Can you ever be certain that you are reasoning correctly?

Discussion of the nature of Ultimate Reality and the path to Enlightenment.
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Fujaro
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Re: When is a fish not a fish?

Post by Fujaro »

DHodges wrote:
Fujaro wrote:Furthermore I do think that I can point to a thing in reality without referring to rigorous A=A. I simply restrict myself to a limited set of properties to define that the thing is A. I can do that logically, and I can do that empirically. So I allow that I might discover empirically that this table in the ninth dimension that I yet can't investigate in no way is the table I thought it was. I would from it suggest another way of dealing with A=A. And I also allow that I might discover empirically that this table has ome proprties in the second spatial dimension I had overlooked up till then. I would allow my own fallible judgement.
What you are saying here doesn't deny A=A, it denies that A="A" - that is, a thing might not be what you think it is.
I haven't suggested it as a denial of A=A. It is in fact my phrasing of Leibniz Law for discerning identity of things on basis of possibly incomplete knowledge of all their properties.
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Re: Can you ever be certain that you are reasoning correctly?

Post by brokenhead »

Kevin Solway wrote:Definitions can't be "debated" in any meaningful sense. For example, if I use the word "God" to mean "the All", then that is a definition that I have made, and it is not open to debate.
You are clearly incorrect here. You could just as easily have defined "God" as anything that can be worn upon the feet. That would also be a definition that you have made. If you insist that because it is a definition then it is not open to debate, it will not be open to discussion, either, since few, if any, people would agree with your definition. If such a definition were not open to debate, then it would have meaning to precisely one person - yourself.

You are saying in any case that if something is not open to debate, then it must be a logical truth. This is a false statement. The converse is true, as we have agreed: if something is a logical truth, then it cannot be debated.

In no case is any definition a logical truth. It is simply a definition. In fact, I do not agree with your definition of "God" as "the All." Any logical statement you subsequently make based upon this definition would either have explicit or implied the form: "If God is the All, then..." I would not consider such a logical statement to be true; however, it would be logically true and therefore a thing in set A.

A logical truth cannot be debated, correct? Fujaro is debating "A=A" in this thread. Therefore, "A=A" cannot be a logically true statement. Rather, a logically true statement would be "If AdefA, then A=A." Fujaro is clearly not debating this. The definition AdefA is what is being debated here.
It's not clear what you are saying here. You are saying one of the following:

1. Anything capable of driving a car, you are giving the label "pebble".

This is a perfectly valid definition. You may use the label "pebble" to refer to other things as well, such as a small rock, but it is common for words to have several different meanings.

or 2. Anything capable of driving a car is a small rock.

The latter is not a definition, but is an empirical claim, and it can be empirically tested. So the latter is not a false definition, as it's not a definition at all.
Furthermore, I cannot rely solely upon logic to arrive at the conclusion that this is a false definition.
You can rely solely on logic to tell you that it's not a definition at all.
But you have said in point 1. that it is not only a definition, but a perfectly valid one.

Suppose what I meant by my definition of "pebble" is precisely statement 1. as you have given it above. You go on to say that it is a perfectly valid definition. If I were to use this definition in a discussion, it would have to be agreed upon beforehand which of several possible meanings we would be connoting when the term "pebble" was employed. If it has to be agreed upon, then the meaning is a priori open to debate and cannot be a logically true statement. Once the definition is agreed upon, it can then become part of a logically true statement in the role of a premise.

You seem unwilling or unable to grasp what I mean by a logically true statement (a thing in set A) as opposed to a thing that is not a logically true statement (any thing in set B, which is anything not in set A.)

Notice that a logically true statement cannot be debated, but its individual parts can be.

Example: "If all houses can fly, and this thing over here is a house, then this thing over here can fly." This is a logically true statement, is it not? As such, it cannot be debated.

But is it a true statement?

Monty Python's Graham Chapman, commenting on the sheep he sees flying overhead: "It is my opinion that they are laboring under the misapprehension that they are birds."
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Trevor Salyzyn
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Re: Can you ever be certain that you are reasoning correctly?

Post by Trevor Salyzyn »

Fujaro wrote: (A) <the real object> <--(pointer 1)--- logical subject A is logical predicate A --(pointer 2)--> <the real object>

While in the idealist view and in the realm of objectivism that concerns thought itself it would be:

(B) <thought of A> <--(pointer 1)--- Logical subject A is logical predicate A --(pointer 2)--> <thought of A>
This is not self-identity. Rather,

(A) <the real object> <--(pointer 1)--- logical subject A is logical subject A --(pointer 1)--> <the real object>
and
(B) <thought of A> <--(pointer 1)--- Logical subject A is logical subject A --(pointer 1)--> <thought of A>

is self-identity.

Dave's criticism, that you are trying to disprove A="A", rather than A=A, is apt.
Leibniz described this problem and formulated Leibniz Law (LL) for the identity of indiscernibles.
Since when does higher order logic disprove lower order logic? Higher-order logical laws, like Leibniz Law, build on lower-order logic! Without self-identity, the identity of indiscernibles is meaningless.
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Re: Can you ever be certain that you are reasoning correctly?

Post by Fujaro »

Trevor Salyzyn wrote:
Fujaro wrote: (A) <the real object> <--(pointer 1)--- logical subject A is logical predicate A --(pointer 2)--> <the real object>

While in the idealist view and in the realm of objectivism that concerns thought itself it would be:

(B) <thought of A> <--(pointer 1)--- Logical subject A is logical predicate A --(pointer 2)--> <thought of A>
This is not self-identity. Rather,

(A) <the real object> <--(pointer 1)--- logical subject A is logical subject A --(pointer 1)--> <the real object>
and
(B) <thought of A> <--(pointer 1)--- Logical subject A is logical subject A --(pointer 1)--> <thought of A>

is self-identity.

Dave's criticism, that you are trying to disprove A="A", rather than A=A, is apt.
I did not posit that pointer 1 and 2 are different. I formulated it in a general sense to stress that mapping to the predicate and mapping to the subject are technically different mappings.
Trevor Salyzyn wrote:
Leibniz described this problem and formulated Leibniz Law (LL) for the identity of indiscernibles.
Since when does higher order logic disprove lower order logic? Higher-order logical laws, like Leibniz Law, build on lower-order logic! Without self-identity, the identity of indiscernibles is meaningless.
Please read. It's not intended as a disprove but as an alternative identity criterion in a objectivistic reality.
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Re: Can you ever be certain that you are reasoning correctly?

Post by Trevor Salyzyn »

Fujaro wrote:I formulated it in a general sense to stress that mapping to the predicate and mapping to the subject are technically different mappings.
Distinguishing the two A's into subject and predicate is incorrect: the predicate is the = sign.
It's not intended as a disprove but as an alternative identity criterion in a objectivistic reality.
But you had suggested that it shows that self-identity is not accurate in all possible worlds. Redefining identity to be a higher-order function does not change the fact that the original, lower-order truth is still valid.
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Fujaro
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Re: Can you ever be certain that you are reasoning correctly?

Post by Fujaro »

Trevor Salyzyn wrote:
Fujaro wrote:I formulated it in a general sense to stress that mapping to the predicate and mapping to the subject are technically different mappings.
Distinguishing the two A's into subject and predicate is incorrect: the predicate is the = sign.
I am distinguishing the grammatical role in the sentence "A is A". Furthermore it doesn't change the argument.
Wikipedia wrote:In traditional grammar, a predicate is one of the two main parts of a sentence (the other being the subject, which the predicate modifies).
Trevor Salyzyn wrote:
It's not intended as a disprove but as an alternative identity criterion in a objectivistic reality.
But you had suggested that it shows that self-identity is not accurate in all possible worlds. Redefining identity to be a higher-order function does not change the fact that the original, lower-order truth is still valid.
Maybe it would help you to distinguish logic as a means and logic as an end.
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Re: Can you ever be certain that you are reasoning correctly?

Post by Trevor Salyzyn »

Fujaro wrote:I am distinguishing the grammatical role in the sentence "A is A".
But you are doing so falsely. A is the subject, "is" is the predicate. The second A is simply a repetition of the subject, a grammatical requirement in English because of the way the predicate "is" modifies the subject. This is quite important to the argument. The only difference between A=A and A is that the former fulfills a grammatical requirement. "A" is not a complete sentence.
Maybe it would help you to distinguish logic as a means and logic as an end.
Why? Whether logic is a means or an end, the truth of self-identity does not change. It's surprising how difficult it is to get a person to affirm the simplest, most elementary, truths. If this amount of disagreement occurs over something so simple, I can only imagine how impossible it must be to argue about most substantial matters.
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Diebert van Rhijn
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Re: Can you ever be certain that you are reasoning correctly?

Post by Diebert van Rhijn »



Indeed A=A is not a sentence like mathematical formulas are. The closest thing in mathematics would be the equivalence relation but then put in a more user-friendly form.
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Re: Can you ever be certain that you are reasoning correctly?

Post by brokenhead »

Trevor wrote:The second A is simply a repetition of the subject, a grammatical requirement in English because of the way the predicate "is" modifies the subject.
If I remember correctly, the second "A" is called the predicate nominative. The function in the sentence is as a direct object, but the verb "to be" is what is called a linking verb and its objects take the nominative case (instead of the objective case), and thus are called "predicate nominatives."

Example:
"Are you the person who authored this post?"
Incorrect reply: "I am him." Correct reply: "I am he."

Say what you want about the Catholics, they do run efficient grammar schools...
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Re: Can you ever be certain that you are reasoning correctly?

Post by Fujaro »

As I said, the one that holds absolute truth, I am not he.
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Re: Can you ever be certain that you are reasoning correctly?

Post by Trevor Salyzyn »

brokenhead, it is not a predicate nominative since it adds nothing to A: it does not rename it, nor does it describe it. Unlike "I am he", the repetition is completely redundant. It is a tautology, something which is grammatically correct in logic, but barely so in English.

If your example had been "I am I", it might have been different.

What should draw your attention is not the repetition of the subject (which is somewhat bizarre, when done so in English), but the use of the word "am" -- the identification itself.
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Re: Can you ever be certain that you are reasoning correctly?

Post by brokenhead »

Trevor Salyzyn wrote:brokenhead, it is not a predicate nominative since it adds nothing to A: it does not rename it, nor does it describe it. Unlike "I am he", the repetition is completely redundant. It is a tautology, something which is grammatically correct in logic, but barely so in English.

If your example had been "I am I", it might have been different.
I'm not sure I agree with you. The sentence "A is A." is not completely redundant, for in logic, it implies "and not something else." For example, in the sentence "A kiss is a kiss," the second occurrence of the word kiss is as a predicate nominative.
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Re: Can you ever be certain that you are reasoning correctly?

Post by Trevor Salyzyn »

brokenhead, we're not dealing with the noun, we're dealing with the verb. Can the verb "to be" imply identity at all? Metaphysics looks at the whole range of being (of which any use of the word "to be" is an example of).

Self-identity by itself is completely redundant. That is the point. It is a tautology, the definition of truth. The sentence "I am he" requires that the tautological "he am he" and "I am I" are assumed. The principle of non-contradiction -- that is to say, the "and not something else" -- is, in some cases, dispensed with. Fujaro has already referenced cases with exceptions to non-contradiction: Leibniz Law, paraconsistent logics, etc.

I have been asking for a similar treatment of self-identity, but no examples have emerged, only misinterpretations.

As to "a kiss is a kiss", you are correct on that point. But that is only so far as the second use of kiss has different contextual connotations than the first. In self-identity, such is not the case.
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Trevor Salyzyn
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Re: Can you ever be certain that you are reasoning correctly?

Post by Trevor Salyzyn »

First link is correct, second link can create confusion, and the last link is flat-out wrong. Do you need links to speak for you?

The predicate of A=A is actually the equal sign. That is what makes it a significant statement. Speaking of the third link, for someone who claims that common interpretations are wrong... well, in the words of Thales, "be careful you are not guilty of what you accuse others of."
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Re: Can you ever be certain that you are reasoning correctly?

Post by Fujaro »

Trevor Salyzyn wrote:First link is correct, second link can create confusion, and the last link is flat-out wrong. Do you need links to speak for you?

The predicate of A=A is actually the equal sign. That is what makes it a significant statement. Speaking of the third link, for someone who claims that common interpretations are wrong... well, in the words of Thales, "be careful you are not guilty of what you accuse others of."
The equal sign is the predicate VERB, the second A is the predicate itself. It's not that difficult even for me with english not as my native language.

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Re: Can you ever be certain that you are reasoning correctly?

Post by Steven Coyle »

Perhaps some latin might help...

Conjugating the verb "I AM" into its Latin counterpart one decodes the latin verb "Sum..."

A = A.
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Trevor Salyzyn
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Re: Can you ever be certain that you are reasoning correctly?

Post by Trevor Salyzyn »

Fujaro wrote:The equal sign is the predicate VERB, the second A is the predicate itself.
I have been stressing that this is a logical sentence; how it is interpreted in relation to English grammatical conventions is a tricky subject. For one thing, it requires that the relation of subject and predicate be interpreted according to the meaning of the sentence, rather than English conventions, and suddenly the only way it makes sense is if the verb is the predicate. But this can't be done in English! Oh no, logic must be wrong!

It's a problem for the language when the verb "to be" is itself called into question... down to how far it can be said to function as a verb at all.

Since you need the air of legitimacy offered by namedropping, Heidegger began his treatise Being and Time in a singular, extended, treatment of how confusing this exact verb, for lack of any other word, is.
Why the perseverance in hostility?
I'm eagerly anticipating the "oh shit" moment when you suddenly switch from "A=A is wrong" to "A=A is so boringly obvious that I can't believe you guys even bother arguing it at all."
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Trevor Salyzyn
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Re: Can you ever be certain that you are reasoning correctly?

Post by Trevor Salyzyn »

Steven Coyle wrote:Perhaps some latin might help...

Conjugating the verb "I AM" into its Latin counterpart one decodes the latin verb "Sum..."

A = A.
Excellent point. In other languages, the grammar is more forgiving. Except in Chinese, where it's completely bizarre.
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Re: Can you ever be certain that you are reasoning correctly?

Post by Steven Coyle »

Thanks.

(insert chinese symbols)
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Re: Can you ever be certain that you are reasoning correctly?

Post by Trevor Salyzyn »

grrr... bloody Chinese language. I read in a book about the history of ideas that self-identity didn't appear in their early philosophy because of the way Chinese grammar works.
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Re: Can you ever be certain that you are reasoning correctly?

Post by Steven Coyle »

Interesting...

With all that abstraction, + a million different ways to say "Hello," I'd probably be a little schizoid, too.
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Re: Can you ever be certain that you are reasoning correctly?

Post by Kevin Solway »

brokenhead wrote:
Kevin Solway wrote:Definitions can't be "debated" in any meaningful sense. For example, if I use the word "God" to mean "the All", then that is a definition that I have made, and it is not open to debate.
You are clearly incorrect here. You could just as easily have defined "God" as anything that can be worn upon the feet. That would also be a definition that you have made. If you insist that because it is a definition then it is not open to debate, it will not be open to discussion, either, since few, if any, people would agree with your definition. If such a definition were not open to debate, then it would have meaning to precisely one person - yourself.
So far as an individual trying to understand philosophical truths is concerned, it is only the individual that matters. It is only when that individual tries to share his ideas with other people that other people would need to share his definitions. And even then, a "debate" over the definitions wouldn't arise. People either find a particular label useful, or they don't. If they find it useful, they will use it. If they don't, they won't.
You are saying in any case that if something is not open to debate, then it must be a logical truth.
No, I'm just saying that definitions, which can't ever be false, are not open to debate.
In no case is any definition a logical truth. It is simply a definition.
It is a logical truth that a thing is itself and not other than itself. This truth is by definition.
In fact, I do not agree with your definition of "God" as "the All."
You might not think that it is useful to use the label "God" for "the All", since you may have used that label for something else, but you can't disagree with the definition itself. You can only say that, for you, it's not useful.
Any logical statement you subsequently make based upon this definition would either have explicit or implied the form: "If God is the All, then..."
It's not a matter of "If God is the All". If I use the label "God" to refer to the All, then that's what the label refers to so far as I'm concerned. And that's all there is too it. It doesn't mean that all other people use the label in the same way that I do.
A logical truth cannot be debated, correct?
Correct.
Fujaro is debating "A=A" in this thread.
Therefore, "A=A" cannot be a logically true statement.
He's not debating that which I refer to by "A=A". Rather, he's debating what he refers to by "A=A". Two different things.

You can only debate something if you can identify it, and if you can identify it then you've already accepted A=A, and so there can be no debate about it. That's what I mean by A=A.

Rather, a logically true statement would be "If AdefA, then A=A."
"If there is a thing, 'A', then that thing is itself and not other than itself."

That's what A=A means.

Fujaro is clearly not debating this. The definition AdefA is what is being debated here.
Ok, so that's the "If there is a thing, 'A'" part.

If a person experiences something, and labels that experience 'A', then there can be no debate about it.

1. Anything capable of driving a car, you are giving the label "pebble".

This is a perfectly valid definition.
Suppose what I meant by my definition of "pebble" is precisely statement 1. as you have given it above. You go on to say that it is a perfectly valid definition. If I were to use this definition in a discussion, it would have to be agreed upon beforehand which of several possible meanings we would be connoting when the term "pebble" was employed.
Yes.
If it has to be agreed upon, then the meaning is a priori open to debate and cannot be a logically true statement.
It would be logically true to anyone who knew what you meant by the words. If you don't know what the words mean in this instance, it is undecidable.
Example: "If all houses can fly, and this thing over here is a house, then this thing over here can fly." This is a logically true statement, is it not? As such, it cannot be debated.
That's right. It is a purely logical statement of fact, in the form of A=A.

It's like saying, "If 2 is 1 greater than 1, then 1 + 1 = 2".
But is it a true statement?
Yes, indeed, it is a true statement. If all houses can fly, then, indeed, any particular house can fly.

But if you said "All houses can fly" instead of "If all houses can fly", then the statement would contain an empirical element, and would not be a purely logical statement of fact. It would contain an element that could be empirically disproven.
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Re: Can you ever be certain that you are reasoning correctly?

Post by clyde »

Kevin Solway wrote:
clyde wrote:If, as you write, an appearance is a cause of a thing and a thing is an appearance, then it follows that:

An appearance is a cause of an appearance; i.e. A is a cause of A!
When I say that "appearance is a cause of a thing" I am implying that the causes of appearance, such as senses, mind, or whatever the causes of appearance might be, are actually causes of the thing in question. So this carries with it much more information than "A thing depends on (or causes) itself", even though the latter is not strictly wrong.
Kevin;

Since you have already stated that “A thing is actually an appearance,” it follows that “the causes of appearance” are “the causes of the thing”.

Of course, the causes of appearance are other appearances; i.e., appearances cause other appearances.

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Re: Can you ever be certain that you are reasoning correctly?

Post by David Quinn »

Fujaro wrote:Not accepting the logical A=A statement amounts to not accepting formal logic itself for it is (with the tenets of the Law of Contradiction and the Law of the Excluded Middle) about the basic tenets of logic itself. It is unclear how to define a logic without LOI. This would surmount to illogic. So as a logical propositional statement it cannot be denied without denying the value of logic itself. I don't deny it and so far, I expect no disagreement with the stances of David and Kevin.

At least Kevin seems to make an additional metaphysical claim when Kevin denies the division between reality and one's own consciousness. But as I identified that stance as a form of solipsism David rejected it decisively. Solipsism is a special form of a metaphysical claim, that of idealism, the idea that there is only one sort of substance: thought. It might be that David and Kevin adhere to other forms of Idealism instead of solipsism (please do elaborate on this).

Kevin and I agree that consciousness is a necessary condition for existence. This is because an existing thing can only exist by virtue of having a form of some kind and, in turn, a form can only exist by virtue of presenting an appearance to an observer.

In other words, without the perspective generated by an observer, there can be no basis for a form to come into existence.

Consciousness isn't the sole creator of existence (i.e. solipsism is untrue), but it is a necessary condition of existence - just as a mirror is a necessary condition for the reflections occurring in its reflecting glass, but not the sole cause of them. However, unlike the mirror where the reflections are caused, in part, by objects external to the mirror, there are no objects or forms external to consciousness.

This is not to say there is nothing beyond consciousness, as nothingness too is a form. In the end, it is impossible for us to describe what is there in any definite sense. All we can say is that what exists beyond consciousness is reality in its unformed and unknowable state, while what we experience within our consciousness is reality manifested as forms. (Keeping in mind that "reality in its unformed and unknowable state" is still just another form and therefore a part of consciousness - but we don't want to get ahead of ouselves).

These conclusions are logically derived from the truth that a form cannot exist without the perspective generated by an observer.

Idealism as a metaphysical stance is contrasted by objectivism which holds that consciousness is not possible without the prior existence of something, external to consciousness, for consciousness to be conscious of: "To be aware is to be aware of something.". This also is a metaphysical claim. It is a stance I as a naturalist adhere to. IMO there is no deductive logic that conclusively can dissolve between some Idealist views and the Objectivist view, but there are phenomena that objectivism can explain but that idealism can not explain.
I can't really describe my stance as either idealistic or objectivist. Although I recognize that consciousness and forms necessarily arise together, that there can't be the one without the other, it doesn't stop from me also recognizing the utility of the standard objectivist view when it comes to practical matters.

For example, if I decide to walk out the door, I fully expect the next room to be waiting there for me, even though I am not directly aware of it at the moment. At the same time, I realize that the room cannot exist in any shape or form without an observer giving it form.

It is a bit like what happens in a dream. I can dream that I decide to walk out a door in the expectation that the next room will be waiting when I get there, even though the waiting room only appears the moment I observe it.

In objectivism there is a different interpretation of LOI than in idealism, for in objectivism thought can be about someting other than thought itself. Ataraxia is right that the definition of reality therefore is relevant for this discussion. I suspect that Kevin and David define reality as all consciousness perceived, but I'd like to have their affirmation or rejection of that.
Reality is utterly everything - which includes all the forms that are perceived in consciousness, as well as the unformed aspect of Reality "beyond" consciousness.

Ataraxia is wrong in linking the truth of A=A to a particular metaphysical stance, whatever it might be. A=A is always true, regardless of what metaphysical views a person might have, or what the metaphysical truth might be.

For example, an objectivist who believes that forms can exist beyond consciousness is, in the very act of positing this belief, affirming that these forms beyond consciousness are indeed what they are and not what they are not.

In other words, the truth of A=A is belief-neutral.

The objectivst worldview supposes that some thought, most conclusively thought that seems to stem from our senses, has its origin in a reality that is not entirely conscious thought. This is a very common view indeed and many philosophers have been proponents of it.
You're right, it is a standard view and, as I mentioned above, a perfectly valid one as far as practical matters are concerned. But it does break down under analysis. It has no ultimate basis.

Fujaro wrote:In this worldview there is a dichotomy between logical validation of LOI and the validation in that part of reality that is not thought. Here a mapping from the physical thing to the logical is needed to interpret LOI in the physical world. Something like this:

(A) <the real object> <--(pointer 1)--- logical subject A is logical predicate A --(pointer 2)--> <the real object>

While in the idealist view and in the realm of objectivism that concerns thought itself it would be:

(B) <thought of A> <--(pointer 1)--- Logical subject A is logical predicate A --(pointer 2)--> <thought of A>

In (B) the tought of A is not discernible from what is labeled with a logical A and therefore LOI always is complete. There are no hidden properties of A. In (A) the problem of non-conclusive indiscernability arises. This is because identity in the physical realm becomes independent from identity in the realm of thought. To make a complete mapping for the real object all properties of A have to be known with infinite precision.

You're confusing the issue of what is there in Nature with our ability or inability to map it.

It doesn't matter whether we can map a thing completely or not. The fact still remains the thing in question is what it is and not something else. Even if it turns out that we can't map a thing completely (and of course, we can never completely map an object with simplified models), the thing still remains exactly what it is - namely, an object that can't be mapped completely with finite models.

A contradiction I perceive in the statements from Kevin and David but cannot resolve is that is claimed by them that LOI holds in all possible worlds. I would counter that by suggesting that the objectivist view imo constitutes a possible world in which the mapping from logical to the physical becomes relevant. So LOI in a logical sense is true in this world but does not hold conclusively for real objects because a fundamental indiscernability is present. In this way LOI can be denied in the possible world of objectivist reality.
Again, you are falsely linking the validity of A=A to our ability or inability to discern objects. They are two separate issues.

So it seems that underlying all the fuzz is the difference in the metaphysical viewpoints adhered to by the debaters. These differences in stances are best shown I think in the flatlander example David gave. For me as a naturalist the 3-D pyramid passing through the flatlanders plane comes closest to what I would call the truth about reality, while David is forced to a dichotomy between flatland and 3-D.
It would only be a dichotomy if I insisted that the object was really a square, or really a pyramid, or really both at once, or whatever. That is, if I insisted that the object has a true, objective, unsullied form underneath the appearances.

Instead, I recognize that no such "objective" form is possible. Things only gain their form in relation to an observer's perspective - in this case, a square from a flatlander's perspective and a pyramid from a 3Der's perspective. And who knows, it might have an entirely different form from a 4Der's perspective.

The disconnection between the flatlanders view and the 3-D view I assess as a great shortcoming in the idealist view.

That might be the case for an idealist, but I am not an idealist. Because of this, I do not perceive any disconnect between the flatlander's view and the 3Der's view.

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