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[Dozebôm Lolumzalìs]

* disagreements about logic (my baby brother says that 1+1=11)

I disagree with this category. By its very definition logic is not disputable. Take your example, for instance. 1+1=11 is not an opinion; it is an utter and complete falsehood. The thing about mathematics is that everything is very black and white -- given the same fundamental set of axioms, everything is either true or false. If you can present a mathematical proof of a statement, that statement is completely and unconditionally true. The same arguments can be applied to logic (which in my opinion falls into the category of mathematics).

Disagreements about logic should reduce to an error in at least one person's thought, yes. I am trying to describe why two people might say "climate change is happening" and "climate change is not happening." One of them might have made an error, or they might have differing judgment despite common data, or - I have hitherto assumed that all participants are acting in good faith and wish to discover truth, and furthermore that they are all perfectly aware of their own reasoning. If this is not true, then there are many more possible reasons for disagreement.

1+1=11 is a possible valid statement, under the laws of language. It corresponds to an invalid statement under the laws of mathematics and the commonly-held axioms, but it is possible to disagree with 1+1=2. Everyone will just think you are stupid or insane, though.

", when preceded by itself in quotes, is unprovable.", when preceded by itself in quotes, is unprovable.

", when preceded by itself in quotes, proves itself.", when preceded by itself in quotes, proves itself.

(Not a spiral truth-untruth, but still on shaky grounds for believing it.)

[Dozebôm Lolumzalìs]

(confound it, sorry for the double-post)

Not true, actually. See Gödel's incompleteness theorem. You can have statements that are true that cannot be proven to be true. Toss in the negative proof of the Entscheidungsproblem and you have that there is no way to actually prove that a statement cannot be proven given a particular set of axioms.

So, it is true that any proposition in mathematical logic can be true or false. It is not true that there necessarily exists a proof of the truth or falsehood of a statement that is true or false, or whether there's a way to tell whether such a proof exists in the first place. Proof implies truth. Truth does not imply the existence of a proof.

(Ispil, does your signature imply that philosophy can turn you to evil? )

Wait, any proposition? Even those that have been proven?

[scriver]

No, for example, you are not allowed to proposition strangers.

[Egan_BW]

What? Nobody told me this rule.

[scriver]

Well that's why you're in the jailhouse now

[Egan_BW]

Wait, you mean I was supposed to stay there? I knocked down the wall and walked out first thing.
Dammit, why doesn't anyone tell me these silly social contracts?

[Dozebôm Lolumzalìs]

(confound it, sorry for the double-post)

Not true, actually. See Gödel's incompleteness theorem. You can have statements that are true that cannot be proven to be true. Toss in the negative proof of the Entscheidungsproblem and you have that there is no way to actually prove that a statement cannot be proven given a particular set of axioms.

So, it is true that any proposition in mathematical logic can be true or false. It is not true that there necessarily exists a proof of the truth or falsehood of a statement that is true or false, or whether there's a way to tell whether such a proof exists in the first place. Proof implies truth. Truth does not imply the existence of a proof.

(Ispil, does your signature imply that philosophy can turn you to evil? )

Wait, any proposition? Even those that have been proven?

I mean the definition of a proposition- a statement which can be either true or false. They are not to be confused with facts, which cannot be false. They are truth-bearers.

Ah, I was going by the layperson's definition ("a statement or assertion that expresses a judgment or opinion"), but you meant the more specific philosophical term, of which I was not aware. Okay.

So if I say something mathematical and it has been proved, it is a fact. Otherwise, it is a proposition. Is that the distinction? And does that mean that propositions become facts after being proven?

Wait, but... You say that "there is no way to actually prove that a statement cannot be proven given a particular set of axioms", but what if I have proven that "'1+1=2' is true" is true? Doesn't that mean that "'1+1=2' is false" is false and thus disproven, then?

[bloop_bleep]

(confound it, sorry for the double-post)

Not true, actually. See Gödel's incompleteness theorem. You can have statements that are true that cannot be proven to be true. Toss in the negative proof of the Entscheidungsproblem and you have that there is no way to actually prove that a statement cannot be proven given a particular set of axioms.

So, it is true that any proposition in mathematical logic can be true or false. It is not true that there necessarily exists a proof of the truth or falsehood of a statement that is true or false, or whether there's a way to tell whether such a proof exists in the first place. Proof implies truth. Truth does not imply the existence of a proof.

(Ispil, does your signature imply that philosophy can turn you to evil? )

Wait, any proposition? Even those that have been proven?

I mean the definition of a proposition- a statement which can be either true or false. They are not to be confused with facts, which cannot be false. They are truth-bearers.

Ah, I was going by the layperson's definition ("a statement or assertion that expresses a judgment or opinion"), but you meant the more specific philosophical term, of which I was not aware. Okay.

So if I say something mathematical and it has been proved, it is a fact. Otherwise, it is a proposition. Is that the distinction? And does that mean that propositions become facts after being proven?

Wait, but... You say that "there is no way to actually prove that a statement cannot be proven given a particular set of axioms", but what if I have proven that "'1+1=2' is true" is true? Doesn't that mean that "'1+1=2' is false" is false and thus disproven, then?

I think he made a typo (he meant "that statement" not "that a statement") and was only referencing the Entscheidungsproblem.

[Zangi]

Is saying something is a chair an opinion?

In the sense that I can put the full weight of my arse on it, without discomfort on my part, sure.  I can opinionate that just about anything, including another human, could be a opinioned as a chair.

[scriver]

 I think we will find if we put that theory to test that non-human chairs will be a lot less opiniated than human ones

[Max™]

Heck with chairs: is opinion?

Is is?

[Egan_BW]

ia ia chairthulo fthagn

[Max™]

T-t-t-team Up Combo! Flawless Victory!

[Dozebôm Lolumzalìs]

(confound it, sorry for the double-post)

Not true, actually. See Gödel's incompleteness theorem. You can have statements that are true that cannot be proven to be true. Toss in the negative proof of the Entscheidungsproblem and you have that there is no way to actually prove that a statement cannot be proven given a particular set of axioms.

So, it is true that any proposition in mathematical logic can be true or false. It is not true that there necessarily exists a proof of the truth or falsehood of a statement that is true or false, or whether there's a way to tell whether such a proof exists in the first place. Proof implies truth. Truth does not imply the existence of a proof.

(Ispil, does your signature imply that philosophy can turn you to evil? )

Wait, any proposition? Even those that have been proven?

I mean the definition of a proposition- a statement which can be either true or false. They are not to be confused with facts, which cannot be false. They are truth-bearers.

Ah, I was going by the layperson's definition ("a statement or assertion that expresses a judgment or opinion"), but you meant the more specific philosophical term, of which I was not aware. Okay.

So if I say something mathematical and it has been proved, it is a fact. Otherwise, it is a proposition. Is that the distinction? And does that mean that propositions become facts after being proven?

Wait, but... You say that "there is no way to actually prove that a statement cannot be proven given a particular set of axioms", but what if I have proven that "'1+1=2' is true" is true? Doesn't that mean that "'1+1=2' is false" is false and thus disproven, then?

Given a proposition, there is no way to prove that a proof or disproof exists without actually finding said proof or disproof. Think about it like the halting problem (since it's just a variation on it). If a proof does exist for a proposition, that makes the proposition true for that given set of axioms. It does not necessarily make it fact. If you want, you can take a look into proof theory to get an idea of what a proof entails in a math-theoretic sense.

There's no distinction between a "mathematical proposition" and a proposition. All that's required is that a proposition is a closed sentence. Closed sentence being the mathematical logic definition of a sentence; that is, a boolean-expressible formula with no free variables. As contrasted with an open sentence, which contains free variables.

That just seems a bit tautological. Isn't "true under these axioms" the same as "proved within these axioms", by definition?

[Max™]

That just seems a bit tautological. Isn't "true under these axioms" the same as "proved within these axioms", by definition?

Not really, but that road leads you off into the dark underbrush of the Reals.