By the way, that Wikipedia article is another example of misinformation in Wikipedia.

We don't need to bring in the question of knowledge in this context, and one cannot know that a false assumption is true. One can assume statements that are false. People make false assumptions often. People even assume contradictory premises fairly often. But one cannot know a statement to be true when it is false. It is not the case that one knows a statement to be true simply by assuming it is true. In particular, one does not know that both lemons are yellow and lemons are not yellow simply by assuming that lemons are yellow and lemons are not yellow.

And the example is unnecessarily overcomplicated, not only with the gratuitous and incorrect remarks about knowing, but also in its logical form. It would be better written:

The assumption "lemons are yellow and lemons are not yellow" implies the conclusion "unicorns exist".

Or, even better, to avoid questions about pluralization and generalization:

The assumption "the Cartier Sunrise Ruby is red and the Cartier Sunrise Ruby is not red" implies the conclusion "the Empire State Building is a unicorn".

And my point is that we don't thereby conclude, independent of the contradiction, that the Empire State Building is a unicorn.

It was claimed in this thread that most philosophers believe it is not the case that there are sentences that are true on the basis of their meaning.

What is the basis for that claim?

Given C, a contradiction, the expression C => P is true. That is because C is false, and whenever the antecedent is false, the implication is true - them's the rules. But it is an elementary and serious error to suppose this shows that P is true. For P to be true, C must first be affirmed. That is, C ^ (C => P) => P, C being true, affirms P. And this is exactly - or should be - what Tones said. — tim wood

In other words you disagree with the Wikipedia quote.
I disagree with the Principle of Explosion itself.
There are no semantics passed from any contradictory premise to any conclusion.
(A ∧ ¬A) only proves FALSE. There is nothing about the semantics of {Cat's are not cats}
that makes {the Moon is made form green cheese} true.

A contradiction doesn't make a false statement true. No one disagrees with that. And it is not the principle of explosion.

A contradiction doesn't make a false statement true. No one disagrees with that. And it is not the principle of explosion. — TonesInDeepFreeze

The POE says the everything is logically entailed by a contradiction and it is simply wrong about this.
A & ~ A proves FALSE and nothing more.

It was claimed in this thread that most philosophers believe it is not the case that there are sentences that are true on the basis of their meaning.

What is the basis for that claim? — TonesInDeepFreeze

That most philosophers were convinced by Quine that the analytic/synthetic distinction does not exist, thus it is impossible to divide expressions {true on the basis of their meaning} from expressions that are true on some other basis such as observation.

This would be amazing if we didn't know the poster's history:

I asked for the basis for the claim. The poster replies by merely repeating the claim.

/

The principle of explosion is that any sentence is entailed by a contradiction. That does not contradict:

If C is any contradiction and P is any sentence, then we have C -> P, but that does not allow inferring P. Rather, we would infer P from (C -> P) & C. But since we never have C, don't have (C -> P) & C so we still don't have P.

If C is a contradiction and P is any sentence, then we have:

(1)

C -> P

and

C |- P

Neither of those give us P, since we would first need to have C, which we don't have.

(2)

((C -> P) & C) -> P

and

(C -> P) & C |- P

Neither of those give us P, since we would first need to have C, which we don't have.

The point of this is to fend against people who don't understand the principle and think that it allows us to make any ridiculous conclusions we wish to make.

The principle of entailment goes very far back in the history of logic. It is in model theory that the principle is given mathematical exactness. The model theoretic version adheres to the general principle: A set of premises entails a conclusion if and only if there are no circumstances in which the premises are all true but the conclusion is false. — TonesInDeepFreeze

I am talking about how the categorical propositions of the syllogism directly encode semantics thousands of years before anyone every heard of model theory. The Tarski Undefinability theorem was before model theory.

The undefinability theorem is from before the method of models was given a fully formal definition. But the basic idea of a model was used long before the formalization of the idea.

And model theory adheres to the ancient notion of entailment: A set of premises entails a conclusion if and only if there are no circumstances in which all the premises are true but the conclusion is false.

And model theory adheres to the ancient notion of entailment: A set of premises entails a conclusion if and only if there are no circumstances in which all the premises are true but the conclusion is false. — TonesInDeepFreeze

That seems to not be restrictive enough. From your idea Donald Trump is Christ is entailed by the Moon is made from green cheese because the Moon is made from green cheese is false.

Notice that the poster switched the discussion from the premise being contradictory to the premise being merely false. We had been explicitly discussing contradictory premises not just false premises.

Consider two sentences D and M.

If D is false, then M might be true or false. The falsity of D does not entail the truth of M. But the conditional D -> M is true.

If D is a contradiction, then M might be true or false. From the mere fact that D is logically impossible we do NOT infer that M is true. But the conditional D -> M is not just true, it is logically true. THAT is the principle of explosion and it does NOT imply that M is true.

If the sentence D is merely false, we cannot infer M. But if we assume the sentence D & ~D then we can infer M. But, again, we would be mistaken to have D & ~D as a premise, so if M is false, thus a mistake to assert it, the mistake of asserting M would occur only by the mistake of asserting D & ~D.

If D is a contradiction, then M might be true or false. From the mere fact that D is logically impossible we do NOT infer that M is true. But the conditional D -> M is not just true, it is logically true. THAT is the principle of explosion and it does NOT imply that M is true. — TonesInDeepFreeze

Yet when D is a contradiction we know that D is not true.

If D is a contradiction of the explicit form P & ~P (or any purely sentential form or even monadic form), then we can mechanically verify that D is a contradiction and thus not true.

But if any of the sentences in the set are quantified, then the set might be inconsistent (yielding a contradiction) though we don't know it's inconsistent, and a given system might not provide a mechanical means to verify whether a given set of sentences is inconsistent or not.

Moreover, the principle of explosion regards entailment, no matter what we happen to know about the truth, falsehood or inconsistency of the sentences. The principle of entailment, even in ancient form, concerns the impossibility of states of affairs whatever our knowledge or lack of knowledge about those states of affairs.

Getting back to the poster slipping from the context of contradiction to falsehood: Yes, all contradictions are falsehoods. But not all falsehoods are contradictions. The point here is that mere falsehood is not what's involved in the principle of explosion.

When the poster tries to justify the slip by noting that all contradictions are falsehoods, he commits the illogic of getting the matter backwards: It's not a matter of all contradictions being falsehoods (which is true) but rather it's that the converse does not hold. The poster can't reason himself out of the proverbial paper bag.

Getting back to the poster slipping from the context of contradiction to falsehood: Yes, all contradictions are falsehoods. But not all falsehoods are contradictions. The point here is that mere falsehood is not what's involved in the principle of explosion. — TonesInDeepFreeze

(A & ~A) proves B is the POE
(A & ~A) proves FALSE is the actual correct inference
Two aspects of the same case.

The poster got it backwards again!

It's not a matter of the conclusion being false but rather that the poster previously tried to slip the discussion from the inconsistency of the premise to the falsehood of the premise.

But to address the latest post anyway:

Let C be a sentential letter and, for some sentence P, we define: C <-> (P & ~P).

Then for any sentence Q:

C |- Q

But since we don't have C, so we don't have Q.

And of course, in any model C is false, while depending on Q, in a given model, Q could be true or false.

It's not a matter of the conclusion being false but rather that the poster previously tried to slip the discussion from the inconsistency of the premise to the falsehood of the premise. — TonesInDeepFreeze

Maybe you are overwhelmed by too many details.
The Principle of Explosion claims this: (A & ~A) proves B

The poster still refuses to recognize that he tried to evade a key point by conflating contradiction and falsehood. That is not a mere detail, but it is central to the matter being discussed.

Meanwhile, yes of course as I have agreed at least a dozen times by now and as in the very post he just now responded to (!):

If C is any contradiction and Q is any sentence, then:

C |- Q

That is a basic result in sentential logic, known to anyone who has studied the subject.

But the poster just keeps posting it over and over though no one disagrees with it.

He might as well say, "2+2 = 4. Ha! Take that!"

Maybe this.
Let CNC stand for "The cat is not a cat," intended here as a false proposition.
Let MGC stand for, "The moon is made of green cheese," also a false proposition.
Let K stand for the implication, (CNC => MGC).
According to the rules, K is true. Period.

But it is a mistake displaying the greatest ignorance to suppose that K in any way proves either CNC or MGC separately.

More to be said but useless to say it if you do not grasp at least this.

If C is any contradiction and Q is any sentence, then:

C |- Q

That is a basic result in sentential logic, known to anyone who has studied the subject. — TonesInDeepFreeze

OK we finally have agreement on one point and I am exhausted that it took this long.
I am not going to bother to extend beyond this point with you because it seems to me that
you may be trying as hard as possible to make sure to avoid any honest dialogue.

Let CNC stand for "The cat is not a cat," intended here as a false proposition.
Let MGC stand for, "The moon is made of green cheese," also a false proposition.
Let K stand for the implication, (CNC => MGC).
According to the rules, K is true. Period. — tim wood

Yet when we make sure to NOT IGNORE the semantics underlying the sentential logic that

That is a basic result in sentential logic, known to anyone who has studied the subject. — TonesInDeepFreeze

said is correct then we understand that there is no semantic connection between
{a cat is not a cat} and {the Moon is made from green cheese} thus {the Moon is made from green cheese} is not entailed by the semantic meaning of {a cat is not a cat}.

We can encode the same thing as a syllogism and see the same thing yet I will not bother with that degree of detail while you two seem to insist on being as disagreeable as possible.

The poster says he's exhausted by the time it took to establish that a contradiction implies any sentence. He could have saved himself that exhaustion by simply reading what I had posted many posts ago. What is exhausted is his own attention span that exhausts itself in less than a minute.

As to honest dialogue, there is no point of dishonesty that the poster can point to in anything I've written, while meanwhile, only a few posts ago, the poster tried to evade a key point by dishonestly conflating contradiction with falsehood, as he continues to dishonestly evade that point.

As to the claim that I am as disagreeable as possible, there is so much confused disinformation written by the poster that there is indeed a great amount to disagree with.

/

The poster cites "semantic connection". That is not a defined term. However, the semantics are clear, as I have mentioned over and over but the poster refuses to recognize:

The notion of semantic entailment is:

A set of premises entails a conclusion if and only if there are no circumstances in which all the premises are true and the conclusion is false.

If the premises include a contradiction, then there are no circumstances in which all the premises are true and the conclusion is false. Therefore, if the premises include a contradiction, then those premises entail any conclusion.

By the way, mutatis mutandis it works the other way too: If the conclusion is logically true, then that conclusion is entailed by any set of premises.

Little doubt the poster still does not understand these points. He was too exhausted to understand them decades ago when he began not learning even the basics of this subject.

On the other hand, there is an alternative approach to logic call 'relevance logic' that does formalize the notion of the content of sentences and bases a different notion of entailment in that regard. And the advantage of that work, in contradistinction with the confused, ignorant and intractable handwaving of cranks is that the work is (as far as I know) rigorous and it is grounded in clear understanding of the subject - both classical and alternative. Some people prefer such approaches as relevance logic though they are more complicated than plain classical logic. But the existence of alternative approaches does not refute that my own reports of classical logic have been correct. And the takeaway here is that study of alternative logics requires prior understanding of the most basic logic that the alternatives include in some parts, reject in some parts and extend in certain ways. We need to be correct in what we say about classical logic if we are to properly critique it or propose a supposedly remedial alternative to it.

The poster cites "semantic connection". That is not a defined term. However, the semantics are clear, as I have mentioned over and over but the poster refuses to recognize: — TonesInDeepFreeze

What is there about the semantic meaning of {cats are not cats} that shows that {the Moon is made of green cheese} ???

The poster asks a question anew. He should read the post to which he is replying.

But I'll say it again in yet different terms:

'shows' in this context does not mean that there is contentual relationship but merely that there are no circumstances in which all the premises are true and the conclusion is false. For an alternative, one would study relevance logics.

The poster's earlier claim that started this part of the discussion was that classical logic is not truth preserving. But it is truth preserving since there are no permitted inferences from true premises to false conclusions. That is the case no matter that classical logic is not a relevance logic.

/

Still interested to know the basis for the claim that most philosophers reject the analytic-synthetic distinction.

Still interested whether the poster will ever admit that he improperly conflated contradiction with falsehood by overlooking that, while contradiction implies falsehood, falsehood does not imply contradiction.

Still interested whether the poster now understands that a footnote is in context of the passage to which it is a footnote.

And meanwhile, the poster still has not understood that he gets both Godel and Tarski exactly backwards. Most specifically that Godel does not claim that the system proves that the there is a proof in the system of the Godel-sentence, but instead proves that the system does not prove it; and Tarski does not use the liar sentence as a premise in any proof (as "this sentence is not provable" is crucially different from "this sentenced is false") but instead proves that the liar sentence cannot be formed in the language.

How to explain cranks? They take a position that they have a system for mathematics that is correct to the exclusion of mathematicians who are terribly wrong. In order to make an impression with that position, the crank postures that he dismantles the work of the mathematicians. But in order to carry out that dismantling, the crank must get the mathematics quite wrong. Not only is the crank's supposed system vague, impressionistic, illogical (to the point of being incoherent) and uninformed, but the crank's remarks about the work of mathematicians are woefully ignorant, confused and disinformational while self-fortified against being corrected on virtually any point ranging from fundamental to incidental, no matter how utterly clear it is that the corrections are sound. And sometimes this goes on for literally decades of spammed repetitions. The only explanation I can think of is that the crank's need is not to understand mathematics or even well conceived alternative mathematics, but rather the crank has a deep need to be taken to be an exceptional, remarkable person who single-handedly has debunked the work of mathematicians. The crank seems to want to live out a kind of hero fantasy and "triumph over conformity" fantasy no matter the facts or logic about the subject, no matter that thereby he makes an abysmal fool of himself though, of course, not in his own hopelessly blinkered mind.

The poster asks a question anew. He should read the post to which he is replying.

The poster asks a question anew. He should read the post to which he is replying. — TonesInDeepFreeze

You have proven to be overwhelmed by the detail of the original thread so I simplified it.
I didn't dumb it down you are very smart. I simplified it so rejecting out-of-hand looks foolish.

I am not overwhelmed by the details here; I am addressing them and contributing them. The poster though ignores not just the details but the most basic aspects of this subject thus making sure that he is not overwhelmed by facts and logic.

Again, the answer begins in the post to which the poster was replying and then elaborated on in my next post. In hopes that it is not ignored yet again:

'shows' in this context does not mean that there is contentual relationship but merely that there are no circumstances in which all the premises are true and the conclusion is false. For an alternative, one would study relevance logics.

The poster's earlier claim that started this part of the discussion was that classical logic is not truth preserving. But it is truth preserving since there are no permitted inferences from true premises to false conclusions. That is the case no matter that classical logic is not a relevance logic.

@PL Olcott - if it helps, @TonesInDeepFreeze is giving you answers that you could find in logic courses at uni. To the best of my ability - which is more limited than Tones' - I think they are accurately portraying the canon to you.

Also to the best of my ability, it doesn't look as if you understand what Tones is writing. "Epistemological antimony" isn't a technical term in any of the proofs you've criticised.

It would be worth reading SEP's articles on the various diagonalisation results. Godel's and Tarski's and Turing's. Moreover, even something like first order logic isn't decidable.

It looks very much like a case where how you're using the words, PL, is not how the literature is using them. And in that regard your ideas - as criticisms of the literature - are off target.

Thank you, fdrake, for those useful words.

Yes, 'epistemological antinomy' is not mentioned as a formal mathematical rubric in Godel's famous paper. The poster abysmally fails to read - fails even to recognize that it has been pointed out to him - that Godel confines the significance to that of analogy not of formal application.

It looks very much like a case where how you're using the words, PL, is not how the literature is using them. And in that regard your ideas - as criticisms of the literature - are off target. — fdrake

I went back to my original post and still stand firmly behind it. The terms that I use
are relevent to truth-maker maximalism yet probably establish brand new ideas in
this field that have no preexisting terms.

It is probably very very difficult for people that know the conventional notions of
Decision Problem Undecidability to have any idea how to apply the words
of this original post to the subject of Decision Problem Undecidability.

To anchor my ideas in Gödel's 1931 Incompleteness I would say that:
G and ~G are not linked by any sequence of truth preserving operations from the
axioms of PA thus are untrue in PA.

G is linked by a sequence of truth preserving operations from the axioms of
meta-math thus are untrue in meta-math.

To people very accustomed to Gödel numbers and diagonalization this may see very strange.
It Is however, the same idea that Wittgenstein had in mind and I know this because I derived his exact same idea about a year before I ever heard of him. https://www.liarparadox.org/Wittgenstein.pdf

Most people very familiar with conventional notions mistakenly conflate boiling ideas down to their bare essence as a simplistic view of these same ideas. That is what everyone here has done.