Reason and Observation

[Tom]

[edit]
I clipped and fused two of my posts from Physics Forums. It's an argument that explains my position of the necessity of observation in determining the truth of premises that refer to existents.
[/edit]

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

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.

If truths about reality are destinations, then deductive reasoning is the car that gets you from one to the other. The argument of this thread boils down to: How far can that car get us?

Is there some limit to the understanding that logic alone can provide? I have answered that question emphatically in the affirmative, on the following grounds:

We have two kinds of logic: deductive and inductive.

The former is concerned with arguments whose premises give absolute support to their conclusions. The problem is that it gives no decision procedure for determining the truth or falsity of propositions with absolute certainty (actually, it's damn near completely silent on the issue).

The latter is concerned with arguments whose premises give probable support to their conclusions. The advantage is that this logic does indeed either lend support to, or outright falsifies, the conclusions that are brought under its analysis.

Since those are the only two kinds of logic at our disposal, I state that absolute truths about reality (known absolutely!) are beyond the capacity of human logic.

 

[Skeptical Greg]

I would agree that nothing can be known absolutely.. But the illusion can be indistinguishable from the real thing.

If reality is an illusion, then our illusion is an illusion.. What can we do?

 

[Tom]

Originally posted by Diogenes
I would agree that nothing can be known absolutely..

I’m glad you said that, because it is precisely what I did not mean.

”Reason-First” People
This argument is directed against the philosophical position of those whom I call “Reason First” people (hereafter, RF). An RF insists on the primacy of (you guessed it) reason in formulating theories about the world. More specifically, they insist on the primacy of reason over observational evidence in determining the truth value of the propositions of such theories. He claims that logic can be used to determine the truth value of a statement, and since logic is certain, then we can have certain knowledge of the world.

Being such a great exponent of reasoning, our RF friend is keen to point out the flaw in the epistemological theory contained in your quote above. He correctly points out that the theory:

P1: Subject S cannot know that p with certainty.

is run aground by the following proposition:

p: Subject S cannot know anything with certainty.

This leads to a paradox, because if S cannot know anything with certainty, then he cannot know that he cannot know anything with certainty, which casts doubt on the theory itself. In other words, this particular theory is its own counterexample!

Our RF friend would conclude that the theory above is logically invalidated, and would find support for his position in that fact. That is, he would say that:

P2: S can know that p with certainty.

follows from the invalidation of the theory.

I would say that our RF friend is not thinking hard enough.

There is a problem with P2. The negation of the statement, “No knowledge can be certain” is not “All knowledge can be certain”. The true negation is, “Some knowledge can be certain.”, which leads to the following epistemology:

~P1: Subject S can know that p with certainty, for at least some p.

My argument aims to make the distinction between those things that can be known with certainty, and those that cannot. I do this by making the distinction between things that are known a priori (mathematics, logic, language, computer programs, and the like) and things that are known a posteriori (rocks, electrons, stars, other persons, and the like). The argument goes further to distinguish between the two types of demonstration of correctness that are unique to each class. Allow me to expand on this point.

A Priori Knowledge
Of the a priori class, I can indeed have certain knowledge. I can know for certain if a calculation has been done correctly, or if a logical argument has been validly derived, or if a mathematical theorem has been correctly demonstrated. I can further know that the conclusions are true, because the premises are true by definition. And since a valid argument which begins from true premises leads to a true conclusion, I conclude that the conclusion is true. This type of demonstration is of the deductive type, and it is certain. In logic, this is what would be called a sound argument and I will define this as a proof.

I would further say that any proven statement is known a priori, since it is entailed by things that are known a priori.

That last part in bold font is important, because I think it is the key to de-paradoxing your above quote.

A Posteriori Knowledge
Of the a posteriori class, I cannot have certain knowledge. I can certainly have a valid argument that leads to a particular conclusion, but since the truth values of my premises cannot be known with certainty, then neither can that of the conclusion. All we can do is make the inductive evidence in support of the premises stronger, and this is done via experiential confirmation. I would define this type of demonstration not proof, but evidence.

Summary of the Distinction
I believe that a great deal of confusion can be cleared up by not mixing up the two standards of demonstration, thereby misapplying them to a class of objects or ideas to which they are inapplicable. In view of my argument, it is nonsensical to ask for a “proof of materialism”, for example, because materialism makes statements about physical objects and processes, which are not known a priori. All we can do is present “evidence for materialism”. One person may decide to accept it, and another may not. Such is the nature of the inductive beast.

In theorizing about existents and their interactions, there are no “proofs”. There are only varying graduations of plausibility, inferred from evidence. Those whom I have encountered who think otherwise seem to believe in some kind of “superlogic”, which has the certainty of deductive validity even when brought to bear on determining the truth value of statements. But in point of fact, no such superlogic exists, and when our RF friend demands a proof of something that is not known a priori, he is making a category error.

Anyway, back to the subject.

Non-Absolutist, Non-Paradoxical Epistemology
I believe that my reconstruction of the theory contained in your quote avoids the paradox by making the above distinctions, and it would read as follows.


P1a. Subject S cannot know that p with certainty, for any item of a posteriori knowledge p.
P1b. Subject S can know that q with certainty, for any item of priori knowledge q.
P2. Statements P1a. and P1b. are items of a priori knowledge.


Goodbye paradox.

But the illusion can be indistinguishable from the real thing.

If reality is an illusion, then our illusion is an illusion.. What can we do?

That's what we're trying to find out here!

edit: fixed some brackets, and filled in an omission

 

[kuroyume0161]

The very short version:

a priori knowledge = formal sets of rules with axiomatic statements as basic building blocks.
a posteriori knowledge = a set of approximate rules inferred from observation in an apparently informal universe.

[Make any necessary corrections as I'm not being nearly rigorous here.]

Heck, even Goedel showed that any formal system is incomplete (there exist propositions in any formal system that can neither be proved nor disproved within that system).

Yes, even after the defeat of Aristotle, people still try to use pure logic to reason about objective reality.

And I agree that this is the problem, especially with certain posters (obviously, you know whom).

Kuroyume

 

[uruk]

Truely a masterpiece Tom. Too bad our RF friend won't understand one bit of it.

 

[Millionframe]

This is very important stuff - thanks for all the info about logic!

 

[Upchurch]

Holy Evaporating Paradox, Batman!

Seriously, Tom, it's a very interesting read. What I gathered in my quick skim was very usefull. I'm going to have to come back to this for a more thurough look.

 

[kuroyume0161]

Same here, Upchurch. I did a cursory skim also, but did a more indepth read of the second part.

Most of this I already knew (but have mostly forgotten over the years). This is basic logic systems explained and some people should have at least this level prior to launching into a "revolution"...

Probably not the definitive or best book on the subject, but "An Introduction to Scientific Research" by E. Bright Wilson, Jr. is a good read on scientific methodology, experimentation, analysis, error quantification, and mathematical/logical support. I haven't read it through completely, but maybe I should.

Kuroyume

 

[Wrath of the Swarm]

Heck, even Goedel showed that any formal system is incomplete (there exist propositions in any formal system that can neither be proved nor disproved within that system).

Any sufficiently powerful system is incomplete, not just any formal system.

 

[Gestahl]

Any sufficiently powerful system is incomplete, not just any formal system.

To be precise, a system powerful enough to make statements about itself... i.e. lexically closed.

 

[Beleth]

I was involved in a very interesting presentation given by the legal department of a rather large company, which pertains to this topic.

It all has to do with what you deduce versus what you discover (or are explicitly told).

The point of the presentation was "Never come out and say that Big Company uses Company X's products, because it could have all sorts of economic and legal implications you don't want to be the source of."

The legal stand is this. You can say that you work at Big Company, and you can say that you have worked with Company X's products, but as long as you leave it at that, and not say that Big Company uses Company X's products, the only link between Big Company and Company X is inferred by the minds of your audience. And from a legal standpoint, that's okay, and no harm is done.

What I'm saying is that it's not only in philosophy and science that there's a difference between reasoning something out and knowing something - there's a legal difference too.

Fascinating posts, Tom. Thank you.

 

[Wrath of the Swarm]

To be precise, a system powerful enough to make statements about itself... i.e. lexically closed.

Specifically, any system of sufficient power to run arithmetic.

 

[Tom]

Heck, even Goedel showed that any formal system is incomplete (there exist propositions in any formal system that can neither be proved nor disproved within that system).

Originally posted by Wrath of the Swarm
Specifically, any system of sufficient power to run arithmetic.

Even more specifically, the three theorems prove that:

1: The Completeness Theorem
First order logic is complete.

2: The First Incompleteness Theorem
Any system that is...

a. Powerful enough to prove each statement in its language.
b. Powerful enough to prove that it proves each statement in its language.
c. Powerful enough to prove the Goedel sentence.

...is inconsistent.

3: The Second Incompleteness Theorem
Any system that is...

a. Powerful enough to prove each statement of the form, "p is provable in this system".
b. Powerful enough to prove that it proves all such statements in (a).
c. Powerful enough to prove the Goedel sentence.

...is inconsistent.

So, Goedel did not just show that sufficiently powerful systems are incomplete, he showed that they are either incomplete or inconsistent.

Incidentally, examples of formal systems that are both consistent and complete are, in addition to first order logic, the Peano arithmetic and Euclidean geometry.

 

[Wrath of the Swarm]

Interesting. I note that's not what Wikipedia says on the matter.
http://en.wikipedia.org/wiki/G%F6del's_incompleteness_theorem

That's essentially what I'd always been told the theorem demonstrates: any sufficiently powerful consistent system cannot prove its own consistency and is thus incomplete.

Perhaps you should log on and change the page, yes?

 

[lifegazer]

First, the prescriptive laws of reasoning (aka logic) cannot be proven "right" within the system of logic itself.

Our ability to reason precedes our logical-constructs.
Our ability to reason is not a logical-construct (a system of logic), since the ability to reason must precede any logical-constructs that are built upon this ability.
I'm not quite sure what you're trying to imply here, but the ability to reason is not a system of logic. Our ability to reason constructs systems of logic.

Second, all arguments rely on unproven axioms (aka assumptions).

Are you stating this as an absolute truth?
If yes, then your axiom is self-defeating because you've just said that all axioms are assumptions.
If no, then your axiom is a belief and not worth the paper it's written on. Either way, your axiom is worthless.

Don't believe everything they teach you Tom. Have the balls to challenge it.

 

[Upchurch]

OMG, did you even read what Tom said, lifegazer? He resolved the very paradox you are attempting to envoke, if you had read more than the first couple of lines.

Please try harder. You absolutely just fell into the steriotype Tom presented; hook, line, and sinker.

 

[lifegazer]

OMG, did you even read what Tom said, lifegazer? He resolved the very paradox you are attempting to envoke, if you had read more than the first couple of lines.

Please try harder. You absolutely just fell into the steriotype Tom presented; hook, line, and sinker.

Did you even read what I said, upchurch?
I destroyed Tom's whole system of logic before he'd written a hundred words of his posts.
If the beginning is incorrect, what hope is there for the end?

 

[Upchurch]

I destroyed Tom's whole system of logic before he'd written a hundred words of his posts.

LOL. Okay, now try reading his second post. The one that talks about the diference between a priori knowledge and a posteriori knowledge. Then talk to us about the "contradiction".

 

[kuroyume0161]

Lifesapper, you really need to read at least ONE book on logic before stepping in it this deeply. This ain't Spock's logic on Star Trek, this is the real formal system which has been in use for thousands of years (since the Greeks developed it).

Tom does not deviate from proper usage or definitions in his posts here. So, all of those deluded mathematicians, logicians, scientists, and anyone else that has used THIS logic in all of this time is incorrect by you, huh?

How do you fit into your house?

Kuroyume

 

[Upchurch]

So, all of those deluded mathematicians, logicians, scientists, and anyone else that has used THIS logic in all of this time is incorrect by you, huh?

Let's make it absolutely clear that this type of formal logic is what is used by philosophers. You know, those same people who lifegazer claims are the only ones capable of exploring the nature of reality.