Interesting question! I think that you seem to think of conjunction (AND, ∧) as akin to addition (PLUS, +) and of logical negation (NOT, ¬) as akin to number-negation (sign-flipping, NEGATIVE, -). If that assumption were true, saying a contradiction would indeed be like saying like nothing at all. But your assumption is flawed, I think. Unlike addition, conjunction isn’t reversible; if you have a proposition (X AND A) and want to find what the orspringly (original) proposition X was, just knowing what A is is not always enough to reconstruct X. — Tristan L
The “domain” of the negation NOT(A) of a proposition A is by definition everything that lies outside the “domain” of A, so to speak, so by definition, there is no overlap between the two. — Tristan L
I don't recall making the claim that conjunction is like mathematical addition but I remember some Boolean logic from high school which makes that claim. — TheMadFool
As for negation being a sign-flipping operation, I admit that's how I read it. — TheMadFool
You're basically talking about complements of sets, right? — TheMadFool
However, I mean this only against the backdrop of sentential logic. — TheMadFool
but E = "God exists" and ~E = "God doesn't exist" are not categorical statements. — TheMadFool
I think that maybe you're confusing the law of non-contradiction with the principle of explosion.Why is the official (logical) explanation for why contradictions are prohibited (ex falso quodlibet) different? — TheMadFool
Actually, conjunction is a bit like multiplication, whereas it is exclusive disjunction (EITHER-OR, XOR) which is a bit like addition. And like multiplication, conjunction isn’t reversible; if you multiply by zero, you always get zero, and if you AND with a false proposition, you always get a false proposition. — Tristan L
And you’re right.
But since conjunction isn’t like addition (see above), you can’t conjoin with the negation of a proposition to undo conjoining that proposition. The logical operations that work together like addition and sign-flipping are XOR and NOT, not AND and NOT — Tristan L
o use your metaphor, stating a contradiction isn’t like first writing “God exists” in the space and then erasing it, but rather like first writing “God exists” in the space and then writing “God doesn’t exist” over it, which makes a mess. — Tristan L
Logicians refer to this as ‘anything follows from a falsehood’, which is the principle of explosion as you mentioned, but rarely explain why this is the case. — Harry Hindu
But they are propositions about categories, or rather, universals (broadthings) more generally. Specifically, they are propositions about Godhood: E is the proposition that there is an x with Godhood, that is, the proposition that Godhood has instantiatedness, and ~E is the proposition that there is no x with Godhood, that is, the proposition that Godhood doesn’t have instantiatedness. — Tristan L
No it can't. It has to logically follow, or be causally related with, the prior statement or its a non sequitur. I did mention this the post you replied to but apparently did not read.(this is the important step because A can be any proposition at all) — TheMadFool
No it can't. It has to logically follow, or be causally related with, the prior statement or its a non sequitur. I did mention this the post you replied to but apparently did not read. — Harry Hindu
"As for the obstinate, he must be plunged into fire, since fire and non-fire are identical. Let him be beaten, since suffering and not suffering are the same. Let him be deprived of food and drink, since eating and drinking are identical to abstaining.”
-The philosopher and polymath Avicenna — Harry Hindu
Yes. I did. Search for the phrase, "non sequitur" on this page. The principle of explosion IS a non sequitur error.I guess everyone has an opinion on the matter but what's your beef with the principle of explosion? Any flaws? You don't mention any — TheMadFool
Then how are you defining, "contradiction"?I love this quote but, on analysis, it, nowhere in its poetic fervor, states a contradiction. — TheMadFool
never really making a point, — TheMadFool
Is the principle of explosion self-evident in the way the principle of non-contradiction is self-evident?To Aristotle, the law of non-contradiction was not only self-evident, it was the foundation of all other self-evident truths, since without it we wouldn’t be able to demarcate one idea from another, or in fact positively assert anything about anything – making rational discourse impossible. — Harry Hindu
Yes. I did. Search for the phrase, "non sequitur" on this page. The principle of explosion IS a non sequitur error. — Harry Hindu
Then how are you defining, "contradiction"? — Harry Hindu
Is the principle of explosion self-evident in the way the principle of non-contradiction is self-evident? — Harry Hindu
(my boldening)At they very least to state ~E = "god doesn't exist" requires one to erase E = "god exists" like so: (god exists) and then write (god doesn't exist). — TheMadFool
there's got to be a sense in which ~p is the opposite of p — TheMadFool
otherwise, to continue with my analogy of blank spaces E = "god exists" and ~E = "god doesn't exist" would simply occupy two different blank spaces and it would be completely ok to do so. — TheMadFool
I suppose, in [...] can't coexist. — TheMadFool
As an attempt to find a common ground between us, I'd like to point out that while I accept that a contradiction is like overwriting a proposition with its negation ("makes a mess"), we should note that this is because the proposition concerned had/has to be erased before the negation could be written down. :chin: — TheMadFool
"As for the obstinate, he must be plunged into fire, since fire and non-fire are identical. Let him be beaten, since suffering and not suffering are the same. Let him be deprived of food and drink, since eating and drinking are identical to abstaining.”
-The philosopher and polymath Avicenna — Harry Hindu
Why do you keep moving the goal posts? I explained it using the way you expressed it in your OP. I already pointed out that A cannot be any proposition under the sun because it has to logically follow. A has to be logically connected to P, and it isn't. You say it is, but how? Do you even know what a non sequitur is? It is defined as a deductive argument that is invalid, therefore you are not adequately applying all the 3. Natural deduction rules to the principle of explosion. Basically, the principle of explosion is a lazy attempt to be logical.Explain it to me with the argument I made:
1. P & ~P.......assume contradictions allowed
2. P............1 Simp
3. P v A......2 Add [A being any proposition under the sun]
4. ~ P.........1 Simp
5. A..........3, 4 DS
Three important facets to the logic above:
1. The propositions themselves
2. The logical connectives (&, v)
3. Natural deduction rules
Have I missed anything?
Explain the non sequitur using one or more of the above. — TheMadFool
p & ~p = Something is something & Something is not that something — TheMadFool
Try thinking of something and it's contradiction in the same moment. That is different than trying to say a contradiction in the same moment, which is impossible. To say a contradiction means that you have to say one sentence and then another that contradicts it in the same moment. It is in saying it that you get the sense of time passing where something is added and then taken away. That isn't what a contradiction is. That is utterly different than thinking of a contradiction, which is done in the same moment with the same thing.It wasn't and thus this thread. By the way, how, in what sense is the law of noncontradiction self-evident? — TheMadFool
Your symbolism is not adequate at representing how the LNC is self-evident, because the symbols appear in different areas of space, not the same area of space, as explained by Aristotle. In order to observe the self-evidence of the LNC, you have to [try to] think of a contradiction, not say or write it.“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
Why do you keep moving the goal posts? I explained it using the way you expressed it in your OP. I already pointed out that A cannot be any proposition under the sun because it has to logically follow — Harry Hindu
Then I don't understand how you can say that the quote I provided doesn't have any contradictions in it. :roll: — Harry Hindu
Try thinking of something and it's contradiction in the same moment. That is different than trying to say a contradiction in the same moment, which is impossible. To say a contradiction means that you have to say one sentence and then another that contradicts it. That is utterly different than thinking of a contradiction, which is done in the same moment. Try thinking of a god that both exists and doesn't exist. Now, use your logical symbols to say the same thing. It takes time to write them out, and the symbol appear in different places than the symbol that they are contradicting. When thinking of a contradiction, you think of the existing and non-existing property in the same moment and in the same visual space - meaning the existing/non-existing god must appear in the same space at the same moment. Remember this quote of Aristotle's: — Harry Hindu
In order not to be wrong, you first have to disjoin E with the trivially true proposition (e.g. 0=0; for our goals, we can speak of the trivially true proposition) to get (E OR 0=0), and only then conjoin the resulting proposition (E OR 0=0) with ¬E to get (E OR 0=0) AND ¬E, which is equivalent to ¬E. — Tristan L
not saying anything is equivalent to saying something trivially true — Tristan L
Do you understand what Aristotle is saying? Take in what Aristotle is saying and then roll it around in your head and then get back to me with how you would paraphrase it.:You speak as if thought is different to speech. It is, quite obviously, but it can be said and it is true that speech is nothing but vocalized thought and thought is simply unvocalized speech. I'm curious though because, if what you say makes sense to you, your brain must work in a radically different manner than mine. Care to share. — TheMadFool
To represent a contradiction with words, you can only represent the opposing ideas separately on a screen or on paper with symbols stretched across space and time. Contradictions are opposing qualities in the same space at the same time. Try to say, "exists" and "not-exists" at the same moment. Do you see the problem now?“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” — Aristotle
Do you understand what Aristotle is saying? Take in what Aristotle is saying and then roll it around in your head and then get back to me with how you would paraphrase it.:
“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”
— Aristotle — Harry Hindu
Try to say, "exists" and "not-exists" at the same moment. Do you see the problem now? — Harry Hindu
While you can say a contradiction, you can't think a contradiction. — Harry Hindu
:ok: Tell me one thing...what is the meaning of trivially true? — TheMadFool
By the way (E v 0=0) & ~E isn't equivalent to ~E. Do a DeMorgan on it and you have (E & ~E) v (~E v 0=0) and you know the rest. — TheMadFool
Saying is not the same as not saying and nothing is not the same as true, trivial or otherwise. Do I have to go Avicenna on you? :smile: — TheMadFool
Try to say, "exists" and "not-exists" at the same moment. — Harry Hindu
It is impossible to think of opposing qualities in the same space at the same moment. If you can do that, then your brain must work in a radically different manner than mine. Care to share. — Harry Hindu
It means being true by the laws of logic and thereby true in a very strong, very necessary way. — Tristan L
Actually, the two are equivalent, and I think that you mean the Distributive Law rather than de Morgan (please correct me if I’m wrong):
(E ∨ 0=0) ∧ ¬E ≣ (E ∧ ¬E) ∨ (0=0 ∧ ¬E) ≣ (0=0 ∧ ¬E) ≣ ¬E
I belive that your second intance of the OR-operator should be an instance of the AND-operator. — Tristan L
Try to say “5 is odd” and “six is even” at the same moment. — Tristan L
And that's the reason why you refer to it as "trivially" true? Something's off. — TheMadFool
:up: It seems you've serendipitously discovered a law of thought viz. One moment, one thought! — TheMadFool
erasing the words "God exists" from the blank space and we return to:(..........), the blank space we started with. — TheMadFool
Contradictions, as they appear to me and as I've delineated above, seem to be simply the act of both affirming and denying a proposition - it basically returns the logical cursor back to its starting point — TheMadFool
Now, contradictions in classical logic (categorical, sentential and predicate logic) are prohibited - they're a big no-no - but, to my utter surprise, not for the reasons I outlined above but, as I've been led to believe, because allowing them makes it possible to prove every conceivable statement true: Principle Of Explosion/Ex Falso Quodlibet. — TheMadFool
Why is the official (logical) explanation for why contradictions are prohibited (ex falso quodlibet) different? — TheMadFool
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