Contradictions!

Reply to TheMadFool 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.

Reply to TheMadFool 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.

Reply to Possibility 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.


Quoting Tristan L

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.

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):

Quoting 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.

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?

Quoting TheMadFool

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.

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.

Quoting TheMadFool

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

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.

Quoting TheMadFool

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

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.

Quoting TheMadFool

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

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.

Quoting TheMadFool

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

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.

Quoting TheMadFool

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

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.

Quoting Tristan L

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.

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

Quoting 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

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: ([s]god exists[/s]) and then write (god doesn't exist).

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

Quoting 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.

: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:

Quoting Harry Hindu

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.

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.

Quoting Tristan L

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.

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.

Quoting TheMadFool

(this is the important step because A can be any proposition at all)

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

Quoting Harry Hindu

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.

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.

Quoting 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

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.

Quoting TheMadFool

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

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

Quoting TheMadFool

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

Then how are you defining, "contradiction"?

Quoting TheMadFool

never really making a point,

Quoting Harry Hindu

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.

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

Quoting Harry Hindu

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

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.

Quoting Harry Hindu

Then how are you defining, "contradiction"?

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

Quoting Harry Hindu

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

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

Quoting TheMadFool

At they very least to state ~E = "god doesn't exist" requires one to erase E = "god exists" like so: ([s]god exists[/s]) and then write (god doesn't exist).

(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.

Quoting TheMadFool

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

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

Quoting 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.

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

Quoting TheMadFool

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

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.

Quoting 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:

Yes, exactly, see the first paragraph.

Reply to TheMadFool 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?

Quoting 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

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?.

Quoting TheMadFool

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.

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.

Quoting TheMadFool

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

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

Quoting TheMadFool

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

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:
Quoting Harry Hindu

“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”

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.

Quoting 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

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.

Quoting Harry Hindu

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

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.

Quoting 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:

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.

Quoting Tristan L

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.

: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.

Quoting Tristan L

not saying anything is equivalent to saying something trivially true

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:

Quoting TheMadFool

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.

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.:

Aristotle:“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”

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.

Quoting Harry Hindu

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

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

Quoting Harry Hindu

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

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?

Quoting Harry Hindu

While you can say a contradiction, you can't think a contradiction.

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?

Quoting TheMadFool

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

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

Quoting 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.

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.

Quoting 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:

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.

Quoting Harry Hindu

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

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

Quoting 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.

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.

Quoting Tristan L

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

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

Quoting 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.

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

@Harry HinduQuoting Tristan L

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

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

Quoting TheMadFool

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

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?

Quoting TheMadFool

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

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.

Quoting TheMadFool

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

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.
[math]\neg E = O[/math] which is saying X is odd, a positive statement.

Quoting 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

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."

Quoting 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.

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.

Quoting TheMadFool

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

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.

Reply to TheMadFool 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.

Quoting Tristan L

0=0

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. In fact, Aristotle made it a point to state it explicitly lest we forget it.

Quoting 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.

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. It appears that when you make an argument, time is supposed to stop at a single instant, a single moment, this moment being occupied by all the propositions in that argument. 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. I recall having come to the conclusion that since a contradiction is defined in temporal terms:

Quoting Harry Hindu

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”

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.

Quoting EnPassant

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

Yes, you're correct. Tristan L showed me the error of my ways. When you're dealing with a proposition and its contradiction, it's more like two propositions swapping places rather than cancelling each other out.

I think I finally understand what I was trying to get at. If you have the time, this is my take:

A proposition may be thought of as occupying "space", in my analogy blank spaces. Suppose there is one like this: (..........). I now assert that E = god exists. Proposition E now occupies the blank space like so: (God exists). If I now claim that ~E = god doesn't exist, necessarily that E can no longer be claimed for the simple reason that ~E means, literally, NOT E. Therefore, I must erase E from the blank space which then transitions from (God exists) to (..........). Back to square one. This is what I was getting at but it appears that the process doesn't end there - my mistake was thinking it does. What happens next is ~E = God doesn't exist, occupies the blank space and (..........) transforms into (God doesn't exist). If I had asserted ~E first and then E, the same process is involved, only the propositions are now switched.

Quoting god must be atheist

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."

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].

Quoting 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.

:ok: :up:

Reply to Pfhorrest :ok:

Quoting TheMadFool

If I had asserted ~E first and then E, the same process is involved, only the propositions are now switched.

Yes. See also Reductio Absurdum as a (dis)proof.

Quoting EnPassant

Yes. See also Reductio Absurdum as a (dis)proof

I don't quite get you there...Mind elaborating a bit?

Quoting TheMadFool

I don't quite get you there...Mind elaborating a bit?

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.

Quoting EnPassant

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.

:ok: :up:

Quoting TheMadFool

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).

This is not the way contradictions are viewed in classical logic, but this is precisely how Aristotle and many philosophers until the rise of classical logic viewed them. The technical name for this view is "negation as cancellation", and the types of logics that use this sort of negation are known as "connexive logics". More info here

You may also be pleased to know that connexive logics are paraconsistent, meaning that EFQ is not valid in these logics. Indeed, Aristotle's syllogistic logic (one of many connexive logics) is paraconsistent, although this fact is not widely known. To convince yourself of this fact, consider the following syllogism:

1. All birds can fly.
2. Some birds cannot fly.
3. Therefore, the moon is made of green cheese.

This is not a valid syllogistic argument, since it violates all the rules of syllogistic reasoning. Now, in the sentential realm, the connexive logician avoids EFQ due to the fact that Simplification is not a valid rule of inference in connexive logic.

Quoting Alvin Capello

negation as cancellation

When you put it like that, it rings a bell. P, a proposition can be viewed as the denial of ~P on the basis that double negation returns the original proposition [P = ~(~P)]. And, ~P is the denial of P. The two do cancel each other because asserting the contradiction P & ~P means that P cancels ~P by denying ~P and ~P cancels P by denying P. It's like integer math with positive and negative numbers: +9 and -9 = (+9) + (-9) = 0.

Quoting 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].

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.

Quoting god must be atheist

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.

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.

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.

3 is prime, 5 is prime, 7 is prime...
[math]P_{1}:[/math] all odd numbers greater than 1 are prime.
3 x 5 = 15 which is composite.
[math]\neg P_{1}:[/math] At least one odd number is composite.
[math]P_{2}:[/math] Not all odd numbers are prime.
[math]\neg P_{1}[/math] forces the positive statement [math]P_{2}[/math]

Reply to TheMadFool Quoting TheMadFool

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.

thank you.

I think it came to focus here: Quoting 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].

I still have questions that need to be clarified.

(1) I state a proposition P.
(2) Then I state ~P.
(3) which amounts to denying P.
(4) this necessarily concludes in not having stated P.

Questions:
A. Is (2) equivalent to (3)?
B. Is P true, or not true?
C. By denying P, do you mean to say that you can prove that P is false?
D. What precisely do you mean by saying "I deny P"?
E. Your stating P, then later denying it: is it a chronological order of your chaning opinion of P, which is independent of the actual truth or falsehood of P, or
F. is it a logical order of progression, in which you give validation why ~P should be held as truth, and not P should be held as true?

You see, I am having problems. Also, you say you go back to the starting point... then you say the starting point is nothing, but your starting point is P.

This is all unclear, and untouchable because it's basically senseless. Sorry, not an opinion on you or on your abilities, but a judgment made only on this... erm... on this... I don't even know what to call it. Series of related thoughts?

NEW ADDITION:

Did you add this to your previous post with a bit of a delay? Because it states precisely what I stated (almost), it is similar enough in meaning so that I would not have made my argument here if I had seen it and read it.
Quoting 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.

That means, that you already preemptively answered my objections, and you and I came to an agreement. No further answer is required from you in this matter. Thanks.

Quoting 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.

What, then, is an example of a trivial truth?

Quoting TheMadFool

The words/concepts you employ must remain the same throughout a proof

True, but when you say

Quoting 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.

I have to counter: The Identity Law isn’t a law about the meanings of terms, is it? The proposition (0=0), which is indeed an instance of the Identity Law, is not the same as the not at all trivial proposition that for all time-points t1, t2 and every x, if I mean x and nothing else by ‘0’ at t1, then I also mean x and nothing else by ‘0’ at t2. The latter is an instance of what I call “the Constancy of Meaning Assumption”, which states that for every name N, all time-points t1, t2, and every x, if I mean x and nothing else by N at t1, then I also mean x and nothing else at t2 by N. It is this Meaning Constancy Assumption (MCA) that we need in a key way when arguing, not the Law of Identity, right?

Quoting 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.

Firstly, as I said above, I think that (correct me if I’m wrong) we need MCA rather than the Identity Law. Secondly, I believe that Aristotle’s additions “at the same time” and “in the same respect” are superfluous if one respects right slottedness (arity) of the relationships involved. For instance, we might say that a tree is green in summer but red in fall, so that the property of greenness belongs to the tree in summer but not in autumn. This fallacy is set right once we see that greenness is actually a binary (two-slotted) relationship, not a one-slotted one, so the tree actually has (the property of being green in summer) both in summer and in fall.

Quoting 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.

:up: So I’m not the only one who has realized that time might throw a wrench into any trial at a rock-solid proof. Great!

Quoting Tristan L

What, then, is an example of a trivial truth?

Nothing springs to mind!

Quoting Tristan L

Meaning Constancy Assumption (MCA) that we need in a key way when arguing, not the Law of Identity, right?

A rose by any other name smells as sweet. For what it's worth here's how I see it...

The Law Of Identity: A = A. In my book, A can mean anything, from letters themselves to entire theories however complex about the world. The essence of this law, what it boils down to, is that if a meaning, whatever that may be, is assigned to a particular symbol, that assignment of meaning must remain the same for the duration of an argument or narrative. Meaning Constancy Assumption seems to be logically equivalent to The Law Of Identity. I have nothing more to say.

The rest of your post went over my head. Above my paygrade for a meaningful reply. Thanks for the engaging conversation.

Quoting TheMadFool

A rose by [...] more to say.

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”.

Quoting TheMadFool

Thanks for the engaging conversation.

The pleasure is all mine :smile:.

Quoting Tristan L

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”.

What do you mean by "the sun is identical to itself"? Is there a danger/risk that it won't be identical to itself? Surely, the reason for formulating The Law Of Identity (A = A) is to prevent the possibility that someone will make the mistake that an A is not-A (violating The Law of Identity).

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.

Where an error can occur is at the level of, the correct word here is, terms which are essentially words that have referents, which as far as I can tell, are individual objects (sun) or entire categories (stars). The problem is a feature of language itself in that the same term, the same word, can have multiple referents (puns??) e.g. a star is a celestial object or, at other times, an actor. Here's where confusion becomes possible, confusion that's detrimental to the soundness of arguments; we're at risk of committing the equivocation fallacy. 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. A referent is just an object specified by the meaning of a term.

Quoting Tristan L

This has nothing to do with the meanings of words.

:chin:

Quoting TheMadFool

What do you mean by "the sun is identical to itself"?

By the sentence "the sun is identical to itself", I mean that the Sun is the selfsame thing as itself. This concerns only the Sun itself and has nothing to do with the word “Sun”.

Quoting TheMadFool

Is there a danger/risk that it won't be identical to itself?

No.

Quoting 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.

True, which is why I hold the Law of Identity to be trivial.

Quoting TheMadFool

Where an error can occur [...] the equivocation fallacy.

I fully forewyrd (agree) with you.

Quoting 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.

This shows that what you mean by the term “Law of Identity” is what I mean by the term “Meaning-Constancy Assumption”. Do you forewyrd?

Quoting TheMadFool

This has nothing to do with the meanings of words. — Tristan L


:chin:

What I mean by “Law of Identity” has nothing to do with meaning. It implies, for instance, that on the level of the Sun itself, the Sun is one and the same as the Sun. The principle that you’re talking about and that is important for the soundness of arguments, which are speechly (linguistic) objects, is what I call “Meaning-Costancy Assumption”. This is the principle that each term should refer to exactly one thing in a fixed way that doesn’t change over time.