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Post by Eugene 2.0 on Apr 9, 2021 9:52:31 GMT
According to Russell an imaginary variable is a variable that describes all the elements in a set as one deity. Non-imaginary or real variables are those to describe a certain element. But it must be obvious that such a criterion isn't enough to be sure whether a variable isn't imaginary. To prove it isn't just a semantic operation, but rather pragmatic. In other words, we must check somehow whether or not this variable describes an element. And it must be noted, that it's impossible to check all the members if we're dealing with endless deities. And from this last point of view is seen that exactly this detail - that our view depends on can we check about this deity - is a method to categories our deities. It's like:
'x - is imaginary' if and only if 'we have a method to check whether or not x - is imaginary' (*)
Surely, we might find another routes of definition, but if we'd try we'd get to this problem since its beginning - because it's a semantic problem. And if we turned we'd get to the pragmatic side of this problem.
So, there should be no objections against (*), and at the same time such a definition has got a reflexive definition, so... it turns us back to the problem since its beginning. And then why do we need to follow semantic side of this problem if pragmatically we can establish it?
That's why the imaginary variables in Type Theory are handle deities. We might as introduce them so to remove. It's optional.
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Post by karl on Apr 9, 2021 13:59:47 GMT
According to Russell an imaginary variable is a variable that describes all the elements in a set as one deity. Non-imaginary or real variables are those to describe a certain element. But it must be obvious that such a criterion isn't enough to be sure whether a variable isn't imaginary. To prove it isn't just a semantic operation, but rather pragmatic. In other words, we must check somehow whether or not this variable describes an element. And it must be noted, that it's impossible to check all the members if we're dealing with endless deities. And from this last point of view is seen that exactly this detail - that our view depends on can we check about this deity - is a method to categories our deities. It's like: ' x - is imaginary' if and only if 'we have a method to check whether or not x - is imaginary' (*) Surely, we might find another routes of definition, but if we'd try we'd get to this problem since its beginning - because it's a semantic problem. And if we turned we'd get to the pragmatic side of this problem. So, there should be no objections against (*), and at the same time such a definition has got a reflexive definition, so... it turns us back to the problem since its beginning. And then why do we need to follow semantic side of this problem if pragmatically we can establish it? That's why the imaginary variables in Type Theory are handle deities. We might as introduce them so to remove. It's optional.
Consider the following statement:
"All members of set A are consistent with every other member of set A."
One cannot rule out the possibility of there existing a consistent set where its impossible to establish whether it's actually consistent. So for such a set, and by what you wrote, consistency is then not a variable that describes all elements in the set, despite the that the set is actually consistent (just not demonstrably so). Or, in other words, not a valid variable.
However, there cannot exist an inconsistent set where it's principally impossible to establish that it's inconsistent, since all that's required is to discover one single contradiction. And if there is no way to deduce a contradiction from the set, then it's not inconsistent. So this means that inconsistent is a variable that describes all elements in the set, as follows:
"Every member of set A is inconsistent with at least one other member of set A."
The reason for this, is that if one finds one contradiction in a set, that contradiction can be used to deduce contradictions with every conceivable statement in the set. In an inconsistent set, all statements are both true and false at the same time.
My own conclusion is that an inconsistent set isn't meaningful to begin with. And in regards to the provability of consistency, I think it's wrong to imagine that we can ever establish what is a valid proof to begin with. So imagine a set we believe to be consistent, but where we can't prove it. Unless our belief is unfounded, we may call that belief as "pragmatic", but that pragmatism may very well just be the beginning of a new way of reasoning things out, as our conception of what constitutes valid evidence evolves in time.
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Post by Eugene 2.0 on Apr 9, 2021 14:05:05 GMT
According to Russell an imaginary variable is a variable that describes all the elements in a set as one deity. Non-imaginary or real variables are those to describe a certain element. But it must be obvious that such a criterion isn't enough to be sure whether a variable isn't imaginary. To prove it isn't just a semantic operation, but rather pragmatic. In other words, we must check somehow whether or not this variable describes an element. And it must be noted, that it's impossible to check all the members if we're dealing with endless deities. And from this last point of view is seen that exactly this detail - that our view depends on can we check about this deity - is a method to categories our deities. It's like: ' x - is imaginary' if and only if 'we have a method to check whether or not x - is imaginary' (*) Surely, we might find another routes of definition, but if we'd try we'd get to this problem since its beginning - because it's a semantic problem. And if we turned we'd get to the pragmatic side of this problem. So, there should be no objections against (*), and at the same time such a definition has got a reflexive definition, so... it turns us back to the problem since its beginning. And then why do we need to follow semantic side of this problem if pragmatically we can establish it? That's why the imaginary variables in Type Theory are handle deities. We might as introduce them so to remove. It's optional.
Consider the following statement:
"All members of set A are consistent with every other member of set A."
One cannot rule out the possibility of there existing a consistent set where its impossible to establish whether it's actually consistent. So for such a set, and by what you wrote, consistency is then not a variable that describes all elements in the set, despite the that the set is actually consistent (just not demonstrably so). Or, in other words, not a valid variable.
However, there cannot exist an inconsistent set where it's principally impossible to establish that it's inconsistent, since all that's required is to discover one single contradiction. And if there is no way to deduce a contradiction from the set, then it's not inconsistent. So this means that inconsistent is a variable that describes all elements in the set, as follows:
"Every member of set A is inconsistent with at least one other member of set A."
The reason for this, is that if one finds one contradiction in a set, that contradiction can be used to deduce contradictions with every conceivable statement in the set. In an inconsistent set, all statements are both true and false at the same time.
My own conclusion is that an inconsistent set isn't meaningful to begin with. And in regards to the provability of consistency, I think it's wrong to imagine that we can ever establish what is a valid proof to begin with. So imagine a set we believe to be consistent, but where we can't prove it. Unless our belief is unfounded, we may call that belief as "pragmatic", but that pragmatism may very well just be the beginning of a new way of reasoning things out, as our conception of what constitutes valid evidence evolves in time.
Oh... I do thank you for commenting. Must say my mind is about to blow now =) I gotta have an extra time to think about it more precisely. |
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Post by Eugene 2.0 on Apr 13, 2021 16:06:36 GMT
According to Russell an imaginary variable is a variable that describes all the elements in a set as one deity. Non-imaginary or real variables are those to describe a certain element. But it must be obvious that such a criterion isn't enough to be sure whether a variable isn't imaginary. To prove it isn't just a semantic operation, but rather pragmatic. In other words, we must check somehow whether or not this variable describes an element. And it must be noted, that it's impossible to check all the members if we're dealing with endless deities. And from this last point of view is seen that exactly this detail - that our view depends on can we check about this deity - is a method to categories our deities. It's like: ' x - is imaginary' if and only if 'we have a method to check whether or not x - is imaginary' (*) Surely, we might find another routes of definition, but if we'd try we'd get to this problem since its beginning - because it's a semantic problem. And if we turned we'd get to the pragmatic side of this problem. So, there should be no objections against (*), and at the same time such a definition has got a reflexive definition, so... it turns us back to the problem since its beginning. And then why do we need to follow semantic side of this problem if pragmatically we can establish it? That's why the imaginary variables in Type Theory are handle deities. We might as introduce them so to remove. It's optional.
Consider the following statement:
"All members of set A are consistent with every other member of set A."
One cannot rule out the possibility of there existing a consistent set where its impossible to establish whether it's actually consistent. So for such a set, and by what you wrote, consistency is then not a variable that describes all elements in the set, despite the that the set is actually consistent (just not demonstrably so). Or, in other words, not a valid variable.
However, there cannot exist an inconsistent set where it's principally impossible to establish that it's inconsistent, since all that's required is to discover one single contradiction. And if there is no way to deduce a contradiction from the set, then it's not inconsistent. So this means that inconsistent is a variable that describes all elements in the set, as follows:
"Every member of set A is inconsistent with at least one other member of set A."
The reason for this, is that if one finds one contradiction in a set, that contradiction can be used to deduce contradictions with every conceivable statement in the set. In an inconsistent set, all statements are both true and false at the same time.
My own conclusion is that an inconsistent set isn't meaningful to begin with. And in regards to the provability of consistency, I think it's wrong to imagine that we can ever establish what is a valid proof to begin with. So imagine a set we believe to be consistent, but where we can't prove it. Unless our belief is unfounded, we may call that belief as "pragmatic", but that pragmatism may very well just be the beginning of a new way of reasoning things out, as our conception of what constitutes valid evidence evolves in time.
I apologize that my answer took so long. 1. Yes, /I hope/ I've got your idea. Of course, to say either this is a variable, or not we have to be sure about its consistency, and at the same time there's a problem appears with it - as it leads to the paradox. Russell and Ramsey, and some other logicians, thought the same - agreeing with you - that such paradox cases can be useful for some reason. Russell claimed is the purpose to do logic, while the others saw some sense in that; like Moriz Schlick who said that is our task is to draw maps of such paradoxes. Anyway, the method of making maps is the endless process that might hold eternally: making the map more and more accurate /and the same is about any models/. 2. Continuing the theme in some other key I'd like to ask you - what do you think of metaphors or tropes? We can't just say 'x is a metaphor', because we have to feel it or to conceive it. In one situation that x can be a metaphor, but not for all the cases (however, it's possible). Such terms as consistency are well logic-formed terms, and the math part of such terms is what tries to stick to some rigorous and unequivocally style... /I don't know how to say it better/ ... Well, such terms (or a contextual part in the terms) are near math, while there might be some other - non-math - parts in such terms. /Such 'non-math' part can be what is close to expression, or phenomenological side, etc./ Considering the last one, I guess, ' x is a metaphor' can be close to ' x belongs to a certain type of non-math terms'. If we'd try to determine formal signs of a metaphor we shouldn't use non-math terms /or non-math parts of the terms/, and therefore we'd never define a metaphor. |
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