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What logical systems categorize A->~A as a contradiction.?

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By the way, as to usefulness, often in mathematical reasoning we prove the negation of a sentence by assuming the sentence and showing it entails a contradiction. And sometimes it is convenient to show that the statement contradicts itself. 

 

So a common argument is of the form:

 

We will show ~P. Assume P. Then, more argument that shows P entails some formula of the form Q & ~Q. So we've shown that P entails a contradiction, thus we've shown that P is not the case.

 

But sometimes it works out even more neatly to assume P, then more argument to show that P entails ~P, thus we've shown that P entails a contradiction (since P entails itself and also its own negation), thus we've shown that P is not the case.

 

This is fairly common in mathematics, and for mathematics that has practical applications in the sciences (and, no, off the top of my head I can't give you one of these proofs as an example, only because I haven't mentally inventoried in that way all the proofs I've read). 

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Valid, as in the logic really follows that way. Sound, as in it is in fact that way. Is this not what "valid" and "sound" means

Those are vague approximations.

 

A formula is valid if and only if there is no interpretation in which the formula is false.

 

An argument if valid if and only if there is no interpretation in which the premises are true and the conclusion is false. 

 

An argument is sound, per an interpretation, if and only if the argument is valid and the premises are all true in the interpretation.

 

A logic system is sound if and only if there is no deduction in that system of an invalid argument.

 

But your mistake was in claiming that there is a formula that is false but still valid. That is impossible. A valid formula is true in all interpretations. 

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you need to say it differently [...]  it's just not crystal clear.

I could not say it with any more crystal clarity.

 

If you don't understand, then you need to get a textbook that gives you the concepts that lead up to this:

 

A contradiction is a formula of the form P & ~P. 

 

And we prove that a formula is equivalent to a contradiction if and only if there is no interpretation in which the formula is true. (So, more casually we also call formulas "contradictions" that are equivalent to a contradiction even though the formula is not itself of the form P & ~P.)

 

P -> ~P is NOT a contradiction, because there IS an interpretation in which it is true, viz. the interpretation that assigns P to false.

 

Indeed, P -> ~P is equivalent to ~P. And ~P is not a contradiction, since it true when P is false.

 

 

[Of course, you could substitute for P to make a contradiction, such as substitute ~(Q & ~Q) for P so that we'd have ~(Q & ~Q) -> ~~(Q & ~Q), which is a contradiction. But we're not talking about substitution here; we're just talking about P as atomic.]

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I could not say it with any more crystal clarity.

Well, most people I've seen aren't very good at explaining logic, and it's not clear except to someone who already understands. I'm curious about an example, since your Hamlet/Trump example is strange and confusing. I understand your explanation, I apply it to your example, then your explanation stops making sense.

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To correlate this to Aristotle's three laws, from what I'm gathering from much of this:

 

The law of idenity: P is P.

 

The law of contradiction : P & ~P

P cannot be ~P at the same time and same respect.

 

I may be mistaken here, but what seems to being said here is: P -> ~P is like the law of excluded middle.

P is either P, or it is not (~P). There is no other alternative.

If P is P, then P (true) -> ~P(false), otherwise, P(false) -> ~P(true)

 

Symbolic notation makes things so much clearer . . . except when it doesn't. :)

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It's not a matter of picking out that particular formula as useful. Rather, it is the whole system of Boolean logic that is useful, and that particular formula is just one that is included in Boolean logic, and it evaluates to saying that P is false. 

 

I pointed out before that mathematicians and computer scientists don't always use English words in the everyday sense but sometimes use these words in a special technical way. With "if then" they mean the material conditional, and when working in basic standard logic take 

 

P -> Q

 

to mean

 

It is not the case that both P and ~Q are true.

 

And even in ordinary English we take 

 

"It is not the case that both P and not-Q are true" to be true when at least at least one of P or not-Q is false. And that is exactly the same as the evaluation of P -> Q.

 

So 

 

P -> ~P is a way of saying ~(P & ~~P)

 

which is equivalent to ~P (which you can easily see by the truth table for "&")

 

Spelling it out:

 

The truth table for "&":

 

P & Q true when both P and Q are true

P & Q false when P is true and Q is false

P & Q false when P is false and Q is true

P & Q false when P is false and Q is false

 

Truth table for "~":

 

~P true when P is false

~P false when P is true

 

So, with P -> ~P evaluating as ~(P & ~~P), it's easy to see that 

 

P -> ~P true when P is false

P -> ~P false when P is true

 

In other words, P -> ~P is equivalent to ~P. 

 

There's nothing the least bit controversial about this.

 

A LEGITMATE controversy though would be to claim that this sense of "if then" suffices for all the other English senses outside the specialized sense in mathematics and computing. 

 

/

 

Are you familiar with the concepts of such things as switching circuits, logic paths, flow charts? These are basic to computing and engineering. They make use of Boolean logic. And while the PARTICULAR formula P -> ~P probably doesn't come up often in practical applications of this, it is not inconceivable that it could come up, and it is a legitimate formula of Boolean logic. And in Boolean logic it "calculates" to ~P. 

 

OK

 

adding to the list, something generated by the symbolic logic system is useless, but the entire system taken as a whole is useful.

 

So now a few final questions for GM:

 

1.  Do you take symbolic logic to be in some way applicable to the fields of the philosophy of existence and the acquisition of knowledge by man, i.e. metaphysics and epistemology or is it applicable only to the special sciences of mathematics and how computational constructs of man can be made to function?

 

2.  Are you familiar with Objectivist metaphysics and epistemology and do you consider yourself an Objectivist?

 

 

3.  On an aside, and keeping entirely to mathematical constructs I was wondering if there were an analogue to actual logic which is like the imaginary numbers, that is to say.... well how do I put it, let me restart:

 

Is there an Alogic which is to logic, what imaginary numbers are to real numbers.

 

The imaginary numbers were fabricated to define imaginary abstractions x which could solve the equation X^2 = negative number, so that any written algebraic equations (presumably of certain form) were solvable.  So if the imaginary numbers are an extension of the real numbers are there systems of logic which added imaginary or odd elements to make the logical system more "complete"/solvable?

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The law of idenity: P is P.

 

The law of contradiction : P & ~P

P cannot be ~P at the same time and same respect.

 

I may be mistaken here, but what seems to being said here is: P -> ~P is like the law of excluded middle.

P is either P, or it is not (~P). There is no other alternative.

If P is P, then P (true) -> ~P(false), otherwise, P(false) -> ~P(true)

Much of that makes no sense.

 

(1) There is no "is" notation in sentential logic. For basic practical purposes, "is" is expressed in predicate logic with identity by the "=" symbol. In predicate logic with identity there is the principle "Ax x=x" ("for any object x, x equals itself). The closest we have in sentential logic is the material biconditional "if and only if": P <-> P (P is true if and only if P is true).

 

(2) It's not a law of contradiction but a law of NON-contradiction: ~(P & ~P)

 

(3) The law of excluded middle is: P v ~P.

 

(4) There's no "P is not P". There is P and there is not-P (~P), but it is not even grammatical to say things like "P is not P". 

 

(5) There's no construction "P(true)". 

 

It seems to me that you're making up your own notation and concepts that you kinda sorta think fit the material here. That's pretty much doomed to not work. If you are curious about the subject, I strongly recommend you read an intro textbook on it. Otherwise, just throwing symbols around is virtually certain not to work.

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It seems to me that you're making up your own notation and concepts that you kinda sorta think fit the material here. That's pretty much doomed to not work.

 

This, I think, is the crux of the problem. It is also the source of a great many math student errors (including my own). It is my observation that people learn most everything intuitively and functionally but not formally. This explains why very few people know what anything is. People can use a table but cannot define the term adequately, for example. Most of the time, a formal understanding is necessary in order to avoid functional errors. The common aversion to formal understandings explains the blizzard of errors we see among men.

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This, I think, is the crux of the problem. It is also the source of a great many math student errors (including my own). It is my observation that people learn most everything intuitively and functionally but not formally. This explains why very few people know what anything is. People can use a table but cannot define the term adequately, for example. Most of the time, a formal understanding is necessary in order to avoid functional errors. The common aversion to formal understandings explains the blizzard of errors we see among men.

 

I think the first error of the OP was the very placement of the post in "Metaphysics and Epistemology"

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So now a few final questions for GM:

And some perhaps not final answers for SL:

 

You wrote, "something generated by the symbolic logic system is useless, but the entire system taken as a whole is useful."

 

That is not what I said. I didn't say that the formula P -> ~P is useless but part of a useful whole. My point is that even IF it were useless itself, it is still part of a useful whole. This does not preclude that it may be useful itself. Actually, it is a useful formula in certain kinds of mathematical deductions; but I refrained from arguing that point since explaining it would involve too much time while a person could find out better about such things by reading an intro textbook.

 

Anyway, about the earlier point: Even IF I could not, offhand, mention a use for that particular formula, that does not preclude it. I don't know whether there has ever been a practical use for the equation:

 

x = (sqrt(43098346702343324.907902384343439)/34389343)*(3248390829+12111429437.750932)

 

and even if no one ever needed to make that calculation, it would not vitiate the mathematics and mathematical notation of which that is a legitimate formula.

 

1. I find that symbolic logic has wide applications in the various fields of philosophy and other studies. And there are various systems of symbolic logic for different fields of philosophical and practical inquiry. However, I do not claim that symbolic logic is perfect, or even good, fit for all questions or matters (recall that I mentioned myself that symbolic analyses may not properly cover certain English language senses in some cases). Especially in this thread, the subject has been sentential logic, which is a very limited, very rudimentary part of symbolic logic. I could not imagine lassoing all philosophical inquiry into sentential logic and I would be highly dubious of trying to lasso all philosophical inquiry into symbolic logic. In any case, it is not legitimately controversial that symbolic logic plays an important role in mathematics and computer science. Even the very WEAKEST argument I could make would be that the very high level of abstract ideation in such studies leads to thinking and innovations that then have practical fallout. 

 

But I am not interested in convincing you of the utility of symbolic logic in various fields of study. I don't wish to argue whether the applications in philosophy are credible from your own philosophic point of view or from an Objectivist view. I hardly doubt that you would find many of the applications to be ill conceived philosphically, but that it is beyond the scope of my time in the this discussion. Especially, it would be beyond my small amount of time for posting to delve into examples from the literature - one can see for oneself by looking at the literature of philosophy in general. I merely wished to point out that your claim that symbolic logic has no practical application at all is not true. 

 

2. I am familiar with the basics of Objectivism, though I am not an expert and I am not an Objectivist. But I'm not opining in this particular discussion about Objectivism. I'm pointing out things about symbolic logic, not Objectivism. 

 

3. I don't think it would be fruitful for me to give you a simple post-length answer to your question about "imaginary objects" since we probably don't have common ground about the very notions of "imaginary" and "object", especially as pertaining to the philosophy of mathematics. But at least a start would be for me to say that I don't have a philosophical point to argue at this particular time. My interest mainly has been to clear up the misunderstandings about symbolic logic I found in this thread. As to the philosophical questions about logic and symbolic logic, there is a vast literature, with many widely divergent points of view. My own philosophical views are not expert and have more to do with providing a context personally for me to regard my (also non-expert) studies in mathematical logic and mathematics, so those views may not be very pertinent to a more general discussion.

 

About imaginary numbers: A common approach in mathematics is to regard complex numbers as ordered pairs of real numbers, thus complex numbers themselves are merely points in the real plane, with the first coordinate considered the "real" part and the second coordinate considered the "imaginary" part, and then the operations on them are defined in a straightforward way. Pretty prosaic actually. 

Edited by GrandMinnow
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I don't know why you think that. It's simple. Simply substitute "Trump is in Hamlet" for "P" in the formulas, while also taking into account that "Trump is in Hamlet" is false.

I'm not asking for some explanation that requires rote memorization of where to put terms. I understand this is formal, so it can't be done by intuition, I'm just telling you that "If Trump is in Hamlet then he was in Hamlet" doesn't make sense to say even if it literally satisfies P -> ~P. See post #49 as to my best explanation of what it'd mean. I asked you to look at the edit part of that post of a simpler way to look at it, not just by substituting any old proposition into P. Sure, you gave a proposition, but it didn't help me. I already substituted in propositions of my own - in other words, you used as an explanation the very thing that is at issue. I'm looking for a natural language explanation. I can write truth tables all day, except that's not learning. I'll get a textbook if I must, except I really don't think the problem is lack of knowledge about logic. I know truth tables and notations (if we're not limited to a regular keyboard).

 

It's one thing to know P -> ~P values on a truth table, except I don't know any circumstances where a  false proposition turns true. If P -> ~P is a way to say "if I am a monkey's uncle, then I will now negate that to make a counterargument and show why I am not a monkey's uncle", then that makes sense.

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"If Trump is in Hamlet then he was in Hamlet" doesn't make sense to say even if it literally satisfies P -> ~P.

"If Trump is in Hamlet then he was in Hamlet" is NOT an instance of P -> ~P.
 
I take it though that your point is to say that "If Trump is in Hamlet then he isn't in Hamlet" doesn't make sense.
 
I've already as much as said that our English sense of "if then" is not always properly captured in sentential logic. Most people would take "If Trump is in Hamlet then he was in Hamlet" as nonsense. I'm not disputing that. But in context of Boolean operations, we are allowed to consider such utterances, especially, as I explained, where "if then" is taken as abbreviation in terms of "and" and "not", and in that context we know exactly how to make sense of such utterances while we don't claim that this sense always exactly matches the way people think about these things in everyday English. 
 
Sorry, Eiuol, I really don't know what more you want from me. (I'm not interested in going back to your #49 edit; just about everything you've said in every other post has a been a jumble of misunderstandings.) 
Edited by GrandMinnow
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And some perhaps not final answers for SL:

 

You wrote, "something generated by the symbolic logic system is useless, but the entire system taken as a whole is useful."

 

That is not what I said. I didn't say that the formula P -> ~P is useless but part of a useful whole. My point is that even IF it were useless itself, it is still part of a useful whole. This does not preclude that it may be useful itself. Actually, it is a useful formula in certain kinds of mathematical deductions; but I refrained from arguing that point since explaining it would involve too much time while a person could find out better about such things by reading an intro textbook.

 

Anyway, about the earlier point: Even IF I could not, offhand, mention a use for that particular formula, that does not preclude it. I don't know whether there has ever been a practical use for the equation:

 

x = (sqrt(43098346702343324.907902384343439)/34389343)*(3248390829+12111429437.750932)

 

and even if no one ever needed to make that calculation, it would not vitiate the mathematics and mathematical notation of which that is a legitimate formula.

 

1. I find that symbolic logic has wide applications in the various fields of philosophy and other studies. And there are various systems of symbolic logic for different fields of philosophical and practical inquiry. However, I do not claim that symbolic logic is perfect, or even good, fit for all questions or matters (recall that I mentioned myself that symbolic analyses may not properly cover certain English language senses in some cases). Especially in this thread, the subject has been sentential logic, which is a very limited, very rudimentary part of symbolic logic. I could not imagine lassoing all philosophical inquiry into sentential logic and I would be highly dubious of trying to lasso all philosophical inquiry into symbolic logic. In any case, it is not legitimately controversial that symbolic logic plays an important role in mathematics and computer science. Even the very WEAKEST argument I could make would be that the very high level of abstract ideation in such studies leads to thinking and innovations that then have practical fallout. 

 

But I am not interested in convincing you of the utility of symbolic logic in various fields of study. I don't wish to argue whether the applications in philosophy are credible from your own philosophic point of view or from an Objectivist view. I hardly doubt that you would find many of the applications to be ill conceived philosphically, but that it is beyond the scope of my time in the this discussion. Especially, it would be beyond my small amount of time for posting to delve into examples from the literature - one can see for oneself by looking at the literature of philosophy in general. I merely wished to point out that your claim that symbolic logic has no practical application at all is not true. 

 

2. I am familiar with the basics of Objectivism, though I am not an expert and I am not an Objectivist. But I'm not opining in this particular discussion about Objectivism. I'm pointing out things about symbolic logic, not Objectivism. 

 

3. I don't think it would be fruitful for me to give you a simple post-length answer to your question about "imaginary objects" since we probably don't have common ground about the very notions of "imaginary" and "object", especially as pertaining to the philosophy of mathematics. But at least a start would be for me to say that I don't have a philosophical point to argue at this particular time. My interest mainly has been to clear up the misunderstandings about symbolic logic I found in this thread. As to the philosophical questions about logic and symbolic logic, there is a vast literature, with many widely divergent points of view. My own philosophical views are not expert and have more to do with providing a context personally for me to regard my (also non-expert) studies in mathematical logic and mathematics, so those views may not be very pertinent to a more general discussion.

 

About imaginary numbers: A common approach in mathematics is to regard complex numbers as ordered pairs of real numbers, thus complex numbers themselves are merely points in the real plane, with the first coordinate considered the "real" part and the second coordinate considered the "imaginary" part, and then the operations on them are defined in a straightforward way. Pretty prosaic actually. 

 

Thank you for your considered response.

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Wait, I lied, I am going to address your edited post after all (it's post 48, not post 49).

 

You wrote:

 

"EDIT: I -think- I figured out the issue, I think the problem is saying something like "Not P is true when P is false" makes more sense in a truth table. Say the value of P is true. In the truth table, I negate P. So if P is negated, then P is false. Whether P is a contradiction is a separate matter entirely. The Hamlet example isn't so good, an example with decision making is probably better."

 

Truth table or whatever: not-P is true when P is false. 

 

And yes, whether P is a contradiction is a separate matter.

 

One thing needs to be clarified:

 

If 'P' is a sentence letter itself, then it is an atomic sentential formula, and is not a contradiction.

 

But if 'P' is being used as a meta-variable ranging over sentential formulas, then P is a contradiction or not according to this:

 

P is a contradiction if and only if P stands for an unsatisfiable sentence (which is equivalent to saying that P is equivalent to a formula of the form Q & ~Q). 

 

So, yes, falsehood and contradiction are different notions. So what? This doesn't vitiate anything I've said.

Edited by GrandMinnow
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"If Trump is in Hamlet then he was in Hamlet" is NOT an instance of P -> ~P.
 
I take it though that your point is to say that "If Trump is in Hamlet then he isn't in Hamlet" doesn't make sense.

Oops, that was a typo.

Indeed, it's not supposed to match everyday English, but you could in some sense translate it into some sentence that makes it clear what is being said. It's not as though there is no implication on natural language, unless it's so rare we'd only see James Joyce say it, i.e. it follows logic normally, but nothing interesting is going on. So in my last post it seems like I made a weird case as an instance of P -> ~P, made into something like a decision (that an operation needs to be performed on P), and not a contradiction.

"if I am a monkey's uncle, then I will now negate that to make a counterargument and show why I am not a monkey's uncle"

Would my example with a monkey's uncle be correct? Of course it's a lot longer than P->~P, and natural language is always a lot longer and prone to some ambiguity so I know it won't be perfect.

Besides, I think I'm satisfied with the answer at this point, even if I'm still uncertain about a few things. I read your most recent post to me as well.

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The "monkey's uncle" expression is of this form:

 

"If Trump is in Hamlet, then I'm a monkey's uncle." Next part: But (obviously) I'm not a monkey's uncle, so Trump is not in Hamlet.

 

That's of the form:

 

P -> Q. But ~Q, so ~P.

 

I mentioned that only to give a fuller context. The P -> ~P formula is different:

 

"If Trump is in Hamlet, then Trump is not in Hamlet." Next part: So if Trump is in Hamlet, then he's in Hamlet and not in Hamlet, which is a contradiction, so Trump is not in Hamlet.

 

That's of the form:

 

P -> ~P. So if P then P & ~P, but ~(P & ~P), so ~P.

 

And that's similar but not exactly the same as the "monkey's uncle" expression.

 

/

 

And it's not "I'll negate I'm a monkey's uncle to show I'm not a monkey's uncle". Actually, to show I'm not a monkey's uncle, the argument would be:

 

I'll show I'm not a monkey's uncle. First suppose I AM a monkey's uncle. Then [some argument showing a contradiction with other given facts from the assumption I'm a monkey's uncle]. Therefore, I'm not a monkey's uncle.

 

For example:

 

I'll prove Joe didn't steal the loaf of bread. Suppose Joe did steal the loaf of bread. Then he would have been recorded by the surveillance camera, but he wasn't recorded by the surveillance camera, so he didn't steal the loaf of bread. 

 

For example, one way to couch Euclid's result that there is no greatest prime number:

 

Suppose there is a greatest prime number. Then [an argument showing that that supposition leads to a contradiction], therefore there is no greatest prime number.

 

/

 

I think you see it now.

Edited by GrandMinnow
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  • 2 weeks later...

.

 

“Philosophy, including Logic, is not primarily about language, but about the real world. … Formalism, i.e. the theory that Logic is just about symbols and not things, is false. Nevertheless, it is important to ‘formalise’ as much as we can, i.e. to state truths about things in a rigorous language with a known and explicit structure.”

Arthur N. Prior*

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I just thought of something that I'd like to add to the discussion here.

 

Modern logic has a rule called disjunction introduction that allows you to add any disjunct to a premise. For example, if I affirm "the grass in my yard needs mowing," then this rule allows me to directly infer "either the grass in my yard needs mowing or Martians are invading the earth." In most contexts, this sort of inference is a violation of Rand's rule that arbitrary statements should be rejected without consideration.

 

As a result, the widespread acceptance of modern logic in academic philosophy has led philosophers to think that the Gettier problem is a serious concern. One version of the Gettier problem occurs when I start with a justified belief I hold that I don't know is false, then arbitrarily introduce a disjunct that I think is false but which is actually true. The result is a true, justified belief which is nevertheless not knowledge. Academic philosophers have spent an enormous amount of effort over the last several decades trying to figure out how this can be possible.

 

So, modern logic isn't just useless to philosophy, it's actively harmful. It's just a set of rules we built computers to obey that produce confusion when they are used as a guide for how to think.

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I just thought of something that I'd like to add to the discussion here.

 

Modern logic has a rule called disjunction introduction that allows you to add any disjunct to a premise.

Who uses a disjunct, specifically, in that way? Also, how does it really have anything to do with Gettier case? Gettier cases are a matter of knowledge needing to be factive and more than justified, i.e. if you are mistaken that something is true, it's not knowledge. Or if you're right by "accident" (despite good methods) it's not knowledge. Looking at a clock that says 2pm makes it justified to believe it is 2pm. On top of that, it is 2pm but the clock is just broken at the 2pm spot, so it's not knowledge. The problem about a Gettier cases isn't arbitrary disjuncts, the problem is treating knowledge as a thing unaffected by your mind.

 

So going back to the whole "A -> ~A" thing, modern logic looks weird or wrong perhaps, but really it's just that way to an untrained eye - and probably doesn't get explained well enough. Besides, of course it will produce confusion if a person doesn't understand a rule. Logic is really "just a set of rules", it's a concept of method. If the rules are for computers to obey, then they're rules for you to obey. If a computer using those rules makes good and useful computations, then the rules they obey are good and useful. So if you obey the rules too, you'll find it good and useful. 

Edited by Eiuol
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Louie said:

So going back to the whole "A -> ~A" thing, modern logic looks weird or wrong perhaps, but really it's just that way to an untrained eye - and probably doesn't get explained well enough. Besides, of course it will produce confusion if a person doesn't understand a rule. Logic is really "just a set of rules", it's a concept of method. If the rules are for computers to obey, then they're rules for you to obey. If a computer using those rules makes good and useful computations, then the rules they obey are good and useful. So if you obey the rules too, you'll find it good and useful.

Logic is an objective method and that's why Oism holds to an ontological view of Logic. Logic is useful because it is the methodological formalization of the law of identy for a conceptual consciousness which can fail to correspond to objective facts....

The fact that Logic is an epistemic method does not make it a "tool" that just "happens" to work. Just like the axioms, the laws of logic "could not be otherwise".

Edited by Plasmatic
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The fact that Logic is an epistemic method does not make it a "tool" that just "happens" to work. Just like the axioms, the laws of logic "could not be otherwise".

Well, yeah, I don't know if you're disagreeing? If it didn't work, it couldn't be any good. It works because it formalizes things methodologically. It's still a tool, it works, and it's objective.

 

EDIT: Your post is a good clarification of what I was saying.

Edited by Eiuol
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