0=0 — Tristan L
Exactly. This observation has led me to the conclusion that that a genuine proof cannot consist of a chain of thoughts, for in that case, it would need the memory to be infallible. I also thought about this when writing mathematical proofs by asking: How do I know that the theorems which I proved on an earlier page and on which I now draw haven’t been tampered with by a hacker or a random glitch in my harddrive and thus rendered false? But that’s likely something for the knowledgelore (epistemology) underforum. — Tristan L
The LNC, as stated in Aristotle’s own words: “It is impossible for the same property to belong and not to belong at the same time to the same thing and in the same respect” — Harry Hindu
Negation can be a positive statement, not just a blank. If I say X is an integer and X is not even I am not saying nothing about X, I am saying it must be odd. Let E = even and O = odd.
¬E=O¬E=O which is saying X is odd, a positive statement — EnPassant
The starting point of a proof or of an argument is never a contradiction. And a contradicion is never a starting point.
I have never seen an argument to start, "Peter is not Peter." Or with "Given the time allotted to finish the project, we can finish the project if and only if we can't finish the project." — god must be atheist
They are different because you made several mistakes in the structuring of your original post. I pointed the mistakes out in my immediately preceding series of posts before this one. — god must be atheist
If I had asserted ~E first and then E, the same process is involved, only the propositions are now switched. — TheMadFool
Yes. See also Reductio Absurdum as a (dis)proof — EnPassant
I don't quite get you there...Mind elaborating a bit? — TheMadFool
Reductio Absurdum makes a conjecture and follows that conjecture through until a contradiction is reached. The negation of the conjecture can then make a positive statement. — EnPassant
I like to view a contradiction in terms of a blank space on a piece of paper on which you write down propositions. Imagine the blank space; (..........). I say, "God exists" and this space gets filled and becomes: (God exists). If I now say "God doesn't exist, this happens:(God exists) - basically you're, if you had an eraser at hand, erasing the words "God exists" from the blank space and we return to:(..........), the blank space we started with.
In essence then a contradiction is to say nothing at all (returning to the blank space after having written down a proposition and then erasing it). — TheMadFool
negation as cancellation — Alvin Capello
I'm not talking about contradictions in the context of arguments. I'm investigating the import of propositions and their negations, specifically that to state a propositions P, then to deny it, ~P, amounts to not stating P [return to the starting point]. — TheMadFool
The starting point of WHAT? It has to be a starting point of something or other, which you haven't named. I can't put words in your mouth. Please state the starting point and state also this is a starting point of what. Thanks. — god must be atheist
1.Start. Nothing as in no propositions have been stated
2. P stated
3. ~P stated.
4. P & ~P stated. P cancels ~P and ~P cancels P. Result = No proposition left. Back to 1. — TheMadFool
I'm not talking about contradictions in the context of arguments. I'm investigating the import of propositions and their negations, specifically that to state a propositions P, then to deny it, ~P, amounts to not stating P [return to the starting point]. — TheMadFool
Now that I realize it, P & ~P, because they cancel each other doesn't amount to a proposition. A contradiction essentially means the person who utters/writes it isn't saying anything at all. If so, any other proposition wouldn't be constrained by the necessity of consistency as there's no proposition in the first place to be consistent with. This is why anything follows from a contradiction keeping in mind that what doesn't follow from a certain proposition is predicated on a resulting inconsistency. — TheMadFool
This is not a trivial truth. It's an instance of a law of thought viz. The Law Of Identity [A = A]. It's basic, I agree, but that doesn't make it trivial. — TheMadFool
The words/concepts you employ must remain the same throughout a proof — TheMadFool
This is where The Law Of Identity, I mentioned above, comes into play. The words/concepts you employ must remain the same throughout a proof, A = A, a perfect example of which is 0 = 0. — TheMadFool
I recall having come to the conclusion that since a contradiction is defined in temporal terms:
The LNC, as stated in Aristotle’s own words: “It is impossible for the same property to belong and not to belong at the same time to the same thing and in the same respect” — Harry Hindu
arguments do have a temporal dimension and one of the ways of offsetting this is The Law of Identity [A = A] which you think is trivial. — TheMadFool
This issue of the temporal aspect of argumentation has been at the back of my mind for quite some time now. Thanks for reminding me of it. — TheMadFool
What, then, is an example of a trivial truth? — Tristan L
Meaning Constancy Assumption (MCA) that we need in a key way when arguing, not the Law of Identity, right? — Tristan L
A rose by [...] more to say. — TheMadFool
Thanks for the engaging conversation. — TheMadFool
It seems that what you mean by “Law of Identity” is not the same as what I mean by “Law of Identity”. Why I mean by that expression is the law that each thing is the selfsame as itself. For instance, the Sun is identical to itself. This has nothing to do with the meanings of words. What you mean by “Law of Identity” is indeed basically what I mean by “Meaning Constancy Assumption”. — Tristan L
This has nothing to do with the meanings of words. — Tristan L
What do you mean by "the sun is identical to itself"? — TheMadFool
Is there a danger/risk that it won't be identical to itself? — TheMadFool
This mistake, in your example of the sun, won't occur at the level of the sun itself - it's not that there's a possibility that sun will suddenly become not-sun. — TheMadFool
Where an error can occur [...] the equivocation fallacy. — TheMadFool
The Law Of Identity is designed to roadblock this fallacy by mandating the constancy of a term with respect to its referent in a given argument i.e. if a specific referent has been applied to a certain term, this term-referent pair must remain fixed throughout. — TheMadFool
This has nothing to do with the meanings of words. — Tristan L
:chin: — TheMadFool
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