0=0 — Tristan L

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.

Exactly. This observation has led me to the conclusion that that a genuine proof cannot consist of a chain of thoughts, for in that case, it would need the memory to be infallible. I also thought about this when writing mathematical proofs by asking: How do I know that the theorems which I proved on an earlier page and on which I now draw haven’t been tampered with by a hacker or a random glitch in my harddrive and thus rendered false? But that’s likely something for the knowledgelore (epistemology) underforum. — Tristan L

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:

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.

Negation can be a positive statement, not just a blank. If I say X is an integer and X is not even I am not saying nothing about X, I am saying it must be odd. Let E = even and O = odd.
¬E=O¬E=O which is saying X is odd, a positive statement — EnPassant

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.

The starting point of a proof or of an argument is never a contradiction. And a contradicion is never a starting point.

I have never seen an argument to start, "Peter is not Peter." Or with "Given the time allotted to finish the project, we can finish the project if and only if we can't finish the project." — god must be atheist

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].

They are different because you made several mistakes in the structuring of your original post. I pointed the mistakes out in my immediately preceding series of posts before this one. — god must be atheist

:ok: :up:

:ok:

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

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

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

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

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

Reductio Absurdum makes a conjecture and follows that conjecture through until a contradiction is reached. The negation of the conjecture can then make a positive statement.

Reductio Absurdum makes a conjecture and follows that conjecture through until a contradiction is reached. The negation of the conjecture can then make a positive statement. — EnPassant

:ok: :up:

I like to view a contradiction in terms of a blank space on a piece of paper on which you write down propositions. Imagine the blank space; (..........). I say, "God exists" and this space gets filled and becomes: (God exists). If I now say "God doesn't exist, this happens:(God exists) - basically you're, if you had an eraser at hand, erasing the words "God exists" from the blank space and we return to:(..........), the blank space we started with.

In essence then a contradiction is to say nothing at all (returning to the blank space after having written down a proposition and then erasing it). — TheMadFool

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.

negation as cancellation — Alvin Capello

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.

I'm not talking about contradictions in the context of arguments. I'm investigating the import of propositions and their negations, specifically that to state a propositions P, then to deny it, ~P, amounts to not stating P [return to the starting point]. — TheMadFool

The starting point of WHAT? It has to be a starting point of something or other, which you haven't named. I can't put words in your mouth. Please state the starting point and state also this is a starting point of what. Thanks.

The starting point of WHAT? It has to be a starting point of something or other, which you haven't named. I can't put words in your mouth. Please state the starting point and state also this is a starting point of what. Thanks. — god must be atheist

1.Start. Nothing as in no propositions have been stated
2. P stated
3. ~P stated.
4. P & ~P stated. P cancels ~P and ~P cancels P. Result = No proposition left. Back to 1.

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

1.Start. Nothing as in no propositions have been stated
2. P stated
3. ~P stated.
4. P & ~P stated. P cancels ~P and ~P cancels P. Result = No proposition left. Back to 1. — TheMadFool

thank you.

I think it came to focus here:

I'm not talking about contradictions in the context of arguments. I'm investigating the import of propositions and their negations, specifically that to state a propositions P, then to deny it, ~P, amounts to not stating P [return to the starting point]. — TheMadFool

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.

Now that I realize it, P & ~P, because they cancel each other doesn't amount to a proposition. A contradiction essentially means the person who utters/writes it isn't saying anything at all. If so, any other proposition wouldn't be constrained by the necessity of consistency as there's no proposition in the first place to be consistent with. This is why anything follows from a contradiction keeping in mind that what doesn't follow from a certain proposition is predicated on a resulting inconsistency. — TheMadFool

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.

This is not a trivial truth. It's an instance of a law of thought viz. The Law Of Identity [A = A]. It's basic, I agree, but that doesn't make it trivial. — TheMadFool

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

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

True, but when you say

This is where The Law Of Identity, I mentioned above, comes into play. The words/concepts you employ must remain the same throughout a proof, A = A, a perfect example of which is 0 = 0. — TheMadFool

I 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?

I recall having come to the conclusion that since a contradiction is defined in temporal terms:

The LNC, as stated in Aristotle’s own words: “It is impossible for the same property to belong and not to belong at the same time to the same thing and in the same respect” — Harry Hindu


arguments do have a temporal dimension and one of the ways of offsetting this is The Law of Identity [A = A] which you think is trivial. — TheMadFool

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.

This issue of the temporal aspect of argumentation has been at the back of my mind for quite some time now. Thanks for reminding me of it. — TheMadFool

: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!

What, then, is an example of a trivial truth? — Tristan L

Nothing springs to mind!

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

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.

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

It seems that what you mean by “Law of Identity” is not the same as what I mean by “Law of Identity”. Why I mean by that expression is the law that each thing is the selfsame as itself. For instance, the Sun is identical to itself. This has nothing to do with the meanings of words. What you mean by “Law of Identity” is indeed basically what I mean by “Meaning Constancy Assumption”.

Thanks for the engaging conversation. — TheMadFool

The pleasure is all mine :smile:.

It seems that what you mean by “Law of Identity” is not the same as what I mean by “Law of Identity”. Why I mean by that expression is the law that each thing is the selfsame as itself. For instance, the Sun is identical to itself. This has nothing to do with the meanings of words. What you mean by “Law of Identity” is indeed basically what I mean by “Meaning Constancy Assumption”. — Tristan L

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.

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

:chin:

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

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”.

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

No.

This mistake, in your example of the sun, won't occur at the level of the sun itself - it's not that there's a possibility that sun will suddenly become not-sun. — TheMadFool

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

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

I fully forewyrd (agree) with you.

The Law Of Identity is designed to roadblock this fallacy by mandating the constancy of a term with respect to its referent in a given argument i.e. if a specific referent has been applied to a certain term, this term-referent pair must remain fixed throughout. — TheMadFool

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

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


:chin: — TheMadFool

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.