Not all proofs use the law. Indeed, the law is not even usually one of the logical axioms. — TonesInDeepFreeze
(1) Trivial. If the law is an axiom then it also provable by the rule of putting an axiom on a line. — TonesInDeepFreeze
they must assume that it is certain implicitly — Amalac
otherwise it is not even possible to talk meaningfully — Amalac
Only if you assume that it is not also the case that the Law is also not provable by the rule of putting an axiom on a line — Amalac
the “proof” circular — Amalac
And the same applies for the rest of the “proofs”. — Amalac
A word is not the same thing as that to which the word refers.
So if Amalac, in the following sentence, means “the word Amalac” then I am not Amalac.
If it means “the person writing this right now”, then I am Amalac.
The only way you can assert that I am both a person and a word is through either denying the Law of Contradiction, or through the fallacy of equivocation. — Amalac
They prove it as a theorem. Of course, our motivation for the system would include proving it as a theorem. — TonesInDeepFreeze
Talk meaningfully" is a large and undefined rubric. — TonesInDeepFreeze
But no difference between Amalac and Amalac. — Banno
Even as Amalac is both a word and you. — Banno
The belief in the law of contradiction is a belief about things, not only about thoughts.
— Russell
His “only” implies that he holds that the belief in the Law of Contradiction is both about thoughts and about things. — Amalac
I think this because logic is about what we can say, and not about the way things are. — Banno
I think this because logic is about what we can say, and not about the way things are — Banno
I think logic is about what we can say, but also about the way things are — Amalac
If we found a situation in which there was an apparent contradiction, what we would do is to re-think how we set out contradictions. — Banno
How do you know that they also don't prove it as a theorem? — Amalac
By accepting the Law of Contradiction as an axiom? — Amalac
By definition, axioms are accepted as true without proof. — Amalac
It cannot be proven — Amalac
Why should a LNC sceptic accept that axiom in the first place? — Amalac
My point is that your proofs only work if someone accepts the Law as an axiom — Amalac
your proofs only work if someone accepts the Law as an axiom, if they don't, then your “proofs” are circular. — Amalac
Without assuming the Law as true without proof, for all I know you may have both asserted and denied everything you've said this far. — Amalac
Logic is about the rules of language. Language is about how the world is. — Banno
That's a very silly question. I don't know it, since it is not the case, and I can't know that which is not the case. Are you trolling me?
Or maybe you meant to type: How do you know that they also don't prove it is not a theorem? — TonesInDeepFreeze
That's one notion. But another definition of 'axiom' is purely syntactical. — TonesInDeepFreeze
The proofs can be mechanically audited whether the auditer knows of LNC as an axiom or not. Indeed, even for everyday reasoning, probably most people haven't even heard of LNC, especially the notion of it is an axiom. And that does not contradict that good reasoning (other than dialethistic) conforms to LNC and sometimes uses it - either as an explicit or implicit principle. — TonesInDeepFreeze
It is fine to have it as a logical axiom, since it is logically true. Sceptics should learn that it is logically true. — TonesInDeepFreeze
First, it is possible for one to assert and deny a proposition. And it is even ubiquitous that people assert propositions that are inconsistent with other propositions. So probably what you mean is that it is not possible to be correct while both asserting and denying a proposition. Then your question seems to be how do we know that contradictions are not the case. But the question of how we know things is different from the question of what axioms we choose. We may know that a proposition is true by reasoning from different axioms that each yield the proposition as a theorem. It is not required that LNC be one of the axioms. If, as we ordinarily do, we require a system that is complete in the sense of proving all validities then it is only required that LNC at least be a theorem even if not an axiom. — TonesInDeepFreeze
What I was aiming at is that logic is the handmaid of what is the case. One of the things we do with language is that when it doesn't seem to show us what is the case we change what we are saying.
DO you agree with that? — Banno
forgot to add that other negation — Amalac
When I resolve the double negative, we get "It is the case that the law is provable by the rule of putting an axiom on a line", which is what I said in the first place. — TonesInDeepFreeze
what's this purely syntactical definition of “axiom” you speak of? — Amalac
But they must still accept it implicitly without proof (that's what I mean by “accept it as an axiom”). — Amalac
As Aristotle said: It is utterly impossible to prove everything. — Amalac
Someone who doubted that the Law of Contradiction was true, would not accept any proof that assumed, without proof, that the Law of Contradiction is true — Amalac
it is trivially true that if you have the Law as an axiom, then you can prove that the Law is true. — Amalac
if I have “God exists” as an axiom, I can prove that God exists. — Amalac
It is fine to have it as a logical axiom, since it is logically true. Sceptics should learn that it is logically true.
— TonesInDeepFreeze
That's just an assertion, the LNC sceptic will demand a proof for it. They want to learn why it's true. — Amalac
Could you please show me a proof of the Law of Contradiction that didn't have it as an axiom — Amalac
That is incorrect. To accept a proof does not require accepting the truth of the premises. — TonesInDeepFreeze
It only proves G is true if all the axioms used in the proof are true. So, since G is an axiom used in the proof, if it is false, then, though we have proved G, we have not proved that G is true. — TonesInDeepFreeze
No, it's not just an assertion. It's a theorem about propositional logic. And it is reducible, in a sense, to a theorem about Boolean algebra. And its proof is reducible to finitary operations, which are reducible to auditing the execution of an algorithm. So (heuristically speaking) we may say that at the root of the question is ability to audit the execution of an algorithm. Of course, it's hard to imagine such an ability in a person who was so delusional that they claimed to witness '0' and '1' written in the same space when only one of them was written in that space. But that is not the same as going all the way back up the chain of reductions I just described to say that LNC must be an axiom. — TonesInDeepFreeze
I recommend 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar. Within about a chapter you could assign yourself the easy exercise of deriving LNC in the natural deduction system there. — TonesInDeepFreeze
it's the one you are refering to here: — Amalac
take LNC as a given in order to derive RAA — Banno
'E!' is being used as a 1-place predicate symbol. — TonesInDeepFreeze
In S4 or S5, or a derivative therefrom, can an individual exist in every possible world without contradiction? — Banno
Bartricks's claim that LNC is true but contingent — Banno
'E!' is being used as a 1-place predicate symbol.
— TonesInDeepFreeze — Banno
one version fo my OP would be to ask if (x) ▢(E!(x)) - if there is some being that exists in all possible worlds. — Banno
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