Is every statement true?

Consider the following argument:

If a statement is true, then it is a member of the set of true statements.

If a statement is false, then it implies a contradiction (as we find in a proof by contradiction). Since anything follows from a contradiction, it follows that the statement is true. Thus the statement is a member of the set of true statements.

Since a statement is true or false, all statements therefore belong in the set of true statements. All statements are true, with the set of false statements being a subset of the set of true statements. A statement thus is either true and true only, or both true and false.

Does this mean that all statements are true?

No, it doesn’t. A false statement doesn’t imply a contradiction. You’d need two statements to have a contradiction.

In a proof by contradiction, we assume a statement, that is actually false although we don’t know that yet at first, and derive a contradiction. The fact that a false statement allows us to derive a contradiction shows that a false statement implies a contradiction. Therefore since anything follows from a contradiction, the statement assumed is true and the statement assumed is false. Thus, it’s true, by conjunction elimination on the previous statement.

Not really, a false statement implies or states something inaccurate. It may cause a contradiction with other knowledge such as one’s own observations but is not in itself a contradiction.

But as far as I’m concerned, nothing is true.

What in the world? Whattttt? What? What are you talking about?
Anything follows from a contradiction? What does that mean? And how does it imply that a contradictory statement is true?

Nigga’s straight-up bonkers.

Firstly, FJ, anything does follow from a contradiction, which is why they are bad. Secondly, please calm the invective down. Forthwith.

browser - I’m trying to make this simple for you - in a proof, there is more than one statement. Think about this. A false statement, by itself, does not imply a contradiction. You can look this stuff up pretty easily.

Firstly, Faust, I asked what that means, and secondly I asked how it implies a contradictory statement is true.
You answered neither, just repeated the statement.

FJ - I am referring to “Nigga’s straight-up bonkers.”

That’s not acceptable.

“Firstly, Faust, I asked what that means…”

Simply put, if A is both true and false, at the same time and in the same place, then truth values, as used in classical logic, have no meaning, so anything goes.

This does not, as I stated, imply that a contradictory statement is true, because there is no such thing as a (simple) contradictory statement to begin with. If I say "I am entirely black and entirely white, I have made two statements. The bare fact of the falsity of a simple statement is not relevant here.

browser is confusing a reductio argument with a mere statement.

Again, I’m trying to keep this simple.

Ah, so if you assume a contradictory statement is true then anything follows. That’s what is meant. I needed that to be clarified to understand where he went wrong in his reasoning.

can be traslated to:
“Since anything follows [if you assume a contradiction is true]…” ← we don’t assume that, so everything he said after that is based on an assumption that we’re not making.

:wink:

No. Anything follows from a contradiction. I have explained the difference between your formulation and mine, above. It may seem like a minor discrepancy, but it is this very discrepancy that has led browser astray.

Not exactly. A true contradiction - a contradiction that is true - isn’t part of classical logic. In classical logic, a contradiction occurs when two propositions are logically “incompatible.” There is no truth value, per se, for a contradiction itself. Remember that contradictions are just relations between claims, so we can, of course, find contradictions.

You’re being really vague and confusing my pants off bro. Be clear.

“Simply put, if A is both true and false, at the same time and in the same place, then truth values, as used in classical logic, have no meaning, so anything goes.”
This is what you mean when you say anything follows from a contradiction, right?

It starts with “if A is both true and false,” it starts with an “if”. That’s key. That “if,” though…for his argument to work, we have to assume that that “if” is valid, that A is both true and false. If A is NOT both true and false simultaneously, then everything that you said after doesn’t stand. The “anything goes” doesn’t stand. You see? That’s what I’m saying.

His whole argument rests on contradiction. It’s circular. We have to assume contradictory statements are true to get to his conclusion that…lol…contradictory statements are true.

I am actually not being vague.

““Simply put, if A is both true and false, at the same time and in the same place, then truth values, as used in classical logic, have no meaning, so anything goes.”
This is what you mean when you say anything follows from a contradiction, right?”

Yes.

“It starts with “if A is both true and false,” it starts with an “if”. That’s key. That “if,” though…for his argument to work, we have to assume that that “if” is valid, that A is both true and false. If A is NOT both true and false simultaneously, then everything that you said after doesn’t stand. The “anything goes” doesn’t stand. You see? That’s what I’m saying.”

Yeah…the problem is that neither of you know the vocabulary of logic, which is what is causing the confusion. I don’t know what it means for an “if” to be “valid”. I think you’re trying to say that the antecedent is true in “If A is both true and false, then every statement is true”, but I’m not sure. What actually happens is this - if A can be both true and false, then truth values mean nothing - every statement is true, and every statement is false, as you wish.

What I think browser may also be doing is trying to make a claim about (Googling…) dialetheism, which is an entire ‘nother can o’ worms.

Let me try another approach.
I’m going to replace “Anything follows from a contradiction” with what you said that actually means, since that’s such a vague phrase. GOD I HATE THAT PHRASE

You see what I did there? I translated his statement into something that I can actually make sense of. That’s important: when you cloud your ideas in vague phrases it becomes hard for anyone else to pick out what’s wrong with it. Now that I translated it, via Faust’s help, to something less vague, it’s quite simple to see what’s wrong with it. “If A is both true and false…” ← but it’s not…so that’s it. That’s where it ends. The proof fails right there.

Anything goes if A is both true and false, sure, but why do we assume that A is both true and false? We don’t.

You see what I’m saying? We have to assume contradictions can be true to get to his conclusion that contradictions are true. Circular. That’s what I’m getting at.

I think it’s something more along the lines of drawing a logical conclusion, or proposition, from a contradiction. If something is asserted as both true and not true, it can become a ‘wild card’, so to speak, in any logical conclusion drawn from it.

In other words something asserted as both true and false says nothing because anything can be said about it, regardless if the statement is true, false, or anywhere in between.

And Faust is right. There must conflicting statements [ex. something is both true and false] in order for a contradiction to take place.

I think this may be a source of confusion as well. The statement which you consider both “true and false” is just false. The only truth to it is that it’s false. But that is a redundant statement about the contradiction itself.

To say a claim is false because it is contradictory is the same as saying it is true that the claim is false. However, the former addresses the claim, whereas the latter addresses the contradiction.

Ahh, I see, FJ. It’s an emotional thing. Okay.

“If a statement is true, then it is a member of the set of true statements.”

I’m fine with that.

“If a statement is false, then it implies a contradiction (as we find in a proof by contradiction).”

This is incorrect. A false statement does not imply anything. It does not imply a contradiction. There is no logical contradiction is this: “I am handsome.” Here’s why. To say that there is a contradiction confuses the technical and everyday (or extended) usages of the word “contradiction”. Logic is about statements only. While the statement “I am handsome” contradicts observation, that makes it only untrue, and not a logical contradiction, but a “contradiction” between the claim and the empirical observation. It’s not a logical contradiction. A logical contradiction is a condition that is logically impossible. It could be true that i am handsome - it just doesn’t happen to be true.

Correct, but a statement never is “in the same place” as an observation. Statements, once subject to verification, can turn out to be false, but a simple statement is never contradictory - for it would need another statement to contradict. Now, paradoxes, which a bit different, can be apparently self-contradictory, but “I am handsome” is not paradoxical. I hope we can leave paradoxes aside, as this would only cloud the waters more.

Again, I don’t think browser made a bad assumption. i think he is confounding two sense of the word “contradiction”, as I stated above.

"

yeah, i think you’re muddying this convo up on purpose.

I assure you, I am not. Logic neither begins nor ends with truth. Logic is concerned only with validity. We can, in a reductio argument, demonstrate a contradiction, which is, of course, the purpose of a reductio argument. A reductio argument is not concerned with truth at all, but with finding that contradiction. Within this context, the phrase “true contradiction” has no meaning. “Contradiction” has meaning.

The context is given by the OP. You’re saying the context is out of context…
I don’t think it’s useful to divorce logic completely from truth like that, because logic doesn’t just exist in a vacuum. It’s used to determine what’s true and what isn’t. Truth and logic are inextricably linked, and if they’re not then logic is pointless and self-referential (I don’t think it is, but you seem to, which makes it extra-weird that you insist on keeping this convo about logic in itself and not about truth, as that would make the whole conversation likewise pointless).

Mind you, I understand that a valid argument doesn’t necessarily provide a true conclusion, I’m not saying that logic is synonymous with truth, but I am saying that it’s really kinda pointless to talk about logic if logic can’t be used to determine what’s true.

What can i tell you? I can’t lie to Jesus, Flannel or otherwise. Logic does not determine what is true. It just doesn’t. You can look this up. It’s a tool, devised for a purpose, and finding out what is true is just not that purpose. This is why I nitpicked about the difference between truth and validity. The purpose of logic is to establish the validity of an argument. here’s a little snippet from Wiki:

“Logic (from the Greek λογική logikē)[1] is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies.”

Logic is not about truth, but about reasoning. I don’t make the news - I only report it.

Logic is self-referential, in that it is based upon axioms that are taken to be self-evident. I’m sorry to be the one to tell you. But I have studied logic for quite a while, and this is pretty much the straight dope.

It’s used to preserve truth, if there is truth to be preserved. It doesn’t determine truth. So there is a relationship between truth and logic - it’s just not the one you claim it to be.

I know this may all seem like nitpicking, but I have seen, over many years now, small mistakes, in nomenclature or in understanding, multiply to the point of nonsense, when people who do not know anything about logic get talking about it. It’s vital to get the basics down first, because logic is not a body of knowledge, it’s a method. I completely understand if you think I am obfuscating, or emphasising trivial elements of logic - many people here have. But you have to be precise about logic, or you get on a very slippery slope, very quickly.