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Gisteron
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29 Jul 2014 20:58 - 29 Jul 2014 21:23 #153864
by Gisteron
Yes, of course. These are the basic logical absolutes or axioms. They go by a number of equivalent names and expressions and are concepts describing fundamental truths that are so transcendant that nothing can be that violates them.
The law of identity states that anything is what it is, or to put it in more formal words, A is identical to A, or any given x is identical to itself. Moreover, it follows, that whenever there are two things, A and B, they are identical by definition if and only if there is no difference between them. In other words A-B=0 <=> A=B. There is a number of other implications, like that A(x)=A(y) => x=y, with A being the truth-value of a statement or a vector with its components being truth-values of statements about the respective variable. Note: A is not a function in general (with injective functions being the exception).
The law of non-contradiction is something like a contraposition of the law of identity. Basically it states that nothing is what it isn't. In other words, !A != A (with !x meaning not-x). Deriving examples is more difficult for this one. The important bit however is, that not everything seeming counter-intuitive is a contradiction. Change throughout space or time or both can occur without violating this law, but in the same context, under the same conditions, with all other things being equal, any particular given is never equal to anything that is not it.
The law of excluded middle is more or less a combination of the former two. It follows, that if everything is what it is, and nothing is what it isn't, then nothing can be both. since A != !A, it cannot be that x = A and x = !A, for that would violate both the law of excluded middle and the law of non-contradiction. I set x as equal to one of the other arguments, but this is also true for x being element of a set which of course renders it not element of the compliment of that set. And of course nothing can be neither, too. Example: "Absolutely everything is either a cucumber or something else." That statement is useless, but it can't help being true
Now, again, one must be cautious with one's definition. Earlier in chat we mentioned how a long cart can be both outside and inside of a bordered area. That is not in violation of the law of excluded middle or non-contradiction, because outside and inside are not absolute terms. In order to be inside something, it is sufficient that any part be inside. However, if every part is inside, then no part is outside, and if any part is outside, than the statement that "all parts are inside" is a false statement, because of the "all" bit.
The example with the cart specifically serves to show that those laws, while they apply universally and transuniversally, only make sense with clearly defined terms and borders. They don't work once you get vague with your language, but then no kind of logic works anymore which is why vagueness is so annoying and a disservice to anyone caring for a productive discussion of any kind.
Here is another illustration, if the way they phrase it helps anyone more:
http://logical-critical-thinking.com/logic/logical-absolutes/
Better to leave questions unanswered than answers unquestioned
scott777ab wrote: In chat you today you mentioned three things, will you please explain to me what these are?
1. The Laws of identity.
2. Non-contradictions.
3. Excluded middle.
Yes, of course. These are the basic logical absolutes or axioms. They go by a number of equivalent names and expressions and are concepts describing fundamental truths that are so transcendant that nothing can be that violates them.
The law of identity states that anything is what it is, or to put it in more formal words, A is identical to A, or any given x is identical to itself. Moreover, it follows, that whenever there are two things, A and B, they are identical by definition if and only if there is no difference between them. In other words A-B=0 <=> A=B. There is a number of other implications, like that A(x)=A(y) => x=y, with A being the truth-value of a statement or a vector with its components being truth-values of statements about the respective variable. Note: A is not a function in general (with injective functions being the exception).
The law of non-contradiction is something like a contraposition of the law of identity. Basically it states that nothing is what it isn't. In other words, !A != A (with !x meaning not-x). Deriving examples is more difficult for this one. The important bit however is, that not everything seeming counter-intuitive is a contradiction. Change throughout space or time or both can occur without violating this law, but in the same context, under the same conditions, with all other things being equal, any particular given is never equal to anything that is not it.
The law of excluded middle is more or less a combination of the former two. It follows, that if everything is what it is, and nothing is what it isn't, then nothing can be both. since A != !A, it cannot be that x = A and x = !A, for that would violate both the law of excluded middle and the law of non-contradiction. I set x as equal to one of the other arguments, but this is also true for x being element of a set which of course renders it not element of the compliment of that set. And of course nothing can be neither, too. Example: "Absolutely everything is either a cucumber or something else." That statement is useless, but it can't help being true
Now, again, one must be cautious with one's definition. Earlier in chat we mentioned how a long cart can be both outside and inside of a bordered area. That is not in violation of the law of excluded middle or non-contradiction, because outside and inside are not absolute terms. In order to be inside something, it is sufficient that any part be inside. However, if every part is inside, then no part is outside, and if any part is outside, than the statement that "all parts are inside" is a false statement, because of the "all" bit.
The example with the cart specifically serves to show that those laws, while they apply universally and transuniversally, only make sense with clearly defined terms and borders. They don't work once you get vague with your language, but then no kind of logic works anymore which is why vagueness is so annoying and a disservice to anyone caring for a productive discussion of any kind.
Here is another illustration, if the way they phrase it helps anyone more:
http://logical-critical-thinking.com/logic/logical-absolutes/
Better to leave questions unanswered than answers unquestioned
Last edit: 29 Jul 2014 21:23 by Gisteron.
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