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.

Far from being like saying nothing, saying a contradiction is making the strongest statement of all. Yes, contradictions are the strongest of all propositions, because the first part of a proposition lets one set of propositions follow, and the second part implies all the rest. This is the explosion which makes contradictions useless.

But far more than just leading to explosion, a contradiction is false by the very wist (nature and essence) of negation. 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. Stating a contradiction is basically asserting that something lies in the overlap, which, as we’ve said, is empty by definition. Hence, all contradictions are false.

Mark, however, that while the boundary between the two domains always cuts everyone in two who tries to stand with one leg in one domain and the second leg in the other domain, so to speak, it need not be fixed at a certain location. Rather, it can bounce around, so to speak – its location can be undetermined. The Law of Not-Contradiction LNC and the Law of the Excluded Middle (LEM) together are weaker than the Principle of Bivalence (PB).

To 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.

So do you now believe in God or don’t you? (just joking :wink:)

That’s at least how I see the matter.

My experience of logic is similar to yours, so I can only offer my thoughts.

The way I see it, logic is based on propositions, which are themselves expressions of a ‘logical’ position in relation to ‘reality’. Without such a proposition, there is no position. It isn’t a blank space, per se, but rather all possible reality. Logic doesn’t create something - all it does is propose a position among all possible reality. It’s a top-down reductionist methodology.

So a contradiction is a bit like an interaction of matter and anti-matter, or the use of imaginary numbers in mathematics. It doesn’t create anything except an imaginary perspective with infinite potentiality, from which we can relate to another position which might have appeared illogical from our own position.

Your philosophy is an enigma to me but as for your statement about how contradictions relate to possibility, all I can say is that once contradictions are allowed, anything, absolutely anything, goes. That's the point of ex falso quodlibet I believe.

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

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.

As for negation being a sign-flipping operation, I admit that's how I read it. Why would you say that's wrong? A few paragraphs below in your post you say this (very insightful I must add):

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

You're basically talking about complements of sets, right? Your reasoning is flawless insofar as categorical logic, with its categories and their respective complements, are concerned, but E = "God exists" and ~E = "God doesn't exist" are not categorical statements.

The rest of your post is no longer relevant. However, I mean this only against the backdrop of sentential logic. If you want to discuss categorical logic, no problem but I fear it'd be going off on a tangent as, I now realize, I seem to have been talking about sentential logic. Perhaps there's a right way to extend the discussion into categorical logic. Any ideas?

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

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.

As for negation being a sign-flipping operation, I admit that's how I read it. — TheMadFool

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.

You're basically talking about complements of sets, right? — TheMadFool

Well, I’m using the language of sets to metaphorically talk about propositions. Of course, sets are extensions of properties, so set-language is actually better suited to talking about properties than about propositions.

However, I mean this only against the backdrop of sentential logic. — TheMadFool

Right, and in sentential logic (witcraft) as in logic broadly, LNC follows directly from the definition of negation, or perhaps we could regard it as part of the definition of negation. If a proposition A isn’t the case, then well, it isn’t the case; if we have NOT(A), we can’t have A.

but E = "God exists" and ~E = "God doesn't exist" are not categorical statements. — TheMadFool

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.

Why is the official (logical) explanation for why contradictions are prohibited (ex falso quodlibet) different? — TheMadFool

I think that maybe you're confusing the law of non-contradiction with the principle of explosion.

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” (Metaphysics, IV). 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.

In the centuries that followed Aristotle, medieval logicians noticed something interesting: if they allowed themselves just one contradiction, they seemed to be able to arrive at any conclusion whatever. 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.

A non sequitur is a logical fallacy where the conclusion does not follow from the premises, so anything does not follow from a falsehood if you apply all applicable logical rules to some proposition. The conclusion is not about, or related in any way to the premise, so even if the premise were true, there is no guarantee that the conclusion will be true or false. Essentially the premise and conclusion would be talking past each other.

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

So, I got mixed up! Thanks for the clarification.

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

I feel I'm getting closer to seeing your point of view on the matter. If I understand you correctly, you're under the impression that my understanding of contradictions (p & ~p) is one that considers the conjunction of the negation of proposition with the original proposition to be an undo operation, which you think is wrong.

This makes sense to me but here's the catch - there's got to be a sense in which ~p is the opposite of p 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. For instance, take E = "god exists" and T = "2 is an even number". I could easily write them down in two different blank spaces as (E & T). Nothing's amiss in doing that - they're not making "opposite" claims. When it comes to E & ~E, there's this oppositeness we have to countenance. 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).

By the way I like the way you described contradictions within the context of my analogy:

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

:up:

I suppose, in my blank space analogy, it boils down to:

1. Propositions about a single entity (god, water, balls, whathaveyou) that are in the same sense are restricted to a single blank space

2. A proposition and its contradiction will, according to 1 above, have to be written in the same blank space but that's impossible - one has to go i.e. one of them will have to give up its seat in a manner of speaking for the other.

In fact, the whole idea of contradictions is basically that (above). Two contradictory propositions are mutually annihilating i.e. they can't coexist.

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:

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

If you want to know

1. P & ~P (assume contradictions allowed)
2. P..................................1 Simp
3. P v A..........................2 Add (this is the important step because A can be any proposition at all)
4. ~P...............................1 Simp
5. A................................3, 4, DS
QED!?

We can prove anything once we allow contradictions.

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

If you're going to take things that way then there's no such thing as individualness, everything becomes a category...something doesn't add up.

(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.

"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

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

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.

"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

I love this quote but, on analysis, it, nowhere in its poetic fervor, states a contradiction. All it does, in my humble opinion, is swing back and forth between a proposition and its negation, never really getting there, never really making a point, the point in fact. It seems to be more about negation if anything. That's, of course, just my opinion. Perhaps you can edify me. Thanks.

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

Yes. I did. Search for the phrase, "non sequitur" on this page. The principle of explosion IS a non sequitur error.

I love this quote but, on analysis, it, nowhere in its poetic fervor, states a contradiction. — TheMadFool

Then how are you defining, "contradiction"?

never really making a point, — TheMadFool

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

Is the principle of explosion self-evident in the way the principle of non-contradiction is self-evident?

Yes. I did. Search for the phrase, "non sequitur" on this page. The principle of explosion IS a non sequitur error. — Harry Hindu

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.

Then how are you defining, "contradiction"? — Harry Hindu

p & ~p = Something is something & Something is not that something

Is the principle of explosion self-evident in the way the principle of non-contradiction is self-evident? — Harry Hindu

It wasn't and thus this thread. By the way, how, in what sense is the law of noncontradiction self-evident?

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

(my boldening)

You express your right feeling for the truth of LNC in the words emboldened by me above. 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. The neutral element of conjuction is the trivially true proposition, so I think that not saying anything is equivalent to saying something trivially true, and erasing is equivalent to disjoining with the trivially true proposition, which yields the latter.

there's got to be a sense in which ~p is the opposite of p — TheMadFool

And indeed there is: (EITHER-OR)ing p with ~p yields the trivially false proposition, which is the neutral element of EITHER-OR.

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

Right, and so, we have what the previous paragraph says.

I suppose, in [...] can't coexist. — TheMadFool

Yes, I think so, or to put it in other words, conjoining a proposition with its negation is of course possible, but yields a necessarily false proposition, namely a contradiction.

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

Yes, exactly, see the first paragraph.

I don’t really want to go into the details there because it isn’t the subject of this thread, but I regard categories, properties, and other universals (broadthings) as themselves things, just as the particulars 1, 2, 3, sin, cos, tan, and so on. I believe that your particular example E, like all ∃-propositions, is a proposition not about something called “God”, but rather about the broadthing of Godhood, which is called “Godhood”; but that’s just a side-remark of mine. If you want a proposition involving no quantification, why don’t you take (9 < 11), for example?

"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

I can’t help but realize that this is of great relevance to my very first thread on this forum, Is negation the same as affirmation?.

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

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.

p & ~p = Something is something & Something is not that something — TheMadFool

Then I don't understand how you can say that the quote I provided doesn't have any contradictions in it. :roll:

It wasn't and thus this thread. By the way, how, in what sense is the law of noncontradiction self-evident? — 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.

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 symbols appear in different places than the symbols 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:

“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

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.

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

What does it mean, "...it has to logically follow."? Are you saying the natural deduction rules that appear in my argument are flawed? Which ones? Where?

Sorry but I didn't move the goal post. I erected it in the first place (recall it was me who provided you the argument). You're unwilling to accept the argument - an instance of ex falso quodlibet - and then proceeded to call it a non sequitur but didn't, obviously because you had better things to do, back it up with an argument of your own.

Then I don't understand how you can say that the quote I provided doesn't have any contradictions in it. :roll: — Harry Hindu

Avicenna probably intended it as such - to do an exposé on the absurdity of denying a contradiction - but, if it's not too much trouble, can you point out the exact location in the quote where a contradiction makes its, what I expect is a grand, entrance.

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

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.

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

:ok: Tell me one thing...what is the meaning of trivially true? 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.

not saying anything is equivalent to saying something trivially true — Tristan L

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:

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

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

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?

While you can say a contradiction, you can't think a contradiction. A contradiction is illogical because it doesn't represent how one thinks. 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.

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

I understand what Aristotle is saying but imagine I don't. How would you explain it to me? Please do.

Try to say, "exists" and "not-exists" at the same moment. Do you see the problem now? — Harry Hindu

Of course, I just gave it a go a moment ago but with a different kind of a contradiction. I tried imagining a square-circle. All that happened was the image in my mind flipped between a square or a circle bit never really getting to a square-circle. So? What's the point?

While you can say a contradiction, you can't think a contradiction. — Harry Hindu

Oh! Now it makes sense. Thanks. You mean to say I can speak/write, for instance, the contradiction "god exists" and "god doesn't exist" - I just did ( :grin: ) - but I can't think it. And...your point is?

:ok: Tell me one thing...what is the meaning of trivially true? — TheMadFool

It means being true by the laws of logic and thereby true in a very strong, very necessary way.

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

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.

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

Please don’t :fear:! Of course saying something isn’t the same as saying nothing, and I even have an original and I believe new solution of an important problem based on an idea which is in a way similar to this one. However, for our purposes, saying nothing can indeed be seen as equivalent to saying the trivial truth. That’s because in a way, saying several propositions is like saying their conjunction, and the empty conjunction is vacuously true.

Try to say, "exists" and "not-exists" at the same moment. — Harry Hindu

Try to say “5 is odd” and “six is even” at the same moment.

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

Well, when I was little, I thought to myself that almight includes the ability to make something be the case and not the case at the same time. This thought gave me a feeling of awe and wonder. Today, it’s still the same.

It means being true by the laws of logic and thereby true in a very strong, very necessary way. — Tristan L

And that's the reason why you refer to it as "trivially" true? Something's off.

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

Yes! Sorry, my brain was probably out on a break that day. :grin:

@Harry Hindu

Try to say “5 is odd” and “six is even” at the same moment. — Tristan L

:up: It seems you've serendipitously discovered a law of thought viz. One moment, one thought!

And that's the reason why you refer to it as "trivially" true? Something's off. — TheMadFool

Of course it isn’t necessarily trivial for us, but for logic (witcraft), any two logically equivalent propositions are basically the same, and since any logically true proposition is logically equivalent to a truly trivial proposition like 0=0 (one whose truth is obvious at once), the logically true proposition is also trivially true from the perspective of logic, isn’t it?

:up: It seems you've serendipitously discovered a law of thought viz. One moment, one thought! — TheMadFool

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.

erasing the words "God exists" from the blank space and we return to:(..........), the blank space we started with. — TheMadFool

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.
$\neg E = O$ which is saying X is odd, a positive statement.

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

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."

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

Contradictions in classical logic are allowed as conclusions. They are used to prove a proposition false. It's called recuctio ad absurdum... if you can show that a proposition is contradictory to itself, then it is absurd, and therefore the proposotion is not valid, it is false.

Why is the official (logical) explanation for why contradictions are prohibited (ex falso quodlibet) different? — TheMadFool

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.

A blank space on the page, not saying anything at all, isn't the same thing as denying everything that could possibly be written in there. The default truth-value of a proposition isn't false, but null. If it were false, then we would start out with an enormous mess of contradictions, because every proposition and its negation would be false: e.g. "God exists" would be false, and "God doesn't exist" would also be false.

If you want a visual analogy like that, you can take your blank page, and write all propositions that are true in green ink, and all propositions that are false in red ink. Since every proposition has a negation, for anything you write in green you'd also have to write its negation in red, and vice versa everything you write in red you'd have to write its negation in green. Erasing a green proposition isn't the same thing as writing a red one.