What Are the Origins and Limitations of Logic?

[Prometheus]

Dude, take a course on the subject.

Dude, what a quaint word.

Your proof proves nothing at all in the sense that I am speaking. Your proof proves that within the rules of logic there are relationships of logic. How incredibly wonderful of you. I said that logic can prove nothing in the real world, and you attempt to prove that relationships of logic can be proven.

Take a look at any of my posts in this thread before you came along with this. I said that logic cannot be used to prove anything in the real world. Your example is off that topic.

 

[quantumdude]

Do understand what a conditional statement is?

Yes, and I understand your symbolic proof, and I understand truth tables. What I am not understanding from your previous discussion is how you can derive necessary truths from an empty set of premises.

When you say, "There are no premises in the above argument,", it begs the question: Why don't you consider your "hypothetical suppositions" premises?

 

[cogito]

Dude, what a quaint word.

Your proof proves nothing at all in the sense that I am speaking. Your proof proves that within the rules of logic there are relationships of logic. How incredibly wonderful of you. I said that logic can prove nothing in the real world, and you attempt to prove that relationships of logic can be proven.

Take a look at any of my posts in this thread before you came along with this. I said that logic cannot be used to prove anything in the real world. Your example is off that topic.

Your claim is false. The above proof proves that a proposition must be either true or false. That applies to every world, including the real one.

 

[cogito]

Yes, and I understand your symbolic proof, and I understand truth tables. What I am not understanding from your previous discussion is how you can derive necessary truths from an empty set of premises.

When you say, "There are no premises in the above argument,", it begs the question: Why don't you consider your "hypothetical suppositions" premises?

Hypothetical suppositions are not premises because their actual truth value has nothing to do with the proof. The premises of an argument are such that their actual truth value is essential to the establishment of the argument's conclusion. Again, take the example of a conditional proof. A conditional proof starts by saying "Let's see what would follow if P was true. Now, if P was true, Q would follow. So, we know that the conditional claim if P, then Q is true, regardless of whether P is, in fact, true".

Hypothetical suppositions are agnostic as to the actual truth value of what is supposed, whereas premises are declarations that some statement has a particular truth value. When, in formal logic, a hypothetical supposition is discharged, it is thereby shown that the truth of the conclusion does not rest upon that supposition.

 

[Les Sleeth]

Hypothetical suppositions are not premises because their actual truth value has nothing to do with the proof. The premises of an argument are such that their actual truth value is essential to the establishment of the argument's conclusion. Again, take the example of a conditional proof. A conditional proof starts by saying "Let's see what would follow if P was true. Now, if P was true, Q would follow. So, we know that the conditional claim if P, then Q is true, regardless of whether P is, in fact, true".

Hypothetical suppositions are agnostic as to the actual truth value of what is supposed, whereas premises are declarations that some statement has a particular truth value. When, in formal logic, a hypothetical supposition is discharged, it is thereby shown that the truth of the conclusion does not rest upon that supposition.

I don't want to put words in Prometheus' mouth, but I understood his statement about logic not producing proof as meaning it doesn't produce proof about external reality. Everybody knows one can produce a proof within the system of logic, but all it tells you is if the logic is correct. Logic without empirical data really can prove nothing about external reality, just as Prometheus said.

 

[Prometheus]

Your claim is false.

Which claim are you referring to?

The above proof proves that a proposition must be either true or false.

This is not true at all, or, if true, it is not meaningful. Propositions are based on words, and whether the words are true or false in the context of logic, they cannot be used to provide a definitive, objective proof about the real world.

That applies to every world, including the real one.

Then I ask you again. Please use your logic example, or another example, to prove on the basis of logic that something exists in the real world. In the example originally provided, for example, you might try to use logic to prove that god does, or does not, exist.

 

[quantumdude]

Cogito, it is certainly the case that we are using two different definitions of the word "premise". In the logic textbook to which I am referring (Logic by Robert Baum), the term "premise" is used to describe not only particular statements, but also schemata whose truth values are unspecified. If you are using the term "premise" to refer exclusively to statements whose truth values are determinate, then that's fine. I just want to get it straight that that's what you mean.

But even so, there are some unresolved points of confusion here.

Hypothetical suppositions are not premises

Well, they certainly do not come from the empty set of premises[/color].

Your initial claim was:

"Formal Logic is sufficient for proving all theorems, which by definition can be proven from the empty set of premises[/color]."

You were asked for an example, but you have not delivered.

because their actual truth value[/color] has nothing to do with the proof.

Doesn't it? Your conclusion is certainly analytically true, but had you not started with a prem---...errr...hypothetical supposition that was analytically false[/color], you would not have arrived at that conclusion.

 

[Prometheus]

The premises of an argument are such that their actual truth value is essential to the establishment of the argument's conclusion.

Perhaps you might enlighten me about a point in logic that is unclear to me. Consider the following:

All blees are blahs
All blahs are blips
Therefore: All blees are blips

I am under the impression that the conclusion is true, given the premises, and that the real life truth value of the premises is irrelevant to the value of this logic. Are you saying that if it is not true that all blees are blahs in real life, then the logic is flawed?

 

[Prometheus]

I don't want to put words in Prometheus' mouth, but

Thanks. This is my point.

 

[quantumdude]

Perhaps you might enlighten me about a point in logic that is unclear to me. Consider the following:

All blees are blahs
All blahs are blips
Therefore: All blees are blips

I am under the impression that the conclusion is true, given the premises[/color], and that the real life truth value of the premises[/color] is irrelevant to the value of this logic.

But you can't have it both ways. Either the premises are given (as true), or they are not. There's no difference between "the truth value" and "the real life truth value".

But in any case, the truth value of any proposition is irrelevant to the validity of any logic. Deductively valid arguments can have false conclusions, provided that at least one premise is false.

Are you saying that if it is not true that all blees are blahs in real life, then the logic is flawed?

I think you need to look at cogito's symbolic proof again. The conclusion of that argument is a tautology[/color], meaning that there is no possible instantiation of that schema that can render its truth value false.

While it may be the case that "All blees are blips" is false, it is certainly the case that "All blees are blips or all blees are not blips" is true.

But as I said, you can only "prove" a tautology if you start with a contradiction. I don't question the validity of cogito's proof, but I do question his claim that he can prove something from nothing.

 

[Prometheus]

But in any case, the truth value of any proposition is irrelevant to the validity of any logic. Deductively valid arguments can have false conclusions, provided that at least one premise is false.

I agree, and this is my point.

But you can't have it both ways. Either the premises are given (as true), or they are not. There's no difference between "the truth value" and "the real life truth value".

I don't want it both ways. Given the premise that all blees are blahs, it is possible to use logic. However, the truth value in real life is irrelevant. In other words, it is not significant to logic whether or not this statement has any truth value in real life. As for truth value at all, that is irrelevant. If this proposition is made, then it is assumed to be true for the purposes of the argument, and for the purposes of the logical argument there is no meaning in questioning whether it might be false in some real world test.

 

[quantumdude]

I agree, and this is my point.

OK

I don't want it both ways. Given the premise that all blees are blahs, it is possible to use logic. However, the truth value in real life is irrelevant. In other words, it is not significant to logic whether or not this statement has any truth value in real life. As for truth value at all, that is irrelevant. If this proposition is made, then it is assumed to be true for the purposes of the argument, and for the purposes of the logical argument there is no meaning in questioning whether it might be false in some real world test.

Right, but one of cogito's points is that his conclusion (unlike yours) is necessarily true. In other words, it is not possible for any statement of the form P V ~P to be false, no matter what statement you insert for the logical variable P. On the other hand, the conclusion of your syllogism is contingent.

 

[Prometheus]

Right, but one of cogito's points is that his conclusion (unlike yours) is necessarily true. In other words, it is not possible for any statement of the form P V ~P to be false, no matter what statement you insert for the logical variable P. On the other hand, the conclusion of your syllogism is contingent.

I have no problem with that. I am still thinking from the standpoint that what he has presented has no ability to demonstrate proofs about the real world. It is an interesting game with words, but that is what I accepted as given at the beginning of this thread. Can he use this to prove that, for example, god exists or does not exist? My goal is not to challenge the tenets of logic, nor to debate it, but to discuss its limitations.

 

[quantumdude]

I have no problem with that. I am still thinking from the standpoint that what he has presented has no ability to demonstrate proofs about the real world.

But he already admitted that he can't do that.

What is the case is that these theorems are true in any possible world, and hence quite uninformative about the particularities of the actual world (i.e., you can't derive any contingent truths about the actual world).

The way I see it, the only issue still in dispute is the "something from nothing" claim.

 

[Prometheus]

But he already admitted that he can't do that.

Then I guess that there never was any real disagreement, although it seemed so.

 

[cogito]

I don't want to put words in Prometheus' mouth, but I understood his statement about logic not producing proof as meaning it doesn't produce proof about external reality. Everybody knows one can produce a proof within the system of logic, but all it tells you is if the logic is correct. Logic without empirical data really can prove nothing about external reality, just as Prometheus said.

And, again, that claim is false. The Law of the Excluded middle applies to external reality. Hence, a proof of the Law of the Excluded Middle proves something about external reality.

 

[cogito]

Which claim are you referring to?


This is not true at all, or, if true, it is not meaningful. Propositions are based on words, and whether the words are true or false in the context of logic, they cannot be used to provide a definitive, objective proof about the real world.


Then I ask you again. Please use your logic example, or another example, to prove on the basis of logic that something exists in the real world. In the example originally provided, for example, you might try to use logic to prove that god does, or does not, exist.

I am referring to your claim about logic not being able to prove anything about the world. The Law of the Excuded Middle applies to the world, hence your claim is false.

Now you seem to be changing what you were asking for. Now you want to a proof that not only applies to the actual world, but a proof that something exists in the actual world. Not all claims that describe the actual world are claims concerning what exists in the actual world. For instance, the claim "For all X, X is identical to itself", describes every world, including the actual, but it is silent on what exists in the actual world.

I will provide a proof that shows that something exists (though it'll involve quantifiers, which I'm not sure you are familiar with), if you admit that the proof I provided earlier describes the actual world. I'll not play a game with you wherein I refute one of your claim just to have you aver that you were claiming something else all along.

As far as a proof of God goes, Kant already proved it is impossible to proof that God exists, and impossible to prove that God doesn't exist.

 

[cogito]

Cogito, it is certainly the case that we are using two different definitions of the word "premise". In the logic textbook to which I am referring (Logic by Robert Baum), the term "premise" is used to describe not only particular statements, but also schemata whose truth values are unspecified. If you are using the term "premise" to refer exclusively to statements whose truth values are determinate, then that's fine. I just want to get it straight that that's what you mean.

No, I am not using 'premise' to refer only to statements with determinate truth values. A premise may have a indeterminate truth value (e.g., the measurement will be "spin-up", Tom will drink a beer tomorrow, etc.). A hypothetical supposition may have a determinate truth value (e.g. (P & ~P)). The distinction I'm drawing is a different one. Premises are suppositions that are not discharged in the course of the proof. The conclusion relies on them in that it is the assumption of their truth that allows the derivation of the conclusion. When a conclusion is derived from a hypothetical supposition that is discharged, it means that regardless of the truth of the hypothetical supposition, the conclusion follows. It means that it doesn't matter if the hypothetical supposition is actually true, because the conclusion doesn't rely for its justification on supposition's being true. In your logic book, look up the use of subproofs, or the use of conditional proofs, and let me know what you find. It could be that this is just a difference of terminology.

Well, they certainly do not come from the empty set of premises[/color].

Your initial claim was:

"Formal Logic is sufficient for proving all theorems, which by definition can be proven from the empty set of premises[/color]."

You were asked for an example, but you have not delivered.

Since they are not premises, the proof follows from no premises. Hence, the proof follows from the empty set of premises. No premises = Empty set of premises. Of course suppositions were used, but these were discharged. Rules of inference were used as well, but these aren't premises, and claiming that they are leads to Carroll's Paradox, which leads immediately to wholesale skepticism.

Doesn't it? Your conclusion is certainly analytically true, but had you not started with a prem---...errr...hypothetical supposition that was analytically false[/color], you would not have arrived at that conclusion.

Yep, it's called an indirect proof, or reductio ad absurdum. You can't work an indirect proof from no premises unless you start with the negation of a theorem.

 

[Les Sleeth]

And, again, that claim is false. The Law of the Excluded middle applies to external reality. Hence, a proof of the Law of the Excluded Middle proves something about external reality.

First, thanks for the logic demonstration, I reread everything you wrote, saw your point, and realized I regularly rely on hypothetical suppositions. Good stuff.

I'll nitpick on your above point however regarding what logic tells us about external reality. Wouldn't you say that to recognize A is either A or not A is a general truth about external reality derived from experience? If we were mere brains in a jar, just exchanging ideas and never experiencing external reality, then we really would not know if Tom could both be 6 feet and not 6 feet. As far as we would know, Tom might simultaneously exist in two modes which walk around at 6' and 4' so that Tom really is both 6' and not 6' (and how about Schrödinger's cat? ).

So logic hasn't revealed anything by itself about external reality; generalized experience first established how reality works, and from that we inferred a general principle of logic which we can use without looking at reality again, or so we assume.

The reason we can assume that relates back to my original post about logic being order embodied in the conscious process of reason, and a product of our consciousness of reality's order. If we weren't able to count on reality operating consistently, then we also couldn't assume order would prevail and that logic could be relied on without fresh observation.

 

[Prometheus]

I am referring to your claim about logic not being able to prove anything about the world. The Law of the Excuded Middle applies to the world, hence your claim is false.

My original point was that logic could not be used to prove existence in the real world, such as the existence of god. I would also accept the existence of trees or whatever. You jumped in the middle of this thread with your statements, leading off on a tangent. I accept you arguments about logic having rules that describe the structure of logic.

For instance, the claim "For all X, X is identical to itself", describes every world, including the actual

Using the logic that you describe, I will accept "For all X, X is identical to itself" as being true. However, I still completely fail to find any real world value to this statement. Are you saying that this statement by itself proves something about the real world, or are you saying that you can substitute X with a substitution that will prove something about the real world? I am not sure what you suggest that this statement proves, and as I see it, it proves nothing.

I will provide a proof that shows that something exists (though it'll involve quantifiers, which I'm not sure you are familiar with), if you admit that the proof I provided earlier describes the actual world.

No to both.

As far as a proof of God goes, Kant already proved it is impossible to proof that God exists, and impossible to prove that God doesn't exist.

I do not believe you. Kant cannot have proven this. You are leaving out the context for his proof in your statement.

 

[XxFREEofFILTHxX]

If we need axiomic requirements for logic, then if axioms are based on false asumptions then what good is logic? Moreover, how do we know if axioms are true, and how do you prove them to be true, or what system do we use to back these asumptions up with? you would have to base it off previous axioms and logicical theories which derived from axioms and logic itself. Because it was clearly said you can't base things off nothing, so what did reason start with?
so what i mean to ask is what is the first asumption which we base our entire system off of?

 

[Prometheus]

If we need axiomic requirements for logic,

Axioms are not only not required for logic, they are not appropriate. It is for this reason that logic is not appropriate to the real world. Logic is a method of establishing relationships. In order to map this to the real world, axioms are applied. The appropriateness of the logic is then dependent upon the value of the axioms. It is for this reason that logic cannot be used to prove the existence of god, for example, because the axioms cannot be proven. Logic can only say that if we accept such and such as axiomatic, we can then use logic to "prove" that god does/does not exist.

if axioms are based on false asumptions then what good is logic?

Logic is about structure, not about content.

Moreover, how do we know if axioms are true, and how do you prove them to be true,

By defiinition, we don't and we don't.

 

[Les Sleeth]

I do not believe you. Kant cannot have proven this. You are leaving out the context for his proof in your statement.

Whether you accept Kant's argument or not, it has been well demonstrated that trying to use logic to confirm or disprove the existence of God is a colossal waste of time. It is the same problem found with two people arguing if a mountain exists which neither of them have seen. The argument inevitably become circular, and nothing is ever resolved until each goes to where the mountain is supposed to be and looks. And then guess what, the argument ends instantly. Either the mountain is there or the mountain isn't. This is exactly what those thinkers realized who started advocating empiricism. Experience and know, that's what we've discovered works.

 

[Les Sleeth]

Because it was clearly said you can't base things off nothing, so what did reason start with?
so what i mean to ask is what is the first asumption which we base our entire system off of?

I say it is experience. That is the source of everything real (i.e., conceptually corresponding to reality) we are reasoning with. Think about it, if you had absolutely NO experience of reality . . . no sights, sounds, smells, feelings . . . and were confined strictly to what you could think, you'd have noticed no trends from which you could generalize to form principles. (Actually we have evidence of this in children who were deprived of experience from being locked in a closet, or something similar.)

 

[Prometheus]

Whether you accept Kant's argument or not, it has been well demonstrated that trying to use logic to confirm or disprove the existence of God is a colossal waste of time.

I never said that I have any problem with Kant's argument. I quite agree that logic cannot prove the existence of god. In fact, I made this very point in my first post of this thread.

What I did not buy is cogito's phraseology, wherein he claimed that Kant made a proof without including the context, which you supplied, of logic.

 

[SlySpy]

How could you refute the law of non-contradiction

And what about this syllogism:

1. An omnipotent being exists
2. This being can create a rock heavier than it can lift (from 1)
3. This being can lift any rock it creates (from 1)
Cl. An omnipotent being cannot exist (from 2 and 3 via contradiction)

This way, by supposing a proposition that does not have to relate to the observed world is it not possible to formulate a negative conclusion about the world?

 

[wuliheron]

How could you refute the law of non-contradiction

And what about this syllogism:

1. An omnipotent being exists
2. This being can create a rock heavier than it can lift (from 1)
3. This being can lift any rock it creates (from 1)
Cl. An omnipotent being cannot exist (from 2 and 3 via contradiction)

This way, by supposing a proposition that does not have to relate to the observed world is it not possible to formulate a negative conclusion about the world?

Check out Fuzzy Logics and Neutrosophy. Fuzzy logic allows for multiple truth values and has proven applications in the real world. Neutrosophy adds a third category of Indeterminate.

 

[Les Sleeth]

How could you refute the law of non-contradiction

And what about this syllogism:

1. An omnipotent being exists
2. This being can create a rock heavier than it can lift (from 1)
3. This being can lift any rock it creates (from 1)
Cl. An omnipotent being cannot exist (from 2 and 3 via contradiction)

This way, by supposing a proposition that does not have to relate to the observed world is it not possible to formulate a negative conclusion about the world?

While I wrote my answer, I see Wuli said some of it, but here it is anyway.

I don't want to challenge the law of non-contradiction, but I will challenge the assumptions used in that syllogism, which I've seen many times, to show how while you’ve demonstrated impeccable logic, you’ve said nothing conclusive about the “world” other than what the syllogism assumes in the first place (the truth of non-contradiction).

To be omnipotent means to be in possession of all the power there is. However, it doesn't tell us if there is a finite or an infinite amount of power to be in possession of; also, all-powerful doesn't mean “omni-capable,” i.e., that the omnipotent being can do anything it wants (analogously, a powerful weightlifter isn't necessarily intelligent).

We know a lot of "power" is packed into matter, so it follows that the omnipotent being uses power to create the rock. If the pool of power being drawn from is finite, then the rock could get so big at some point that the power used up creating the rock doesn't leave enough for lifting, and so an omnipotent being in a finite power pool could create a rock that was not liftable. If, on the other hand, the pool of power being drawn from is infinite, then the rock could never get so big that there wasn't enough power left to lift the rock, and in that case the omnipotent being could not create a rock that was not liftable.

Thus we can see that we can’t draw conclusions about reality without sufficient facts, and facts are given to us by experience. As I pointed out before, we couldn’t even assume non-contradiction if we’d not observed reality behaving that way.

 

[SlySpy]

Check out Fuzzy Logics and Neutrosophy. Fuzzy logic allows for multiple truth values and has proven applications in the real world. Neutrosophy adds a third category of Indeterminate.

I am aware of fuzzy logic, but I'll have to look into neutrosophy. And even then, wouldn't an argument refuting the law of non-contradiction require that very law? If you could, can you show me an example where this can be done without relying on the law of non-contradiction (to be consistent.)

 

[wuliheron]

I am aware of fuzzy logic, but I'll have to look into neutrosophy. And even then, wouldn't an argument refuting the law of non-contradiction require that very law? If you could, can you show me an example where this can be done without relying on the law of non-contradiction (to be consistent.)

Every type of logic is based on reductio ad absurdium, reduction to the absurd. Even the law of noncontradiction is based on this principle. However, at one time people considered the idea that the Earth is round absurd. The only way around this problem is to use emperical evidence to prove or disprove such logical axioms.

There are numerous examples of logics that work in the real world, and all of them incorporate variations on the law of noncontradiction. As occurred with the advent of quantum mechanics we can protest all we want and claim it makes no sense whatsoever, but there will always be Neil's Bhor around to remind us to "Shut up and calculate".