The Limits of Reason: Exploring the Rationality of Logic

[Les Sleeth]

What's "Given"

Originally posted by drag
I'd like to point out that we are complex
biological machines and we are born adapted to
this world. It would be strange to assume that
the adaptations are only regarding our physical
traits - we have complicated brains that are
also adapted to this Universe. For example,
brains of our size could be created with just
the preferable adaptive state to deal primarily
with chess games - like a separate computer program.
Of course, our brains appear to differ from computer
programs because we appear to be able to include
and learn new possibilities but we still have
some basic processing principles "set in".

I am not sure I understand your point, but I think it is that it's not just the ability to feel that is built into us, but also some reasoning ability as well; and that beyond the "test" of trust I gave for reason (which is dependent on external interaction with reality) there should be internal elements that tell us something about reason as well.

That's a very tough question to ponder. I've been trying to remember my first cognitive moments, my first awareness of "me." One I have I don't know if it's a dream or not, but it is such a vivid flash of memory I've always suspected that it's real. The memory is of being held by someone in the delivery room and seeing a woman (my mother I assume) in a prone position, and a dark man with a thick, black moustache. Later in life I asked my mom what her doctor looked like, and she described the man I saw! I have other flashes too, meaningless events, but things I remember as an infant.

Anyway, in terms of my reasoning ability, I don’t remember any thoughts as an infant. I mostly remember being interested, fascinated, happy. It is really difficult to say whether I ever would have learned to reason properly if the external world weren’t there with which to interact. In fact, some of the worst thinkers at this site seem to be those who don’t pay enough attention to external circumstances. I suspect the order of the universe plays an important role in teaching a human how to use the brain. What I find very interesting is that the part of me which needed no “outside” training to be intrigued, happy, fascinated, etc., is still there inside me. And as much as I enjoy using my brain in all the various thinking opportunities of life, I value that “given” inner part even more.

 

[Mentat]

Originally posted by Iacchus32
And yet, if there was no idea (conceived of the mind = essence) in the first place, there would be nothing concrete to "brag about" in the second place.

This assumption is false, and is leading you toward a (probably) wrong conclusion. If there is no god, then the universe (and everything within it, obviously) came about without a thought. Action can take place without thought/imagination/planning/conscious awareness/etc...

Which is very interesting (credit to Lifegazer), for it suggests our whole notion of material existence is brought about by abstract thought (or, as Tom would say, the application thereof) and, since we all live in the world collectively (or so materialists claim), then we all must be part of the same "collective mind" as a whole ... Only question is, whose mind is it?

And this is probably lifegazer's problem too. Let me repeat, a thought is not required, for an action to take place.

In which case let me restate what I said to Tom:

I guess this has something to do with you telling me in the other thread that the "idea of God" was abstract and that nothing would become of it. And yet, what I'm telling you is that this whole world is built upon nothing "but" abstractions. Therefore it all must have begun with a single "axiom" or idea. Based upon the idea of God perhaps?

So you see that's the whole point, because if God does exist, then this becomes the axiom (idea) by which everything (materially) becomes manifest.

But this conclusion is based on the aforementioned false premise.

 

[wuliheron]

Originally posted by drag

wuliheron, you have an interesting point,
but don't we have the criteria of
apparent consequences to compare our basic
abstract ideas to ? You can't fully trust
anything, but I do not see how the apparent
consequenses of our abstract thought justify
an opinion that says something like - "our logic
is merely a function of physiological and
phsycological evolution (biology in other words)
and is not the result of our attempt to grasp
reality". After all, what is the evolutionary
advantage of creatures with thought "frames" that
are disconnected from reality ?

The hidden assumption in your statement is that merely reacting to nature or reality confirs a survival advantage. For ants it might, but we are much more complex than that. In particular, we are much more adaptable than ants and inhabited every continent on the planet before agraculture was invented. The map is not the territory, but maps are still incredibly useful no matter how far divorced from reality they might be in many respects.

In a very real sense, our specialty is pretense or the suspension of disbelief. Lying is an artform requiring method acting, we immerse ourselves in our suspension of disbelief while still retaining the ability to directly connect to reality if necessary. In order to do this we must first be capable of lying to ourselves.

As children we all play make-believe, but as we grow older we begin to treat our make-believe more seriously and call it fact. Thus we are incredibly adaptable as animals go. Faced with a life or death situation, however, our brains shut down all the pretense and we react with incredible speed.

 

[Iacchus32]

Originally posted by Mentat
And this is probably lifegazer's problem too. Let me repeat, a thought is not required, for an action to take place.

Are you referring to the possibility that everything we see around us which is man made just sprang up aribitrarily and at random? This is not possible.

If on the other hand, we were to take mankind out of the picture, thus leaving only the natural world, I would say the likelihood of things happening at random are more plausible. Yet even here, everything seems to have its own rhythm, and behaves in accord with its "own season," suggesting that things really don't happen arbitrarily (not as rule).

Whereas with mankind (at least in the west), it's all about "control" over the environment which, requires a tremendous amount of thought and effort.

 

[quantumdude]

Originally posted by LW Sleeth
You may assume too much. A child is born, and if emotionally unhampered, knows a priori how to smile and be fascinated. Since joy and interest are part of reality, it seems there is a priori knowledge of reality. Possibly it would be more accurate to say we cannot know anything external to ourselves a priori.

Yes, that's what I meant.

Both points seem true, but maybe the issue is a chicken-egg sort of thing. All humanly-created objects are preceded by reason, yet an idea in this universe would never come to fruition without physical effort.

I don't see any chicken-egg issue. A building derives its concreteness from the materials from which it is made, not because a group of men acted on a thought.

Nonetheless, I can't see how it can be denied that we can have creative thought without action, but no creative action without thought.

Of course, it can't be denied.

 

[quantumdude]

Originally posted by Iacchus32
So "abstract" is equal to idealism, and "concrete" is equal to materialism.

Not at all. Everyone acknowledges a difference between things in one's personal imagination and things outside of it, even if one does regard the outside world as "in the imagination of god".

And yet, if there was no idea (conceived of the mind = essence) in the first place, there would be nothing concrete to "brag about" in the second place.

You keep repeating this, and I keep acknowledging it.

Which is very interesting (credit to Lifegazer),

Please. That is hardly his idea.

for it suggests our whole notion of material existence is brought about by abstract thought (or, as Tom would say, the application thereof)

You have to be kind of careful here, because a "notion" of anything, including a material world, is abstract. A notion is just a thought. I say "careful" because you are about to try to derive a consequence on the actual world from the notion of it, and I don't think that is warranted.

and, since we all live in the world collectively (or so materialists claim), then we all must be part of the same "collective mind" as a whole ... Only question is, whose mind is it?

This is where you go looney, and it is because you have made the jump from "notion of material existence" (which is abstract and in the mind) to "actual material existence" (which is taken as concrete) to conclude that we are all of one mind.

Just from reading your posts, I can assure you that we are not.

So tell me, what's the difference between a concrete idea and a "solid idea?" Say like 1 + 1 = 2?

You're still confused: I already told you that there is no such thing as a "concrete idea". Mental constructs are not concrete.

Is this what an axiom is? (I just looked up axiom in the dictionary for the first time by the way.)

That's not an axiom of arithmetic (the axioms are in ring theory), but it is an abstract notion, which is the only thing that is really important to this part of the discussion.

So you see that's the whole point, because if God does exist, then this becomes the axiom (idea) by which everything (materially) becomes manifest.

That's a big "if", and it's one that can't be proven, observed, or obtained by introspection. Since those are the only ways I can think of to obtain knowledge, I see no point in contemplating that "if".

 

[quantumdude]

I'm going to ask that we forget about god in this thread. It really has nothing to do with the topic.

Continuing along this line...

Originally posted by Tom
First, the prescriptive laws of reasoning (aka logic) cannot be proven "right" within the system of logic itself.
Second, all arguments rely on unproven axioms (aka assumptions).

All systems of logic can be put into one of two categories:

1. Deductive
2. Inductive

I explained all this in detail in my Logic Notes thread, but let me give a rundown here.

Deductive Logic
An argument is deductive if its premises necessarily imply its conclusions. With a mandate to construct such a system of logic, one is led directly to a formal structural language that strongly resembles mathematics. It contains rules for types of inferences that can always be trusted. This should not be misunderstood to mean that deductive logic can be used to derive absolute truths about reality. In fact, deductive logic is completely silent in this regard. It should be understood as follows:

I may not know whether the premises are correct, but I do know for certain that: If the premises are true, then the conclusion must be true.

That conditional statement expresses the only idea of which we can be confident using only deductive logic. Deductive logic does not contain a procedure for testing the truth or falsity of propositions (except for some propositions about deductive logic, of course).

Inductive Logic
An argument that is not deductively valid is inductive. The premises of an inductive argument provide only partial support for its conclusion, and as such the conclusions of inductive arguments are accepted only tentatively. This may prompt one to ask, "Why bother with inductive logic?" Good question. The answer is that it is impossible to reason about anything that cannot be known a priori without inductive logic. So, the price we pay for inductive reasoning may be the lack of absolute support for the conclusion, but the benefit is that we obtain the ability to say something meaningful about reality. In other words, inductive logic provides a means to judge the truth or falsity of propositions, but only in a probable (as opposed to absolute) sense.

The discipline of implementing these two kinds of reasoning to learn about reality is called science.

 

[wuliheron]

In turn, Tom, the science of logic can be described as both an epistemological and ontological pursuit depending upon the context. Is science or logic discovering reality, creating reality, describing reality, all the above, or none of the above? It just depends upon the context.

Hence, I would dispute your assertions that:

[We can know for ] certain that: If the premises are true, then the conclusion must be true. [using deductive logic]

This tautological assertion denies the existence of genuine paradoxes and the indeterminate which can neither be said to be true or false, but whose existence can be logically demonstrated. Therefore I would modify this statement as valid only within the context logic itself and with the assumption that paradoxes and the indeterminate are axiomatically false.

 

[quantumdude]

Originally posted by wuliheron
In turn, Tom, the science of logic can be described as both an epistemological and ontological pursuit depending upon the context. Is science or logic discovering reality, creating reality, describing reality, all the above, or none of the above? It just depends upon the context.

I would say that science describes what we observe. Is that reality? I don't know, but it's the only thing I have access to.

Hence, I would dispute your assertions that:

[We can know for ] certain that: If the premises are true, then the conclusion must be true. [using deductive logic]

This tautological assertion denies the existence of genuine paradoxes and the indeterminate which can neither be said to be true or false, but whose existence can be logically demonstrated.

Actually, the above description does allow for the paradoxes of logic. Remember that a paradox is a type of statement to which we cannot assign a truth value. When I say, "If the premises are true..." I am not ruling paradoxes out, I am simply narrowing the scope of the condtional to exclude them.

Note that the scope is also set to exclude false statements, but it does not rule them out, either.

For a complete description of deductive validity, we could say:

When an inference is deductively valid, then
1. If the premises are true, then the conclusion must be true.
2. If at least one of the premises is false, then the conclusion must be false.
3. If at least one of the premises is undecidable (aka paradoxical), then the conclusion is undecidable.

 

[wuliheron]

Originally posted by Tom
I would say that science describes what we observe. Is that reality? I don't know, but it's the only thing I have access to.

In other words, it becomes an epistomological or ontological pursuit depending upon the context.

Actually, the above description does allow for the paradoxes of logic. Remember that a paradox is a type of statement to which we cannot assign a truth value. When I say, "If the premises are true..." I am not ruling paradoxes out, I am simply narrowing the scope of the condtional to exclude them.

Note that the scope is also set to exclude false statements, but it does not rule them out, either.

For a complete description of deductive validity, we could say:

When an inference is deductively valid, then

1. If the premises are true, then the conclusion must be true.
2. If at least one of the premises is false, then the conclusion must be false.
3. If at least one of the premises is undecidable (aka paradoxical), then the conclusion is undecidable.

Because a truth value cannot be assigned does not mean paradoxes don't possesses a truth value which classical logic simply cannot determine. Therefore the argument is tautological and the clearest way out of the tautology is to create a new axiom.

 

[Iacchus32]

Originally posted by Tom
This is where you go looney, and it is because you have made the jump from "notion of material existence" (which is abstract and in the mind) to "actual material existence" (which is taken as concrete) to conclude that we are all of one mind.

You're the one who brought up the Problem of Other Minds in the other thread, while it was also emphasized (by both you and heusdens I believe) that by sharing our experiences, we can be more conclusive about their actual existence, if in fact we all concluded the same thing. In other words we're speaking about a "collective experience." Whereas what could that possibly mean if we all didn't share the "same ideals?"

Consider "the entity" of the United States government, which rules over the entire nation. Couldn't this be perceived as the "one mind" which consolidates the interests of the entire nation? Or, if we were to say, "We the people of the United States," aren't we referring to a "collective experience" under one "collective mind?" Me thinks so.

 

[ahrkron]

Originally posted by wuliheron
Because a truth value cannot be assigned does not mean paradoxes don't possesses a truth value which classical logic simply cannot determine. Therefore the argument is tautological and the clearest way out of the tautology is to create a new axiom. [/B]

What argument are you referring to?
I don't see any tautological argument there.

 

[wuliheron]

Originally posted by ahrkron
What argument are you referring to?
I don't see any tautological argument there.

The tautology is that the argument Tom put forth that If the premises are true, then the conclusion must be true. is based on the the definition of truth which Aristotle first established using reductio ad absurdum. In other words, using the concept of the absurd Aristotle defined the absurd as definitively false and then went on to declare that everything must be either true or false, which rules out the possibility of genuinely self-referential and self-contradictory paradox to which classical logic cannot assign any truth value.


Actually, I woke up this morning with the solution to this connundrum in my mind. Instead of creating a new axiom to compound the tautology, it is more elegant to take out the word "must" and replace it with a vague conditional term like "may". This also removes the essentially black and white fundamentalist charcter of classical logic and places it the category of paraconsistent logic.

 

[ahrkron]

Originally posted by wuliheron
The tautology is that the argument Tom put forth that If the premises are true, then the conclusion must be true. is based on the the definition of truth which Aristotle first established using reductio ad absurdum.

No, actually, the statement is independent of how you define "truth".

Also, the statement you are referring to is not an argument, but only one of the three conditions for a deductive argument (that contains the mentioned premises) to be valid.

btw, why do you say that Aristotle defined truth using reduction ad absurdum (sp?)? I do not know much about how (and if) he tried to define "truth". Can you provide a reference?

My impresion is that he may have used reduction ad absurdum in some arguments, but that is much different than defining truth itself.

Also, the very technique of reductio ad absurdum relies on the assumption that, when you have a self-cotradictory statement, it has to come from either a faulty reasoning or a false hypothesis, which means that it rules out contradictions as valid end points of a deductive argument.

Actually, I woke up this morning with the solution to this connundrum in my mind. Instead of creating a new axiom to compound the tautology, it is more elegant to take out the word "must" and replace it with a vague conditional term like "may".

I take it you are referring to the definition of validity given by Tom. I don't think this is a good solution, since the concept of validity is extremely helpful as it is. Also, when you use "may" instead of "must", you need to supply a way to decide if it is indeed the case that the resulting conclusion "may" be true, which means you need to have the stronger version anyway.

This also removes the essentially black and white fundamentalist charcter of classical logic and places it the category of paraconsistent logic.

I don't think it has to do with paraconsistent logic. Maybe with modal logic (in which the "degree of credibility" of statements is added to the description).

 

[Mentat]

Originally posted by Iacchus32
Are you referring to the possibility that everything we see around us which is man made just sprang up aribitrarily and at random? This is not possible.

If on the other hand, we were to take mankind out of the picture, thus leaving only the natural world, I would say the likelihood of things happening at random are more plausible. Yet even here, everything seems to have its own rhythm, and behaves in accord with its "own season," suggesting that things really don't happen arbitrarily (not as rule).

Whereas with mankind (at least in the west), it's all about "control" over the environment which, requires a tremendous amount of thought and effort.

No, no, I wasn't saying that all man-made things were in fact just random occurances. I was merely illustrating the fact that not all actions are thought-out first. Some actions don't require any thought at all.

 

[Mentat]

Originally posted by Tom
I'm going to ask that we forget about god in this thread. It really has nothing to do with the topic.

Continuing along this line...



All systems of logic can be put into one of two categories:

1. Deductive
2. Inductive

I explained all this in detail in my Logic Notes thread, but let me give a rundown here.

Deductive Logic
An argument is deductive if its premises necessarily imply its conclusions. With a mandate to construct such a system of logic, one is led directly to a formal structural language that strongly resembles mathematics. It contains rules for types of inferences that can always be trusted. This should not be misunderstood to mean that deductive logic can be used to derive absolute truths about reality. In fact, deductive logic is completely silent in this regard. It should be understood as follows:

I may not know whether the premises are correct, but I do know for certain that: If the premises are true, then the conclusion must be true.

That conditional statement expresses the only idea of which we can be confident using only deductive logic. Deductive logic does not contain a procedure for testing the truth or falsity of propositions (except for some propositions about deductive logic, of course).

Inductive Logic
An argument that is not deductively valid is inductive. The premises of an inductive argument provide only partial support for its conclusion, and as such the conclusions of inductive arguments are accepted only tentatively. This may prompt one to ask, "Why bother with inductive logic?" Good question. The answer is that it is impossible to reason about anything that cannot be known a priori without inductive logic. So, the price we pay for inductive reasoning may be the lack of absolute support for the conclusion, but the benefit is that we obtain the ability to say something meaningful about reality. In other words, inductive logic provides a means to judge the truth or falsity of propositions, but only in a probable (as opposed to absolute) sense.

The discipline of implementing these two kinds of reasoning to learn about reality is called science.

Isn't there a paradox, when it comes to deductive logic (as you've defined it)? To say that the acceptance of premises A and B means that you must accept Z, is to create another premise (premise C), which states: If you accept A and B, you must accept Z. But, let's say that I were to accept premise C, what then? Well, there is no premise that says that by accepting A, B, and C I must accept Z, so we make one (premise D): If you accept A, B, and C you must accept Z. However, there is no premise that says that accepting A, B, C, and D means I must accept Z...and so on, and so on, forever.

I guess Deductive Logic still works, but there is that little paradox, isn't there?

 

[Mentat]

Oops, I see the paradox has already been brought up. Sorry about that.

 

[drag]

Greetings !

Mentat,
I believe that if you read Tom's second part -
about inductive logic, you'll notice he says
that inductive logic is probable - never
certain.

Tom,
I believe that what wuliheron basicly means,
and I'm referring to your enitial "short"
definition version, is that not only should you
recognize the probable nature of inductive logic,
like you said, but also that you can have different
rules (different types of logic) when it comes
to your discription of deductive logic.
This also means that some things may be true/false/
indeterminate for the same sets of premises.
There should, theoreticly, be no limmit to
the possible range of such approaches.

Originally posted by LW Sleeth
That's a very tough question to ponder. I've
been trying to remember my first cognitive moments,
my first awareness of "me." One I have I don't
know if it's a dream or not, but it is such a
vivid flash of memory I've always suspected that
it's real. The memory is of being held by someone
in the delivery room and seeing a woman (my mother
I assume) in a prone position, and a dark man
with a thick, black moustache. Later in life I
asked my mom what her doctor looked like, and
she described the man I saw! I have other
flashes too, meaningless events, but things I
remember as an infant.

You actualy remember yourself when you were
born ?! Com'mon !

Live long and prosper.

 

[Les Sleeth]

Originally posted by drag
You actualy remember yourself when you were
born ?! Com'mon !

I don't know, I could have dreamed it too. My point was that I believe I remember early moments in my life where I wasn't thinking, and that another apparently inborn cognitive part of me was functioning just fine.

 

[wuliheron]

Originally posted by ahrkron

No, actually, the statement is independent of how you define "truth".

The statement itself, If the premises are true, then the conclusion must be true, explicitely defines truth as having true premises. Furthermore, it implicitely defines truth as either applying to everything or not being that which is false.

Also, the statement you are referring to is not an argument, but only one of the three conditions for a deductive argument (that contains the mentioned premises) to be valid.

btw, why do you say that Aristotle defined truth using reduction ad absurdum (sp?)? I do not know much about how (and if) he tried to define "truth". Can you provide a reference?

Sorry, I don't have any links to the subject. Aristotle incorporated the Law of the Excluded Middle and the Law of Noncontradiction as the foundations of his logic. Each of these he demonstrated using reductio ad absurdum which can also be derived from both principles.

This was a significant deviation from other logics at the time like that of Heraclitus which either implied everything was ultimately true or in Zeno's case, absurd. All of these logics, however, were ultimately based on the use of reductio ad absurdum. The situation was analogous to a bunch of politicians today arguing that each other's views are more absurd.

Aristotle's logic became the first to take a fundamentalist stance of everything either being true or false. It is widely hailed as the first formal logic, but has been described as a blunt instrument leading to authoritarianism and is not considered very applicable in modern science, but for the time was revolutionary.

Also, the very technique of reductio ad absurdum relies on the assumption that, when you have a self-cotradictory statement, it has to come from either a faulty reasoning or a false hypothesis, which means that it rules out contradictions as valid end points of a deductive argument.

No, that is the law of the noncontradiction from which reductio ad absurdum can be derived. Zeno of Elias was the first to highlight the extreme use of reductio ad absurdum in the west. He argued that the universe is indivisible, indestructable, immortal, and unchanging. His famous paradoxes he proposed as proof that his own beliefs that change and motion are impossible are no more absurd than anyone else's beliefs.

I take it you are referring to the definition of validity given by Tom. I don't think this is a good solution, since the concept of validity is extremely helpful as it is. Also, when you use "may" instead of "must", you need to supply a way to decide if it is indeed the case that the resulting conclusion "may" be true, which means you need to have the stronger version anyway.

You are correct in assuming the classical definition Tom espouses is useful, but today a great deal of attention is focused on creating many valued logics, the simplest of which assert there is truth, falsehood, and the indeterminate, infinite, or synergistic product of truth/falsehood and the most complex being an infinitely valued logic. The more advanced versions are statistical in nature and incorporate nested matricies. Despite any absurdities they may present from a classical view, they have proven incredibly useful for hundreds of years now.

Two valued logics reflect the way the human mind works but Aristotlian logic is built on denial of its own fundamental assertions. What it leaves out is pointedly the more qualitative human attitudes and affects that define words like absurd. For all these reasons and more the science of logic can be described as a pragmatic science of the absurd, but the art of logic is much more.

I don't think it has to do with paraconsistent logic. Maybe with modal logic (in which the "degree of credibility" of statements is added to the description).

Oh really, why do you think that?

 

[quantumdude]

Originally posted by Iacchus32
You're the one who brought up the Problem of Other Minds in the other thread, while it was also emphasized (by both you and heusdens I believe) that by sharing our experiences, we can be more conclusive about their actual existence, if in fact we all concluded the same thing. In other words we're speaking about a "collective experience." Whereas what could that possibly mean if we all didn't share the "same ideals?"

The similarity of experience comes from the facts that:

1. We are looking at the same universe.
2. We are constructed similarly.

Consider "the entity" of the United States government, which rules over the entire nation. Couldn't this be perceived as the "one mind" which consolidates the interests of the entire nation?

No, it couldn't. If you want proof, just ask one senator what another senator is thinking. He won't be able to tell you, I promise.

Or, if we were to say, "We the people of the United States," aren't we referring to a "collective experience" under one "collective mind?" Me thinks so.

Me thinks not, for the same reason.

 

[ahrkron]

Originally posted by wuliheron The statement itself, If the premises are true, then the conclusion must be true, explicitely defines truth as having true premises.

Not even taken by itself,

but remember that the statement was one of three conditions that, if met, define what a deductive argument is:


When an inference is deductively valid, then
1. If the premises are true, then the conclusion must be true.
2. If at least one of the premises is false, then the conclusion must be false.
3. If at least one of the premises is undecidable (aka paradoxical), then the conclusion is undecidable.


What you are doing is similar to the following:

Tom says "a lung doctor is reliable if: 1. when he says you're ok, your lungs are ok, 2. If he says you're bad, there's something wrong with you, ..."

Then you blame Tom for defining "OK" as having to do with the lungs, when his aim was at the concept of "reliability", and he provided no general criteria for the definition of "OK" (truth).

 

[quantumdude]

Originally posted by wuliheron
The tautology is that the argument Tom put forth that If the premises are true, then the conclusion must be true. is based on the the definition of truth which Aristotle first established using reductio ad absurdum.

OK, I think I see where there could be some confusion. First, the "if...then" is not an argument, but a statement (but that's a minor point). The major point here is that statement is not a tautology. It only makes sense to refer to statement schema (the skeleton of a statement, expressed in terms of variables) as "tautologies".

The schema for my statement is: p-->q, which is not tautological. It would only be so if p=q, which it does not here (p and q have different subjects).

Like Ahkron, I don't know about the reductio ad absurdum definition of truth, and I also don't see how it is important.

In other words, using the concept of the absurd Aristotle defined the absurd as definitively false and then went on to declare that everything must be either true or false, which rules out the possibility of genuinely self-referential and self-contradictory paradox to which classical logic cannot assign any truth value.

We don't have to stick with Aristotle. Propositional logic does indeed accommodate paradoxes. Statements are either true, false, or undecidable.

The statement itself, If the premises are true, then the conclusion must be true, explicitely defines truth as having true premises.

No, it does not define truth at all--It defines deductive validity. The definition of truth has do be done separately.

Furthermore, it implicitely defines truth as either applying to everything or not being that which is false.

No, it doesn't. Like I said, it simply limits the scope of statements being considered to those statements that are true. It does not rule out undecidable propositions at all. In fact, Goedel's work on undecidable propositions was done in this very same propositional logic, which uses the above definition of deductive validity.

 

[quantumdude]

Originally posted by ahrkron
What you are doing is similar to the following:

Tom says "a lung doctor is reliable if: 1. when he says you're ok, your lungs are ok, 2. If he says you're bad, there's something wrong with you, ..."

Then you blame Tom for defining "OK" as having to do with the lungs, when his aim was at the concept of "reliability", and he provided no general criteria for the definition of "OK" (truth).

Yes, that's it.

Here's what I was thinking. Look at the whole process as a computer program.

The first function is a sorting routine that puts statements in one of three sets. Set 1 is for true statements, Set 2 is for false statements, and set 3 is for undecidable statements. In the sorting function is the definition for "true", "false", and "undecidable". The returned results of the function are sorted statements.

The second function is a function for determining deductive validity. It contains as a definition: If the premises are all from Set 1, then the conclusion must also be from Set 1.. The second function does not know or care how the statements were sorted; it only cares what Set they are in.

Hopefully it is clear now that:

1. It is not the case that the second function defines truth.

2. It is not the case that the second function rules out Set 3 statements (aka paradoxes).

 

[wuliheron]

Originally posted by ahrkron

What you are doing is similar to the following:

Tom says "a lung doctor is reliable if: 1. when he says you're ok, your lungs are ok, 2. If he says you're bad, there's something wrong with you, ..."

Then you blame Tom for defining "OK" as having to do with the lungs, when his aim was at the concept of "reliability", and he provided no general criteria for the definition of "OK" (truth).

The other way around, I am debating the validity of his assertions on the basis of semantics and emperical evidence. Read the rest of this post for more details.

Originally posted by Tom

OK, I think I see where there could be some confusion. First, the "if...then" is not an argument, but a statement (but that's a minor point).

For me, it is THE point. Whether you view it as an argument or statement of fact depends upon your point of view. Of course if you take the point of view it is a simple statement of fact it supports classical logic, but then it becomes a tautological argument.

The major point here is that statement is not a tautology. It only makes sense to refer to statement schema (the skeleton of a statement, expressed in terms of variables) as "tautologies".

The schema for my statement is: p-->q, which is not tautological. It would only be so if p=q, which it does not here (p and q have different subjects).

Again, you are using dialectical logic to assert the validity of dialectical logic. You are saying that truth is not falsehood and then attempting to use this fundamental assumption to prove that truth must lead to truth using these rules and falsehood must lead to falsehood.

Like Ahkron, I don't know about the reductio ad absurdum definition of truth, and I also don't see how it is important.

We don't have to stick with Aristotle. Propositional logic does indeed accommodate paradoxes. Statements are either true, false, or undecidable.

As Kurt Godel proved, any system must be based at least in part on axioms that can only be taken on faith. Hence even logical systems are ultimately based on faith as much as anything else and the roots of this faith can be traced in the west to the pervasive use of reductio ad absurdum in ancient Greece.

Faith in the vague concept of the absurd, in both east and west, is the foundation of all logical systems. In order to prove the validity of the foundations of a particular logic, Godel proved using propositional logic, one must go outside of that particular logic and use something else. Because all systems of logic are based on rules and the natural language concept of the absurd, ultimately semantics and emperical evidence are the only systematic means of varifying their validity.

Because no single logic system has yet been devised that is applicable to all we observe, emperical evidence does not support one logic over another. Nor has any semantic or linguistic approach proven universally applicable either. What we can say, however, is that mathematics, logistics, and linguistics are all converging towards some kind of statistically oriented logic or logics as capable of explaining all we observe at least well enough to narrow our definition of the absurd. In physics, this is known as a Theory of Everything.

 

[quantumdude]

Originally posted by wuliheron
For me, it is THE point. Whether you view it as an argument or statement of fact depends upon your point of view. Of course if you take the point of view it is a simple statement of fact it supports classical logic, but then it becomes a tautological argument.

A few points:

First, no matter how you look at it, logical constructions of the form If p then q. are statements and not arguments. An argument--by definition--has at least two statements.

Second, I don't think that there is any such thing as a "tautological argument". Tautologies are types of statements, not arguments.

Third, statements of the form If p then q. are not tautological unless p=q, which is not true in this case.

Again, you are using dialectical logic to assert the validity of dialectical logic. You are saying that truth is not falsehood and then attempting to use this fundamental assumption to prove that truth must lead to truth using these rules and falsehood must lead to falsehood.

No, I am not proving anything. I am defining deductive validity. Also, I am not saying anything about the definition of "truth" or "falsehood".

Ahkron has it right with his analogy, and I go into it further with my 'computer program' explanation. The definition of deductive validity is completely independent of the definition of truth.

As Kurt Godel proved, any system must be based at least in part on axioms that can only be taken on faith. Hence even logical systems are ultimately based on faith as much as anything else and the roots of this faith can be traced in the west to the pervasive use of reductio ad absurdum in ancient Greece.

I understand the implications of Incompleteness, but they have no bearing on the definition of deductive validity, unless you want to prove that logic is "right", which is not what I am doing here.

The rest of your post deals with the question, "How can I know which logic is correct?" which is not what I am getting at here. I was dividing logic into two categories (in response to FZ's question), basically to show that logic cannot be used to get to any "absolute truths" about reality, and as you point out, about itself, either.

 

[wuliheron]

Originally posted by Tom
A few points:

First, no matter how you look at it, logical constructions of the form If p then q. are statements and not arguments. An argument--by definition--has at least two statements.

Correct, an argument has at least two statements. In this case the two statements can be expressed as p is not q and, therefore, if p then q.

Second, I don't think that there is any such thing as a "tautological argument". Tautologies are types of statements, not arguments.

Third, statements of the form If p then q. are not tautological unless p=q, which is not true in this case.

Again, I agree but with the qualification that whether a tautology is a statement or an argument depends on the intent of the individual. Possession is nine tenths of the law, but intent is the last tenth and the hardest to prove.

This goes to the heart of our difference of opinion here. For me, any system of thought or finite definition is suspect of being an argumentative tautology until proven otherwise. Call me extremely skeptical if you want, but this how I think. Of course, this way of thinking in itself is suspect of being an argumentative tautology in its own rite. That's life.

Ahkron has it right with his analogy, and I go into it further with my 'computer program' explanation. The definition of deductive validity is completely independent of the definition of truth.

So... if the concept of validity has nothing to do with truth what does it have to do with? Falsehood? Indeterminacy? Paradox? What good is it to logic if it doesn't apply to these things?

I understand the implications of Incompleteness, but they have no bearing on the definition of deductive validity, unless you want to prove that logic is "right", which is not what I am doing here.

The rest of your post deals with the question, "How can I know which logic is correct?" which is not what I am getting at here. I was dividing logic into two categories (in response to FZ's question), basically to show that logic cannot be used to get to any "absolute truths" about reality, and as you point out, about itself, either.

Not just which logic is correct, but if any logic is correct in any final analysis. Quantum Mechanics implies everything is utterly random when push comes to shove. Logic may simply be maps we use to because they are particularly expedient from our limited viewpoint. This situation implies, paradoxically, that logic may possibly be used to achieve some sort of absolute truth about reality.

In this manner the argument always leads back to its own foundations and to individual interpretation.

 

[RuroumiKenshin]

First of all, I would like to point out this is my first post on this thread and I have not read this thread, so forgive me if I am repeating something.

Logic is:

The study of the principles of reasoning, especially of the structure of propositions as distinguished from their content and of method and validity in deductive reasoning.

A system of reasoning: Aristotle's logic.
A mode of reasoning: By that logic, we should sell the company tomorrow.
The formal, guiding principles of a discipline, school, or science.
Valid reasoning: Your paper lacks the logic to prove your thesis.
The relationship between elements and between an element and the whole in a set of objects, individuals, principles, or events: There's a certain logic to the motion of rush-hour traffic.
Computer Science.
The nonarithmetic operations performed by a computer, such as sorting, comparing, and matching, that involve yes-no decisions

I know the last definition is for computers, but are "yes-no" descisions what the aim of logic is?

logic

n 1: the branch of philosophy that analyzes inference 2: reasoned and reasonable judgment; "it made a certain kind of logic" 3: the principles that guide reasoning within a given field or situation; "economic logic requires it"; "by the logic of war" 4: a system of reasoning [syn: logical system, system of logic]

 

[quantumdude]

Originally posted by wuliheron
Correct, an argument has at least two statements. In this case the two statements can be expressed as p is not q and, therefore, if p then q.

You've lost me.

p=The premises are true.
q=The conclusion must be true.

My compound statement is

If the premises are true then the conclusion must be true.
or If p then q.

Your compound statement is

The premises are true is not the conclusion must be true.

Which is not true, and indeed makes no sense.

Again, I agree but with the qualification that whether a tautology is a statement or an argument depends on the intent of the individual. Possession is nine tenths of the law, but intent is the last tenth and the hardest to prove.

Again, this is not a matter of proof, but of definition.

Definition: A tautology is a statement schema for which it is not possible for the statement to be false, regardless of the truth values of the independent logical variables.

That's what I mean by tautology. If you mean something else, then we aren't speaking the same language.

Tom:
Second, I don't think that there is any such thing as a "tautological argument". Tautologies are types of statements, not arguments.

Third, statements of the form If p then q. are not tautological unless p=q, which is not true in this case.

Wuli:
This goes to the heart of our difference of opinion here. For me, any system of thought or finite definition is suspect of being an argumentative tautology until proven otherwise.

These second point is a matter of definition, which you consistently seem to confuse for a matter of proof. There is no proof or disproof of a definition.

The third point is a matter of proof, and here it is:

If p is true and q is false, then the compound statement p-->q is false. If p is true and q is true, then the compound statement p--q is true. Since the truth value of p-->q is contingent on the truth values of p and q seperately, it is not a tautology.

So... if the concept of validity has nothing to do with truth what does it have to do with? Falsehood? Indeterminacy? Paradox? What good is it to logic if it doesn't apply to these things?

First, the concept of validity has only to do with the structure of an argument. This is what I have been saying all along. When using valid logic: If the premises are true, we can be sure the conclusion is true. Unfortunately, deductive logic does not provide us with the means to determine if the premises are true. That brings me to...

Second, there is a class of logic that applies to truth and falsehood, and that is inductive logic.

 

[quantumdude]

Not just which logic is correct, but if any logic is correct in any final analysis. Quantum Mechanics implies everything is utterly random when push comes to shove.

QM would seem to imply that if one cannot rid oneself of the notion that subatomic particles really do behave like little BB's. But if one can accept that these constituents of matter are really excitations in a quantized field, then there are constraints and symmetries in the theory, and it is readily seen that everything is not utterly random.

I mean, think about it: If "everything is utterly random" were the true implication of QM, there would not be so many courses devoted to it. It would be one 2-minute lecture in which everyone writes those 4 words down in their notebooks and collects their PhD in physics.

Logic may simply be maps we use to because they are particularly expedient from our limited viewpoint. This situation implies, paradoxically, that logic may possibly be used to achieve some sort of absolute truth about reality.

That would be a hard sell, because as I said the only logic that we have for judging the truth or falsity of a proposition is inductive logic, and that logic does not tell us absolutely whether or not a proposition is true.

In this manner the argument always leads back to its own foundations and to individual interpretation.

Well, everyone does have their own interpretation, but it is important for everyone involved to know and agree on the definitions of a subject. Definitions are created to facilitate communication, but when one insists that definitions be justified (as if they were contingent propositions), then communication gets all muddled up.

 

[quantumdude]

Originally posted by MajinVegeta
I know the last definition is for computers, but are "yes-no" descisions what the aim of logic is?

More detailed answers can be found in this thread and in my Logic Notes thread, but roughly speaking:

The aim of deductive logic is to evaluate reasoning.

The aim of inductive logic is to evaluate the truth of a proposition.

 

[wuliheron]

Originally posted by Tom

You've lost me.

Your compound statement is

The premises are true is not the conclusion must be true.

Which is not true, and indeed makes no sense.

Exactly, it makes no sense. I'm saying that the premises may be the conclusion, that these are merely words and the distinction could be entirely artificial and ultimately a tautology.

Again, this is not a matter of proof, but of definition.

Definition: A tautology is a statement schema for which it is not possible for the statement to be false, regardless of the truth values of the independent logical variables.

That's what I mean by tautology. If you mean something else, then we aren't speaking the same language.

Well... yes and no. Here is a good article on general semantics which focuses particularly sharply on this issue:

http://www.general-semantics.org/Basics/SM_logic.shtml

Essentially the argument by modern scientific standards is that for a theory to be considered scientific it must be disprovable. You must be able to falsify the theory. Hence, a tautology by this definition is not scientific. Nor is it logical by the standards of logic when analyzed semantically.

First, the concept of validity has only to do with the structure of an argument. This is what I have been saying all along. When using valid logic: If the premises are true, we can be sure the conclusion is true. Unfortunately, deductive logic does not provide us with the means to determine if the premises are true. That brings me to...

Again, check out the article on semantics. I would argue that this is tautological and that the real purpose of deductive logic is not prove validity of structures, but falsehood. Logic is based on the concept of the absurd, hence the most concise way of describing logic is as a science of the absurd that attempts to prove the falsity of things.

Sorry if this is confusing and distracting you from making your point. However, it is another way of approaching the issues of logic and reason. An approach I might add that is not dependent upon any single kind of logic.

 

[Iacchus32]

Originally posted by Mentat
No, no, I wasn't saying that all man-made things were in fact just random occurances. I was merely illustrating the fact that not all actions are thought-out first. Some actions don't require any thought at all.

If it doesn't require thought then I would say it was a random act. Even so, I don't believe we are here strictly by random. There's too much evidence against it, even if that meant we are here by means of our "own devices." Or, would that be "device-iveness?"

 

[quantumdude]

Originally posted by wuliheron
Exactly, it makes no sense. I'm saying that the premises may be the conclusion, that these are merely words and the distinction could be entirely artificial and ultimately a tautology.

That doesn't help. You said that the two statements:

p-->q
and p is ~q

formed an argument. It is clear to me that the second statement has nothing to do with anything here.

What is your point here? That one can construct nonsensical statements if one wishes? I already knew that!

Well... yes and no. Here is a good article on general semantics which focuses particularly sharply on this issue:

http://www.general-semantics.org/Basics/SM_logic.shtml

Essentially the argument by modern scientific standards is that for a theory to be considered scientific it must be disprovable. You must be able to falsify the theory. Hence, a tautology by this definition is not scientific. Nor is it logical by the standards of logic when analyzed semantically.

I will read the article, but there really is no "yes and no" issue here. A tautology is exactly what I said it is (at least in logic textbooks it is). Also, I am aware that tautologies are not falsifiable, but that really has nothing to do with anything here either. If your point is that "logic is not scientific", then you don't have to convince me because I already knew that. No one claims that logic is science.

But also, we are getting sidetracked because, as I said before, the statement If the premises are true, then the conclusion must be true is not a tautology. That error is carrying through this discussion, and it is getting combined with other errors to make things really hairy. The next mistake is coming up...

Again, check out the article on semantics. I would argue that this is tautological and that the real purpose of deductive logic is not prove validity of structures, but falsehood.

You are confusing the methods of science and deductive logic here. Falsifiability is necessary to science precisely because of the inductive element involved in scientific logic (IOW, no proposition can be proven true absolutely, but it can be proven false absolutely). The inductive element is necessary because we have no a priori understanding of physical reality. Deductive logic, on the other hand, is a mental construct of which we do have a priori understanding. More specifically, deductive logic is that mental construct which was developed for analysis and construction of structurally correct arguments.

That's what it is, and that's what it was designed for. Again, if you disagree with that definition, then we simply ain't talking about the same thing.

 

[Iacchus32]

Originally posted by Tom
The similarity of experience comes from the facts that:

1. Are looking at the same universe.
2. Are constructed similarly.

In other words we're all in this together. Meaning we all share the same universe "collectively." By which we can conclude that we have a similar capacity to reason, "collectively." Of course we may not always agree? ...

No, it couldn't. If you want proof, just ask one senator what another senator is thinking. He won't be able to tell you, I promise.

Then who's running the country? A runaway brain? ... Actually that might be closer to the truth. Hmm...

Me thinks not, for the same reason.

Of course I'm speaking more in a "reprentative sense." Maybe we should confer with the House of Representatives instead?