:

Let 'T' stand for transfinite induction on e_0.

(1) PRA+T |- Con(PA)

In other words, by finitistic means plus transfinite induction on e_0 we prove that PA is consistent.

vs

(2) PRA |- Con(PRA+T) -> Con(PA)

In other words, by finitistic means we prove that Con(PRA+T) implies Con(PA).

It is not, at least to me, apparent that those are not constructive (I don't know whether they are or are not constructive). However, I did find a paper that mentions that T is constructively challenged by some writers (actually, it's not T that's discussed but an alternative assumption used in place of T).

But you also say:

Gentzen proved that the inconsistency of PA implies the inconsistency of PRA + transfinite induction on the ordinals. — sime

That is of the form ~Con(PA) -> ~Con(PRA+T). But, at least prima facie, that does not intuitionistically imply Con(PRA+T) -> Con(PA). Yet, the latter version is the one we more often see. So I don't know why you stated Gentzen as ~Con(PA) -> ~Con(PRA+T).

The one we talked about:

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

And I mentioned it most presently only about an hour ago and as you responded to me right after.

You made a general statement about it. You made your own claim about mathematics and mathematical logic. And your claim is incorrect. My point about that pertains irrespective of Wittgenstein. Whether people are talking about Wittgenstein in particular, or inclusive of other tangents, it is worthwhile to point out that certain specifics mentioned in context or standalone are in error.

I have explained more than once already why it is not the case that an otherwise consistent system can be made inconsistent by retracting the LEM. It is not required that I propose couching Wittgenstein into formal syntax and semantics to make that point.

I have nothing to say about that.

Though, while probably not specifically apropos of Wittgenstein himself, one can look up the subject of paraconsistent logic.

I don't bristle against being corrected on matters of logic.

I don't know what scope you have in mind by 'meaning' but I take the LEM in its utterly ordinary sense:

Syntactically: P v ~P

Semantically: Every sentence in the language is either true in the model or it is false in the model (where 'or' is the inclusive or; while the 'but not both' clause for exclusive or is demanded by the law of non-contradiction: ~(P & ~P)).

Of course, you persist petulantly. No, you were not correct. You gave the impression that I had just learned the quote function, which is not correct.

And you were incorrect about LEM, as I amply explained. And It's not contrarianism simply to explain that the connection you claimed between the LEM and paradox is incorrect and is a basic misunderstanding of logic.

And, yes, as I mentioned, I knew that not using the quote function had disadvantages, including the one you just mentioned.

This incident with you, in line with ones with other posters with misconceptions about logic and mathematics (not in philosophy, but in the mere rote, technical facts) confirms my thought that on forums such as this, it is virtually impossible to post corrections and explanations without there being posters who will bristle personally about that.

As far as I can think it through, your first paragraph seems reasonable and good added information to my own remark..(Though when I said 'by certain means', that of course encompasses whatever means do fail, such as the constructive ones you mention.)

As for the second paragraph, I don't know what you mean by an ordinal that is not well-founded. Any ordinal is well-founded by the membership relation, of course.

I am well versed in the quote feature, as seen in my many posts in other threads. But I had been experimenting with not using it lately in order to avoid too much of a personal tit-for-tat kind of conversation. That is, to make my remarks general toward the points no matter who might have asserted them, though I recognize the disadvantages of not using the quote feature. Then, at a point, it became too cumbersome to reply without quoting, and then my experiment manifestly failed when you resorted to the personal claim that I lack "good will".

It's not that I bar myself from making personal remarks - indeed, I may liberally say quite what's on my mind about a poster's qualities - it's just that lately I was in the mood for experimenting with ways that might avoid my receiving them and then replying with them.

So, congratulations are in order now also for your utterly petty, sophomorically sarcastic, and incorrect snipe. Well done.

maybe with a little good will you would be able to understand what I am saying. — Olivier5

I'm informing you that it is a basic misconception to think that the LEM is a consideration in the way you have claimed. That is not ill will.

I wouldn't call it pointless to point at one consequence among many. — Olivier5

It's one consequence not just among many but among all. The supposed connection with the LEM does not hold.

If a system is inconsistent, then the system contradicts every statement in the system, not just the law of excluded middle. So it is pointless to adduce the law of excluded middle in this way. There seems to be expressed a very basic misunderstanding of logical and mathematical theories: It is a crucial and basic point that needs to be understood: A consistent system cannot be made inconsistent by retracting axioms.

To be clear, Fitch's paradox is not a conclusion that the truth of all statements is known, but rather the conclusion is that it is not the case that the truth of all statements is knowable, since by a modal argument, if the truth of all statements is knowable then the truth of all statements is known, while it is not the case that the truth of all statements is known.

Anyway, I don't know what was intended by mentioning Fitch's paradox in connection with my remark about the second incompleteness theorem.

Again, retracting the law of excluded middle does not provide contradictions. Intuitionist mathematics eschews the law of excluded middle. If classical mathematics is consistent, then perforce intuitionist mathematics is consistent.

Godel's 2nd incompleteness theorem is not that certain systems can't be proven consistent, but rather that if they are consistent then they can't be proved consistent by certain means. For example if PA is consistent, then PA cannot be proved consistent by PA itself, though it can be proven consistent with, for example, Gentzen's proof. Moreover, the 2nd incompleteness theorem is about formal provability of consistency and does not itself say anything directly about knowledge, which is a philosophical issue, not covered by the mere mathematics of Godel's proof..

/

The true arithmetical statements that are unprovable include ones that are not trivial, and especially are not of the form of logical truths (since, of course, all logical truths are provable). One can easily look up the subject of substantive mathematical statements (ones worth knowing whether they are theorems) that are undecidable in the pertinent systems.

/

It was said, "[...] inconsistencies in math are analogous to "This sentence is not true."

A system is inconsistent if and only if it has a theorem of the form P & ~P. Whatever is meant by "analogous", we know that there are inconsistent systems having nothing to do with the liar paradox.

/

Russell's paradox is couched in terms of sets when discussing set theory, but the basic paradox does not require any notion of sets whatsoever. That is illustrated by using "shaves" rather than "member of". The result is that for any 2-place relation whatsoever, call it 'R', it is not the case that there is an x such that for all y, we have y bears R to x if and only if y does not bear R to y. Symbolically:

~ExAy(Ryx <-> ~Ryy)

There is no mention of the notion of 'set' there.

/

Among the salient uses of the liar paradox for mathematical logic is Tarski's theorem. That theorem is that systems of a certain kind that also can form their own truth predicate can thereby form the liar paradox so that such systems are inconsistent.

I don't know all the mathematics for engineering, but I don't imagine that reliably building bridges or other common practical endeavors depend on settling the liar paradox. But that in itself does not entail that foundations and seeking consistent axiomatizations do not have bearing on mathematics for the sciences.

should math allow contradictions? I.e. should we get rid of the law of excluded middle in math — Olivier5

Retracting excluded middle wouldn't allow contradictions. The logic is monotonic: we don't get additional theorems from subsets of a consistent axiom set. To have contradictions, we have to add axioms that prove contradictions. (To prevent contradictions from entailing all statements, normally a paraconsistent system is used that retracts the law of explosion.)

It's probably fair to say that the import of foundations for the mathematics for the sciences is mostly theoretical as opposed to practical. But some of the central concerns of foundations involve natural questions, such as: Is there an algorithm to determine whether or not any given mathematical statement is a theorem? Is there an algorithm to determine whether or not any given mathematical statement is true in a given model? Is there an algorithm for determining whether any given program and input halts? Is the algorithm that determines whether any given Diophantine equation has solutions? Those are questions of clear intellectual interest even if nothing else.

Moreover, whether any given result in foundations has immediate practical consequences, the concepts and methods of foundations provide the context for the study of the branches of mathematics that do have great practical consequences. The field of computability is built from symbolic logic and mathematical logic. The invention of modern computing itself was born in this context.

Moreover, one may wish to evaluate for themself whether certain results in undecidability bear upon certain matters in combinatorics and cryptography. And, without doubt, one of the most pressing practical questions in mathematics is the P vs NP question, of which a resolution would have enormous practical import. There is currently a million dollar prize for a solution, which might be the largest offered prize now in mathematics. A solution is sought not only to satisfy theoretical curiosity but rather for the great economic impact of the question. Also, for example, Turing's theorem on the unsolvability of the halting problem is often said to entail that there can be no universal program debugger, so researchers don't have to waste efforts looking for one, but instead various other concepts and approaches are devised.

/

It was posted: "[...] people talk about paradoxes as if they undermine the validity of mathematics."

Certain paradoxes, if allowed as formal contradictions, vitiate the systems in which the contradictions are derivable. The common response in foundations is to provide systems in which the contradictions are not derivable (or at least are not shown to be derivable).

It was posted: "Godel's proof of his theorem has always seemed goofy to me. I don't understand how the claim that one odd, trivial contradiction proves that math is incoherent in any meaningful way makes sense."

I don't know what is meant there by "goofy". And Godel's proofs are not proofs of contradictions. And I don't know any informed writer on mathematics who has claimed that Godel's work makes mathematics incoherent.

It was posted: "[...] allowing contradictions in math is equivalent to dropping the law of the excluded middle"

No it is not.

Indeed.

One my choose to hold that a proposition does not exist until it is has been expressed. But even if we restrict to the set of propositions that have been expressed, we have not vitiated Fitch's argument.

It is not the case that for all propositions p that have been expressed we have p -> Kp. Therefore, as Fitch shows in the proof, it is not the case that for all propositions p that have been expressed we have p -> LKp.

this was stated by TMF — InPitzotl

Which is exactly why I am asking why it is being stated.

SEP is clear that 'Kp' means "We know that p is true" and that the proof uses sentences. And Wikipedia is even more explicit (I don't automatically trust Wikipedia, but it is lucid on these points). There is no need to trust me; one can read for oneself.

The distinction between propositions and sentences is an involved subject in philosophy and logic. [See, for example, Chuch's 'An Introduction To Logic' for one widely referenced treatment.] But no matter, it is not the case that the intended interpretation of the 'K' operator is that of knowing the existence of a propostion or sentence, but rather it is that the proposition or sentence is known to be true. Ignoring that point leads to incoherence.

One needs to read the expositions such as at SEP.

As I said, the general topic regards propostions, but the formal portion of the argument uses sentences. One should read the expositions. And, again, it doesn't matter whether propositions or sentences, the critical point stands that the intended interpretation of 'Kp' is not "We know that p exists" but rather "We know that p is true".

I am not sure that all discussants here understand:

(1) 'Kq' stands for "q is known to be true" and it does not stand for "q is known to be a sentence (or proposition)".

(2) Fitch is not arguing that for all p, we have p -> Kp. Fitch is arguing that if for all p, we have p -> LKp, then for all p, we have p -> Kp, but since it is not acceptable that for all p, we have p -> Kp, it is not acceptable that for all p, we have p -> LKp.

Why is
q -> Kq
being stated?
— TonesInDeepFreeze
Because p -> Kp was stated. — InPitzotl

Who stated it? To be clear, Fitch does not hold that p -> Kp.

No one believes that as a generalization for all q.
— TonesInDeepFreeze
Apparently some people do. It's an antirealist position; the p doesn't exist until it's proposed, and it isn't true until you say it is, or some such thing. — InPitzotl

What specific quotation or reference is given by anyone (other than a flagrantly errant poster) that p -> Kp? The supposed antirealist notion is p -> LKp.

"We don't know that the earth is round"

and

"We believe that the earth is flat" [...]

The differences are so easy to point out that I don't see the sense in asking about it. — TonesInDeepFreeze

There is no mistake in that.