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

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Some of these points have been mentioned, but I'd like to summarize:

 

(1) P -> ~P, as ordinarily understood, is a formula of symbolic logic, which is a context that may differ from the Objectivist notion of logic. One cannot understand such formulas of symbolic logic without first studying the textbook basics of symbolic logic (the Kalish-Montague-Mar book that was mentioned is indeed a fine introductory text). It is not meaningful to discuss such formulas with only a "kinda sorta" vague understanding mixed with Objectivist terminology that might not apply to the specialized terminology of symbolic logic.

 

(2) Symbolic logic has different systems. The usual context of such a formula is what is called 'classical sentential logic' (or 'classical propostional logic'). A less usual context is intuitionistic sentential logic. And there are others. But for purposes of basic discussion, I'll keep to the context of classical sentential logic.

 

(3) Reidy is incorrect that (P -> ~P) -> (P & ~P) is a theorem. What is, for example a theorem, is (P -> ~P) -> ~P. (Also, his mention of the completeness theorem in this context is wrong.)

 

(4) The letter 'P' here is a variable that ranges over "statements" (more precisely 'formulas'). It does not range over other objects. So conflating '->' with 'is' makes no sense.

 

(5) The symbol '->' stands for the Boolean function that maps <P Q> to 0 when P is mapped to 1 and Q is mapped to 0, and maps <P Q> to 1 otherwise. (And '0' can be interpreted as 'false' and '1' as 'true'.) 

 

In this sense, 'P -> Q' is understood as 'if P then Q' (this is called the 'material conditional'). P -> Q is false when P is true and Q is false, and it is true otherwise. 

 

This 'if then' is not claimed to correspond to all other English language meanings of 'if then'. It does correspond usually, but not always, and it is not meant to always correspond. In particular, some people find it wrong, or at least odd, that P -> Q is true when P is false. But in context, it is not intended that this sense of 'if then' corresponds always with certain other ordinary English senses, though it does correspond in a basic way, in the sense of the following analysis:

 

We do not need '->' in symbolic logic. We could take it as a defined symbol in this way:

 

P -> Q

by stipulative definition of our specialized symbol '->' is merely an abbreviation for 

~(P & ~Q).

 

And it is easy to see that, even in virtually all everyday English contexts, ~(P & ~Q) is false when P is true and Q is false, and it is true otherwise, just as we said for P -> Q. 

 

Thank you.  With regard to the original post can you explain exactly, and step by step, what

 

A->~A

 

means?

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No, P -> ~P does not just mean "some true proposition entails another true proposition that's not the same as the first proposition."

 

Rather, it means, "If P is true then P is false".

 

And it is not a contradiction. Rather, it boils down to saying "P is false". It does NOT say that there is a statement P such that P is both true and false. '->' is NOT the same as '&'. It says that P implies its own negation, so P itself implies a contradiction since P implies itself (of course) but also it implies its own negation. So P is false since P implies a contradiction. 

 

Again, P -> ~P is not a contradiction. What is a contradiction is to assert both P and P -> ~P.

 

Please, I wish all the people in this thread who are opining about this subject without FIRST learning the basics from a textbook, would indeed first read the chapters of a texbook and then come back to discuss it.

 

Premise 1: P->~P means "If P is true then P is false"

Premise 2: P->~P is not a contradiction

 

Conclusion:  The statement "If P is true then P is false" is not a contradiction. 

 

How can this not be a contradiction, I mean if P IS true how can it BE false?

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Thank you.  With regard to the original post can you explain exactly, and step by step, what

 

A->~A

 

means?

The same as P -> ~P, as I explained in my post.

 

I used the letter 'P' instead of the letter 'A' only because I don't like using 'A' as it can sometimes get caught in a sentence where 'A' is also the indefinite article in English. But that's not of substantive importance. You may regard 'P -> ~P' just as you would regard 'A -> ~A' and while also considering that 'A' in "A is A" would be understand as standing for and object but in "A -> ~A" it stands for a proposition. 

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Perhaps it is unfortunate that we call this "logic" instead of "information mechanism" or "data responsive functionalism".  There is nothing wrong with determining, conceiving, cataloguing the behavior of computers.

 

I do ask though what in any real context would

 

A -> ~A be applicable even in a computer program?  Here I assume A is exactly the same on either side, else a different symbol would have been used on one side.

 

Imagine an algorithm having the following pseudocode:

 

100  Fetch A (whatever it is) from Memory A

110 Translate/Validate A as something from which Truth can actually be determined (optional code)

120 Evaluate Truth of A generating T/F of A data

130 Store T/F data in Memory T/F of A

140 Fetch contents of Memory A and T/F of A Memory and run "Implication", generating Implication Data

150 Store Implication data in Memory Implication

160 Fetch Memory Implication to Verify implication

170 Translate Memory implication, to determine implication T/F value associated with A

180 Store Implication T/F value associated with A in temporary Memory Temp T/F

190 Checking validity of calculation by comparing Memory T/F of A with Memory Temp T/F

 

For step 190 to come up with a mismatch, clearly the "logic" of evaluating A must be different from that evaluating the "implication" of A to the point of contradiction.  

 

Again here the As are exactly the same but somehow truth or falsity can apply differently?

 

Independent of my example when could it ever be applicable? 

Perhaps you would set up a program that way if A does something that you don't ever want to happen. You could think about an anti-virus program that way, albeit a bit abstractly. "Infected -> not infected."

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Premise 1: P->~P means "If P is true then P is false"

Premise 2: P->~P is not a contradiction

 

Conclusion:  The statement "If P is true then P is false" is not a contradiction. 

 

How can this not be a contradiction, I mean if P IS true how can it BE false?

Because it does NOT say that P is true. Indeed it says P is NOT true.

 

It is NEVER the case that we have both P is true and P -> ~P is true.

 

It is NEVER the case that P is true and P is false. 

 

P -> ~P does NOT say that P is true and P is false.

 

What is says is

 

IF P is true then P is false. 

 

Consider theses examples:

 

(1) If Donald Trump is a character in Hamlet, then Hamlet was written by Shakespeare.

 

(2) If Donald Trump is a character in Hamlet, then I'm a monkey's uncle.

 

(3) If Donald Trump is a character in Hamlet, then Donald Trump is not a character in Hamlet.

 

(1) is true in symbolic logic. (Ordinary symbolic logic puts aside that there's no obvious connection between Trump not being in Hamlet and Shakespeare being the author of Hamlet. If you want to require a connection, then you'd use another form of symbolic logic called 'relevance logic'.)

 

(2) is true, since Trump is not in Hamlet. This is everyday English. People say it all the time; it's a well understood way of saying that Trump is not in Hamlet (given that it is true that I'm not a monkey's uncle) and the "IF THEN" is TRUE: 'IF Trump is in Hamlet then I'm a monkey's uncle" is true, because the first part is false. 

 

(3) is just a special version of (2). It is TRUE again because Trump is not in Hamlet. Of course, yes, it would be contradictory to say BOTH: "Trump is in Hamlet" and "If Trump is in Hamlet then he's not in Hamlet". But we're not saying both. 

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Also by what definition of "contradiction" is the statement:

 

"If P IS true THEN P IS false"

 

not a contradiction?

A contradiction is a statement of the form:

 

P & ~P

 

I.e., a contradiction is a statement of the form:

 

P is true and P is not true

 

or

 

P is true and P is false.

 

Or a contradiction is a statement that is equivalent to a statement of the form P & ~P. For example

 

P & (P -> ~P)

 

is a contradiction, since it is equivalent to 

 

P & ~P.

 

/

 

But (P -> ~P ) is not equivalent to P & ~P. Instead it is equivalent merely to ~P.

 

/

 

And again, P -> ~P does NOT assert there is a P that is both true and false. 

Edited by GrandMinnow
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Because it does NOT say that P is true. Indeed it says P is NOT true.

 

It is NEVER the case that we have both P is true and P -> ~P is true.

 

It is NEVER the case that P is true and P is false. 

 

P -> ~P does NOT say that P is true and P is false.

 

What is says is

 

IF P is true then P is false. 

 

Consider theses examples:

 

(1) If Donald Trump is a character in Hamlet, then Hamlet was written by Shakespeare.

 

(2) If Donald Trump is a character in Hamlet, then I'm a monkey's uncle.

 

(3) If Donald Trump is a character in Hamlet, then Donald Trump is not a character in Hamlet.

 

(1) is true in symbolic logic. (Ordinary symbolic logic puts aside that there's no obvious connection between Trump not being in Hamlet and Shakespeare being the author of Hamlet. If you want to require a connection, then you'd use another form of symbolic logic called 'relevance logic'.)

 

(2) is true, since Trump is not in Hamlet. This is everyday English. People say it all the time; it's a well understood way of saying that Trump is not in Hamlet (given that it is true that I'm not a monkey's uncle) and the "IF THEN" is TRUE: 'IF Trump is in Hamlet then I'm a monkey's uncle" is true, because the first part is false. 

 

(3) is just a special version of (2). It is TRUE again because Trump is not in Hamlet. Of course, yes, it would be contradictory to say BOTH: "Trump is in Hamlet" and "If Trump is in Hamlet then he's not in Hamlet". But we're not saying both. 

 

What possible meaning do you attribute to "then" in (3)?

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What possible meaning do you attribute to "then" in (3)?

What I'm saying in this thread is merely in context of symbolic logic. In that context, I don't give meaning to "then" but rather to "if ... then ...".

 

And I stated the meaning in a post earlier. I summarize more simply:

 

The meaning of "if ... then ..." is

 

If "P then Q" is true when P is true and Q is true.

If "P then Q" is false when P is true and Q is false.

If "P then Q" is true when P is false and Q is true.

If "P then Q" is true when P is false and Q is false.

 

Or, I define "If P then true" to stand for "It is not the case that both P is true and Q is false". And the truth table for that is exactly the same as the one I just mentioned.

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So when P stands for "Trump is in Hamlet" and also Q stands for "Trump is in Hamlet", we have this:

 

"If Trump is in Hamlet then Trump is not in Hamlet", which is true in the case that Trump is not in Hamlet and false in the case that Trump is in Hamlet, and since it's the case that Trump is not in Hamlet, we have that "If Trump is in Hamlet then Trump is not in Hamlet" is true, and from this the logic does NOT allow the inference of the contradiction "Trump is in Hamlet AND Trump is not in Hamlet". And it doesn't even allow the inference of the false statement "Trump is in Hamlet" but rather it does provide the inference of the TRUE statement "Trump is not in Hamlet".

 

"Trump is not in Hamlet" is true.

 

"If Trump is in Hamlet then Trump is not in Hamlet" is true.

 

And the contradiction, "Trump is in Hamlet and Trump is not in Hamlet" cannot be derived from this.

Edited by GrandMinnow
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Tell me if I get this correct:

 

1. "then" as such has no meaning (in this context of its use in symbolic "logic")

2  Only "If.. then..." has meaning (in the context)

3. The only meaning (in the context) of If.. then... is determined by a "truth table"

 

Whatever standard about a chosen "truth" table which is required for it to meet the standard of a valid "logic", it does not include avoidance of statements:

 

"If Trump is in Hamlet then Trump is not in Hamlet"

 

as meaningless contradictions?

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Please, I wish all the people in this thread who are opining about this subject without FIRST learning the basics from a textbook, would indeed first read the chapters of a texbook and then come back to discuss it.

It's just one method of discussion to state in no uncertain terms what my thoughts are, it isn't to say I know infallibly that it's true. I prefer to be corrected entirely about misunderstandings than not attempt an answer. Your explanation makes better sense to what I was reaching toward, that P -> ~P is not a contradiction because it's something to do with statements/formulas/etc instead of objects per se. But am I wrong to say "~P" isn't strictly a negation and might mean other things too? It looks like if you said "If P is true, then P is false, which would also tell us this implies P & !P, so we have a contradiction". If I'm understanding you, you're just saying P -> ~P is something false but not itself a contradiction.

 

Can you give an example translated to a more concrete context?

Edited by Eiuol
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Tell me if I get this correct:

 

1. "then" as such has no meaning (in this context of its use in symbolic "logic")

2  Only "If.. then..." has meaning (in the context)

3. The only meaning (in the context) of If.. then... is determined by a "truth table"

 

Whatever standard about a chosen "truth" table which is required for it to meet the standard of a valid "logic", it does not include avoidance of statements:

 

"If Trump is in Hamlet then Trump is not in Hamlet"

 

as meaningless contradictions?

The scare quotes aren't needed. Whether one thinks symbolic logic is logical in any given sense of the word 'logic', the field of study is commonly known nevertheless as: symbolic logic. And that includes the commonly understood notion of: truth table. 

 

Your points 1 - 3 are basically correct, but I wouldn't use the word 'valid' since it is itself a technical word in symbolic logic that has a special meaning.

 

Yes, "if then" in this context refers to the material conditional, which is interpreted through that truth table.

 

And yes, symbolic logic doesn't regard statements of the form "If Trump is in Hamlet then Trump is not in Hamlet" to be contradictions or meaningless. 

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But am I wrong to say "~P" isn't strictly a negation and might mean other things too? It looks like if you said "If P is true, then P is false, which would also tell us this implies P & !P, so we have a contradiction". If I'm understanding you, you're just saying P -> ~P is something false but not itself a contradiction.

I'm sorry, but that is mixed up.

 

(1) ~P is a negation. A negation is defined as any formula (loosely speaking 'statement') of the form ~P. In other words, when we say 'negation' we mean nothing more than putting the negation sign in front of a formula. 

 

(2) What do you mean by '!P'? Anyway, it is not the case that P -> ~P implies P & ~P.

 

(3) I'm not saying simply that P -> ~P is false. I'm saying it is false when P is true and it is true when P is false. And it is not a contradiction. 

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I'm sorry, but that is mixed up.

1. ! as I've seen is usually a negation and ~ is "not". Guess we need some LaTeX here! I already knew this.

 

2. You wrote "[P -> ~P] says that P implies its own negation, so P itself implies a contradiction". So I don't see what I mixed up.

 

3. "Rather, [P -> ~P] boils down to saying "P is false". " So it looks like you mean something else by 3 compared to your quoted statement here. "If Trump is in Hamlet is true, then Trump is in Hamlet is false" or the other way isn't itself a contradiction, but it looks like the interpretation is that something isn't quite right. Do you have a better example.

Edited by Eiuol
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You wrote "[P -> ~P] says that P implies its own negation, so P itself implies a contradiction". So I don't see what I mixed up.

 

Yes, P -> ~P implies P -> (P & ~P), so P -> ~P implies that P implies a contradiction. 

 

That is, asserting P -> ~P implies a contradiction when we ALSO assert P. But if we don't assert P, then we don't assert a contradiction by merely asserting P -> ~P.

 

(P -> ~P) -> ~P

 

and

 

~P -> (P -> ~P)

 

so

 

(P -> ~P) <-> ~P

 

and ~P is not necessarily a contradiction, since it is true when P is false.

 

However, P & (P -> ~P) IS a contradiction:

 

(P & (P -> ~P)) <-> (P & ~P)

 

/

 

What was mixed up in your post is that you said:

 

"you're just saying P -> ~P is something false."

 

I'm not saying that P -> ~P is false. I'm saying it is false when P is true and it is true when P is false. And it is not a contradiction.

Edited by GrandMinnow
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The scare quotes aren't needed. Whether one thinks symbolic logic is logical in any given sense of the word 'logic', the field of study is commonly known nevertheless as: symbolic logic. And that includes the commonly understood notion of: truth table. 

 

Your points 1 - 3 are basically correct, but I wouldn't use the word 'valid' since it is itself a technical word in symbolic logic that has a special meaning.

 

Yes, "if then" in this context refers to the material conditional, which is interpreted through that truth table.

 

And yes, symbolic logic doesn't regard statements of the form "If Trump is in Hamlet then Trump is not in Hamlet" to be contradictions or meaningless. 

 

I could not have said it better nor proved the point so succinctly myself.

 

Thank you for this. 

 

I have my own system of logic by which I have evaluated "symbolic logic" and its "validity" as regards the realm of reality.

 

 

There is nothing wrong with building mental constructs divorced from reality as long as you do not commit the error of thinking or claiming that they apply to reality, nor that any errors arising from such constructs imply any sort of problem (read "paradox") in real actual logic as applied to actual reality.

 

There are no paradoxes in reality only errors of logic, which is not logic at all.

Edited by StrictlyLogical
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What specific point do you feel is proved?

 

It was mentioned that symbolic logic is used for computing and other technology. Sentential logic, a part of symbolic logic, is really nothing more than Boolean logic. It it has been used - as a basis - by scientists, mathematicians, and engineers for decades more than a hundred years, and by computer scientists since the advent of the modern computer. The computer you use and a great amount of the other technology you use makes use of this material as core concepts. I would think you consider such things as your computer and the tasks it performs to be real. 

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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. 

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I'm not saying that P -> ~P is false. I'm saying it is false when P is true and it is true when P is false. And it is not a contradiction.

When I say "something false", I mean it couldn't in fact be the case, i.e. it may be totally valid, but something is wrong somewhere so it's not sound. "if Trump was not in Hamlet, then he was in Hamlet", of course it could be true, as in "Trump was not in Hamlet; but his not being in Hamlet means he was in Hamlet (with some fancy Sherlock-style reasoning to prove it)." But I'd go on to say "It entails that Trump was AND wasn't in Hamlet, and that part is a contradiction. Where was Trump really the day Hamlet was performed? His alibi for the night of the murder doesn't hold up!"

 

If P = True, and ~P = False, then I don't understand how further reasoning won't show P -> ~P isn't a contradiction. If your reasoning without a doubt shows P -> ~P is true, some premise you used to demonstrate P or ~P would be wrong. Is there a case you can show where P -> ~P is sound?

 

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.

Edited by Eiuol
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When I say "something false", I mean it couldn't in fact be the case, i.e. it may be totally valid, but something is wrong somewhere so it's not sound.

 

I don't understand what you're saying. You're using terms 'valid' and 'sound' in a way I can't reconcile with their use in symbolic logic.

 

If something is false, then it is not the case. And it is NOT valid. 

 

And, no, P -> ~P does NOT entail P & ~P. It is plainly, flat out incorrect to say otherwise.

 

And you ask about this case:

 

P is true and ~P is false, and how in that case P -> ~P is not a contradiction.

 

Simple:

 

"P is true" and "~P is false" are equivalent.

 

So we're simply talking about the case where P is true.

 

In that case P -> ~P is false.

 

And when we have BOTH that P is true and the claim P ->~P, we have a contradiction.

 

As I've said about three times already:

 

P & ( P -> ~P) is a contradiction.

 

But when P is FALSE, then P -> ~P is true.

 

You're making me repeat myself:

 

P -> ~P is true when P is false

P -> ~P is false when P is true.

 

P -> ~P is not a contradiction, since it is true in the case when P is false. (A formula is a contradiction if and only if it is false no matter what the sentence letters (such as 'P') are assigned as truth values. )

 

But P & (P -> ~P) IS a contradiction, since it is false no matter whether P is assigned true or false.

 

/

 

I can see that you have some basic confusions about this subject and its terminology. If you are interested in the subject, I strongly recommend getting a good intro textbook.

Edited by GrandMinnow
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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? I'm not confused on the terminology as much as I think you're only writing an explanation that works for somebody who already sorted all of this out about P -> ~P.

 

You're making me repeat myself

Right, it's not helping, so you need to say it differently. Does my edit in my post rephrase it all correctly? I need some examples besides the Hamlet one if I still am missing your point. I kinda-sorta see what you're saying, it's just not crystal clear.

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