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

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Hello,

I'm new to this forum and discussing philosophy, and would like to hear some thoughts. A post on the philosophy stack exchange forum had the following posed

As of this moment, I haven't seen an objectivist weigh in. Unless I'm reading this wrong, it seems to be throwing theories around right in the face of the objectivist philosophy, and the axiom of existence exists (A is A). Isn't objectivism a logical system? Am I missing something here? Has everyone on that forum drunk the anti-axiomatic Kool Aid?

Thanks,

Justin

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this is basic propositional logic. an implication is only false is the first term is true and the second term is false.

http://en.wikipedia.org/wiki/Material_conditional

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O.k. so if I understand correctly, essentially "A->~A" is a proposition, and NOT the same as saying "A is not A", which is simply stating a contradiction- that if A exists, then not A exists (and therefore a contradiction).

I think I was equivocating the use of the boolean value of A in propositional logic with the axiom of A is A (in which case A is always true).

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I don't know about other systems, but as I learned sentential logic from the Kalish & Montague text, this is not a contradiction.  A contradiction is a statement of the form P & ~P (P and not-P), or one which entails such a statement.  That is to say

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

is a theorem.  This in turn is to say, given a completeness proof, that the latter statement is not false under any assignment of true or false to the variables.  We have only one variable, P, and the statement turns out to be false where P is false.

Edited by Reidy

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Can any statement which can never be true in reality have any positive implications in reality? (assuming we do not dispense with reality when we make statements)

Edited by StrictlyLogical

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I haven’t opened my logic books on this, but under a conception of logic as the art of noncontradictory identification and as bound to the axioms existence exists, existence is identity, and consciousness is identification of existence, I think the standard rule in which a falsehood, say A, implies the truth anything we please should be rejected. In an Objectivist setting of logic, I should think “if A, then not-A” always false. (In tune with #5.)

It seems to me also that we suppose it to be false when we compose an indirect proof that A is false, wherein by assuming it true, then joining it to other premises we already accept as true, we deduce not-A as true conclusion. If the conjunction of the premise A and the conclusion not-A in such a proof is not taken as a falsehood, shall we throw out all our mathematical proofs of A’s that are false, where the proofs are by the indirect method? (I see this concern has occurred to others at the other site.)

Peter, in #4, what are the axioms and inference rules for arriving at that theorem?

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The Kalish & Montague system uses no axioms, only inference rules.

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Hello,

I'm new to this forum and discussing philosophy, and would like to hear some thoughts. A post on the philosophy stack exchange forum had the following posed

As of this moment, I haven't seen an objectivist weigh in. Unless I'm reading this wrong, it seems to be throwing theories around right in the face of the objectivist philosophy, and the axiom of existence exists (A is A). Isn't objectivism a logical system? Am I missing something here? Has everyone on that forum drunk the anti-axiomatic Kool Aid?

Thanks,

Justin

I'm a computer science major, so I've had to do a ton of proofs in modern symbolic logic. Modern logic has legitimate applications in computer science for understanding how computer programs and circuits work, because we've built programming languages and computer hardware to operate according to this specific set of rules that we came up with. However, outside of computer science, I don't think it is useful for anything. What you're doing here is like trying to figure out how to prove the rules of chess - there is no proof and it doesn't even make sense to ask for proof, it's just what we've decided to work with.

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Also, when offering a conditional (IF X ...), is not the author implying, for the validity of the "foundation" upon which the conditional may be uttered, the very fact that that X must be possible?

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What you're doing here is like trying to figure out how to prove the rules of chess - there is no proof and it doesn't even make sense to ask for proof, it's just what we've decided to work with.

That's not fair to logic as a field, as it isn't a game like chess as whatever rules you want, and it isn't so narrow as to be only part of computer science. It's more like the other way around, that modern logic has practical applications and was needed for the development of computer science. It's really abstract, so it's not obvious how it's useful and not just an academic game. I've barely read about logic as a field, though I'd still say it has a lot to say about being rational, or what it takes to be rational, and how it's so abstract that its principles apply to anything related to making decisions.

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Lots of answers & perspectives... thank you. However, I'm still ignorant, and need something like Rand's Razor....

If "A is A" (i.e., whatever A is, exists), then am I wrong in claiming:

"If A then not A" Is a contradiction

?
Thanks again

Justin

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I see a serious equivocation here.  In "A is A", A is a variable ranging over objects.  In "If A then not A", it ranges over statements.  You can't expect one of these to tell you much about the other.

I also see a misreading.  "A is A" means that an object is itself. It does not mean what you say it means in #11.  Any problems you might raise are the result of these two confusions.

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Peter’s distinction in #12 is one important to keep track of, but there are other things people mean by those expressions. To say that “A is A” means an object is itself has been agreed upon by thinkers who then go on to attach different meanings (narrow [this is only this and not any other] or broader including the narrow [this is only this and only what it is specifically]) to the principle that an object is itself. The expression “If A then not A” in a logic text of propositional logic would mean A to stand for propositions or statements. But in other contexts, we could use that conditional formula as shorthand for fact about objects, not the more reflective conditional about truth in our statements about objects. When I say “if this material is fissile, then it is not fissile,” then I know I’ve made a mistake about the material, about the material’s A is A, about its identity in the broader sense including not only A’s thatness, but A’s whatness. Identity, broad and narrow, is a fact of existence, of objects, we sensibly affirm as well as being a principle of order in our better thinking.

“The same attribute cannot at the same time belong and not belong to the same subject in the same respect” (Metaph. 1005b19–20). Under that formulation, Aristotle seems to be anchoring noncontradiction in fact of objects.

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Lots of answers & perspectives... thank you. However, I'm still ignorant, and need something like Rand's Razor....

If "A is A" (i.e., whatever A is, exists), then am I wrong in claiming:

"If A then not A" Is a contradiction

?

Thanks again

Justin

As far as I am concerned the statement "If A then not A" as a statement about anything whatever is a meaningless incoherent contradiction.  Now such a thing as an "expression" in symbolic logic or mathematics may not be the same as a statement, but as a statement it amounts to, both in content and consequence, a complete zero, a non-statement, complete silence, or non-thought like the jabbering of a parrot.  We take words to represent concepts, and logic to be the manner by which we form, think about, and validate knowledge conceptually.  Interpretations of logic as mere playing with words cannot reliably give you anything more than that.  Symbolic logic was developed and advocated for the most part (correct me if I am wrong) by rationalists and nominalists who knowingly or unknowingly are only playing with symbols divorced from reality and concepts... and hence in a real sense divorced from (although perhaps parallel or similar looking to) actual logic.

Another issue I have with the OP is "implication".  If you take implication to mean logically implied in reality, consequential due to the identity of things, based on certain premises, then a zero, a nothing, a non-statement can have no forward "implications".  Something meaningless cannot have any implications in a conceptual logical reality based sense.  A zero is not a magic utterance that can make ANYTHING literally possible.  At the very most an attempt at a conclusion based on certain premises, if it contains a contradiction, i.e. it actually cannot be formed or if formed is a meaningless utterance, does imply that something in the premises is false or (unfortunately) the process of logic was incorrectly followed.

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

"If a statement's truth implied its untruth..." game over.  Such a P is impossible, one cannot pose an IF in respect of its existence.  Any proffered statement P itself would be invalid and utterly meaningless it could not be true OR false let alone true AND false.

Unless one wants to play games... then of course:

"If a statement's truth implied its untruth, that would imply that the statement is false and the statement is true."

"Identity? Concepts? What are those? Statements are immune to such things... being purely a string of symbols to play at."

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That's not fair to logic as a field, as it isn't a game like chess as whatever rules you want, and it isn't so narrow as to be only part of computer science. It's more like the other way around, that modern logic has practical applications and was needed for the development of computer science. It's really abstract, so it's not obvious how it's useful and not just an academic game. I've barely read about logic as a field, though I'd still say it has a lot to say about being rational, or what it takes to be rational, and how it's so abstract that its principles apply to anything related to making decisions.

Okay, what practical applications does modern logic have outside of computer science (and perhaps mathematics)?

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"If a statement's truth implied its untruth, that would imply that the statement is false and the statement is true."

"Identity? Concepts? What are those? Statements are immune to such things... being purely a string of symbols to play at."

I agree, but I want to point out how things change if we view modern logic as a set of rules that computers operate according to, instead of a set of statements about reality. In a program, you could set up a line of code such that a boolean variable A changed from true to false if it was set to true initially. Alternatively, in Java at least, you could wrap the boolean variable A in an object whose sole purpose is to set the variable back to false whenever it was set to true for some reason. Neither of these would be very useful programs, mind you, but they are at least coherent interpretations of the statement.

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Okay, what practical applications does modern logic have outside of computer science (and perhaps mathematics)?

Nothing in philosophy can be answered that way. The obvious connections are those fields, but the best answer I can give is "everything". It's about knowing more, and knowing more about logic means more knowledge of rationality. A lot of philosophy appears impractical at first glance; "practicality" isn't even so important here.

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Nothing in philosophy can be answered that way. The obvious connections are those fields, but the best answer I can give is "everything". It's about knowing more, and knowing more about logic means more knowledge of rationality. A lot of philosophy appears impractical at first glance; "practicality" isn't even so important here.

I haven't seen it used for anything productive by philosophical standards either. When philosophers use symbolic logic it seems to obfuscate their arguments more than clarify them. The assumption is that since symbolic logic is useful in mathematics it must be useful everywhere else, which is not necessarily true.

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As far as I am concerned the statement "If A then not A" as a statement about anything whatever is a meaningless incoherent contradiction.

Not always, if "A" is a proposition. "If I am human, then I am an animal." That is true. But humans aren't the same as animals. Different concepts. A -> ~A. The law of identity is different. You liked Boydstun's post, and I think he stated what the idea is better, so I don't understand what you're arguing against.

Symbolic logic was developed and advocated for the most part (correct me if I am wrong) by rationalists and nominalists who knowingly or unknowingly are only playing with symbols divorced from reality and concepts

That's guilt by association, as the problem with them wasn't logic, it was that their underlying beliefs said that their symbols weren't related to reality. But the symbols were related to reality! And computer science shows it, otherwise computers wouldn't work so well.

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Not always, if "A" is a proposition. "If I am human, then I am an animal." That is true. But humans aren't the same as animals. Different concepts. A -> ~A. The law of identity is different. You liked Boydstun's post, and I think he stated what the idea is better, so I don't understand what you're arguing against.

That's guilt by association, as the problem with them wasn't logic, it was that their underlying beliefs said that their symbols weren't related to reality. But the symbols were related to reality! And computer science shows it, otherwise computers wouldn't work so well.

1.  You misunderstand, I am not arguing against anything Boydstun said.

2.  In your example, if A is a proposition "If I am human, then I am an animal" then A -> ~A still does not make any sense.  If the proposition is "True" (assuming it can be) how can it at the same time be false?

3.  "But humans aren't the same as animals"... this is an irrelevant observation... so WHAT if humans aren't animals?  A is not B ? If A is a species and B is a genus, A can be a type of B or be an instance of B, or a concrete which is a member of A can also be a member of B (e.g. you are a human and you are an animal).   and Yes, you are a human and you are an animal.  But you cannot ever be what you are not.  What is your point?  This is incoherent.

4.  "The law of identity is different"  - Eiuol April 27, 2015  - You said it.

only for those things which are not themselves brother...

Edited by StrictlyLogical

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I agree, but I want to point out how things change if we view modern logic as a set of rules that computers operate according to, instead of a set of statements about reality. In a program, you could set up a line of code such that a boolean variable A changed from true to false if it was set to true initially. Alternatively, in Java at least, you could wrap the boolean variable A in an object whose sole purpose is to set the variable back to false whenever it was set to true for some reason. Neither of these would be very useful programs, mind you, but they are at least coherent interpretations of the statement.

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?

.............................

Whether A is a statement, a proposition, or anything,

A->~A

purports to say that however you evaluate A on the left, THAT EVALUATION implies evaluation on the right which is opposite to that on the left, ALL while assuming the subject of evaluation e.g. proposition and the method of evaluation (determine truth) is the same. It essentially purports to say that doing the exact same thing is contradictory.

My point here is that when A is meaningless and cannot be evaluated, making any statements about its implications are as meaningless as the original statement.

Edited by StrictlyLogical

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Not always, if "A" is a proposition. "If I am human, then I am an animal." That is true. But humans aren't the same as animals. Different concepts. A -> ~A. The law of identity is different.

I think this is somewhat confused.  My understanding of symbolic logic (limited though it may be) is that A and ~A are the A and Not-A of the law of identity.

"If I am human, then I am an animal" is another statement entirely and would not be expressed as A -> ~A but A -> B: let "human" = A; let "animal" = B; A implies B; if I am human, then I am an animal.  A -> ~A would accordingly mean "If I am human, then I am not human."  That's not right.

My point here is that when A is meaningless and cannot be evaluated, making any statements about its implications are as meaningless as the original statement.

This is true.  We can recognize errors in form and in content as being distinct kinds of errors.

Similarly, you may remember from some math class coming up with an answer to an equation like -2.5 or something, which may follow from the equation provided perfectly well, but will not serve to answer a word problem dealing (for instance) with children.  Working with the abstract numbers of math can be very useful, but we must always be careful in their application.  I suspect that symbolic logic is a similar kind of abstraction, with analogous potential missteps.  It might yet be useful.

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1. Okay.

2. It doesn't mean "true, therefore false". "True" isn't a proposition, at least not in this context. A proposition is a statement that is true or false. So A -> ~A only would mean that some true proposition entails another true proposition that's not the same as the first proposition. ~A doesn't mean "The negation of A" here. Of course, a negation is a proposition too, so A -> ~A is a contradiction in that case. Post #12 makes this whole point clear as well.

3. Logic can specify how to treat relations, members of sets, and so on. Strictly speaking, all I said is the concept "human" is different than the concept "animal". You could say "A is a member of B" but it'd be strange to say "A, as a member of set B, is B." The thing about logic here is that it gets rid of ambiguity in natural language. Sometimes "is" makes sense as "is a member of" (A human is an animal).

4. Point 2 should clear this up.

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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.

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A -> ~A only would mean that some true proposition entails another true proposition that's not the same as the first proposition. ~A doesn't mean "The negation of A" here. Of course, a negation is a proposition too, so A -> ~A is a contradiction in that case.

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.

Edited by GrandMinnow

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