Truth and logic certainly have an interesting relationship, but what is it? We already use logic to help us arrive at truth, and even to judge when we have arrived at truth, but are they the same thing? In my view, the differences are important, but especially mathematicians and physicists, who have grown very accustomed to relying on logic to find truth, tend to lose track of the differences.

Let's start with definitions. By logic, I just mean a formal system for connecting the truth value of various statements to the assumption of a "true" set of axioms or postulates. What's more, we all (or most) tend to use a similar such formal system, that logicians are careful to elaborate and distinguish in ways that might not be essential for us to track in most cases. By truth, I just mean whatever gibes with our experience. Hence, we have a way of judging truth, it is what "works" in our reality. Further, we should qualify that we are interested here in objective truth, the truth that we can all agree on (or all those who are not mentally disabled in some way). Subjective aspects of truth is another matter that we should probably just steer clear of for the purposes of this thread.

So where's the discussion? The discussion is about the connection between logic and truth. We know that the truth of the theorems will always be contingent on the truth of the axioms, but will they also be contingent on some aspect of the logic? In other words, is there a difference between a logical truth that is connected by logic to a true axiom or postulate, and an actual truth that is judged as true on its own merits, similarly to how the axioms or postulates were judged true in the first place?

Possible issues that might emerge include the liar's paradox ("this statement is not true", http://en.wikipedia.org/wiki/Liar_paradox) and Godel's incompleteness theorem (http://en.wikipedia.org/wiki/G%C3%B6...eness_theorems).

Is the induction axiom a logical truth?

Peano axioms wikipedia article, axiom number 9: "If K is a set such that 0 is in K, and, for every natural number n, if n is in K, then S(n) is in K, then K contains every natural number."

S(n) is the "successor" function so that, for instance, S(S(S(0))) = 2.

Isn't logic just a tool we use to investigate or argue about the truth of statements? I'm not sure how truth can be contingent on the system of logic. Obviously, Godel shows that there are some true statements that can't be proved in a given system of logic (although it isn't clear to me if there are ever any "significant" truths that can't be proved or just specially constructed things such as those Godel used in his proof). But they are still true (by definition).

So is the question (or part of it) whether we can prove those truths in some other way (e.g. in a different logic system which is incomplete in different ways)? Or can we somehow "intuitively" know them to be true?

In which case are they axioms and don't need proving? Although some axioms can be problematical; the classic example being Euclid's paralle postulate. By changing this you end up with different formal systems with their own version of "true" geometry.

It might be more worrying if Godel had also shown that any sufficiently complete system will include "false positives" (prove as true things that aren't).

All axioms are logical truths, as any axiom can prove itself (trivially). I think what you are asking is, is it what I'm calling an actual truth, or a truth by experience. Finding the latter is often the goal of mathematics, and when it is, the hope is that all the axioms are themselves actual truths. Telling if these are actual truths is always a bit tricky, but I'm willing to grant that all the axioms of Peano arithmetic are actual truths, they seem quite "truthy." That may itself be a difficult question, but I'm more interested in how logic connects the actual truth of axioms to the actual truth of other things. (I think you also mean S(S(0)) = 2, but I get your drift.)

Right, you can accept that extraneous statement as another axiom. He showed that you can never "complete the set."If I understand your concern, there, I think you should start worrying.

He showed that a complete system will be inconsistent, that you'll not only be able to prove a statement but also its negative.

Surely you're not asking how to make a proof, I know, but wouldn't the axioms provide the connections too?

Maybe I need an example of what your are concerned about.

I see truth as whatever reality already exists within the universe, regardless of how it came to be, and that is what we are trying to find out. To discover the truth, we use logic and reasoning. Different types of logic are the tools we use while reasoning is the set of all logic we might use to discover some truths. If the conclusions fit the facts as we know them, then we say we have come closer to the truth. If certain forms of logic fit many truths, then we adopt those types of logic into our set of reasoning in order to help us find other truths. That goes for physics and the same goes for mathematics. Different forms of math and types of theorems are the logic and the whole set of mathematics is the reasoning we have at our disposal which is used to discover truths. Some types of logic can be built up upon others as well. That's just the way I see it, anyway.

__________________
Let's put together the pieces of The Grand Puzzle . (website - now revised)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

It would be an example of an actual truth that is not a logical truth, or a logical truth that is not an actual truth. No one has an example in Peano arithmetic (we only know there is either one or the other). We do have an example in common language though, this particular version of the liar's paradox: "this statement is not a logical truth" is an example (I would argue) of an actual truth that is not a logical truth. The connection between this and Peano arithmetic was made by Godel, but since you cannot formalize the difference between a truth and a logical truth using mathematics, you end up not knowing which kind of example you have in the mathematics.

I went back through your OP to see what you meant by "actual truth". How does "this statement is not a logical truth" "work in our reality"? What if it's a logical falsehood? How would we know if we didn't know what you meant by "actual truth" and "logical truth"?

in my reality, there is a person called KenG who goes around thinking about stuff, and posting on forums; is that true, in the same way for everyone, specifically, is that true for KenG, in the same way?

I think that we each have our own perception of reality, which has to be different for each person. I've come to the conclusion that there just isn't an objective reality.....so there is no objective logic..

__________________

What about in geometry? Euclid's axiom that parallel lines do not meet at infinity has less 'fit' to relativistic space than Riemannian geometry.

Riemannian geometry 'enabled Einstein's general relativity theory, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.'

Euclid's axioms satisfy logical truth, but perhaps not actual truth, if actual truth is seen as a functional representation of matter.

Logic is:

...an analytical theory of the art of reasoning whose goal is to systematize and codify principles of valid reasoning....

...It is formal in the sense that it lacks reference to meaning. Thereby it achieves versatility: it may be used to judge the correctness of a chain of reasoning... solely on the basis of the form (and not the content) of the sequence of statements that make up the chain. ¹

A chain of reasoning whose value is T for all possible assignments of {T,F} to its prime components is a tautology. That is, it's valid. This provides a mechanical way to decide whether a given chain is valid, namely, the computation and examination of its truth table.

_______________________
¹ Sets, Logic and Axiomatic Theories, Robert R. Stoll, Freeman & Co., 1961.

__________________
Everyone is entitled to his own opinion, but not his own facts.

Yes, these are the important questions. See Cougar's post too. What I mean by actual truth is truth that requires meaning to understand (I think this came up in a thread awhile ago, I dimly recall), and logical truth is more of a symbolic connection, a tautological form of truth. The latter is bulletproof, but that's its flaw as well as its strength. Meaning is sloppy, difficult... and important.

So when I said that actual truth is truth from experience, I was referring to meaning, because that is also where meaning comes from (and maybe some imprinted workings of our brain, let's not get into neurology and genetics). Which is not to say there's no reasoning involved, because to have meaning we have to make sense of things, but it's not formal symbolic logic, it's a more manipulative form of reasoning. Understanding that is one purpose of the discussion here. But when I say that "this statement is not logically true" is a statement with an actually true meaning, it's because its truth value does not follow from a formal symbol table, expressly because such formal associations don't include self-reference. But self-reference has meaning, so if we look at the meaning of the statement, we can recognize it as an actual truth-- that statement does not conform to a logical tautology. That confers upon it the meaning of a true statement, though it cannot be a logical truth because it isn't bulletproof, and that's what makes it true.

Yes, this shows that truth-by-experience can be different things in different contexts, just as experience has different contexts. Cougar's quote pointed to the versatility of logical truth, but I think truth-by-meaning is the more versatile in a different way. Logical truth is versatile in the sense that it is the same thing in many different situations, but truth-by-meaning is versatile in the sense that it conforms itself to the different situations.
Yes, that's exactly why the connection between logic and truth is so interesting and so important-- logical truths whose axioms might at first glance seem to have no connection with actual truth might end up surprising us by leading us to a new actual truth. Can the surprise run the opposite way too, where an axiom that does seem to be actually true could connect via logic to something that doesn't? That would be the way Peano arithmetic could have a logical truth that was not an actual truth, an outcome many mathematicians find fairly unpalatable.
Quite so, and indeed one might even say that they satisfy actual truth in some contexts-- and not others.

I would make some careful distinctions between the concepts of truth, provability, axioms and logic in the various (and, I would say, separable) domains of pure mathematics, applied mathematics, and science (physics in particular).

Please do!

Well, since I believe that pure mathematics has a universality independent of any physical realisation, I suspect that my view won't carry much weight here.

That sounds as though I'm taking a Platonist view, but I am only saying that (I think) pure mathematics takes the form of "if I say x, then I am compelled to say y" (where x and y are, roughly, axioms (and established theorems) and deduced theorems).

Isn't that undecidability: you can't prove a statement true or false, which is not the same as proving true something known[*] to be false.
[*] known somehow - which is perhaps Ken G's point - although I'm not quite sure what the point is - or at least I can't decide ...

You can have a universal deduced theorem which flows perfectly logically from its axioms but does not describe reality. Euclid's axioms are universal as logical truth, but with the question how logic relates to actual truth, the internal coherence of axioms is not sufficient to say anything about their truth.

The theme of actual truth versus logical truth raises the question of mathematics as a logic of nature, suggesting a lower value for logic that has no link to actuality, and the comparison with physical realisation as the locus for emergence of the truth of logic.

There is a nice line at http://en.wikipedia.org/wiki/Platonic_idealism: Do axioms need to be created by pure logic, or are they discovered in the universal world of logical ideas?

Indeed Euclid's geometry quite possibly does not describe any part of the physical world. But I would say that it is a valid system independently of that. In fact I've seen it formalised in such an abstract way that it would barely seem to require any of our intuitions about space.

That isn't questioned, it is what is being called logical truth.
Also granted-- that is what logic does. It is the relationship to actual truth that is under question. It sounds like you are saying that you accept without comment that the two have to always be one and the same, but I thought that "this statement is not a logical truth" demonstrates that there is at least some difference, since actual truth relies on meaning. Would you not agree that formal mathematics is suspicious of meaning, relying instead on formal logic? After all, in Peano arithmetic one must, and does, prove that 1+1=2, whereas reliance on meaning would preclude the need for that exercise.

This is another interesting direction to take the basic OP question. We have two issues for the interaction between pure logic and experiential truth (or what we have been calling meaning), one relates to the actual truth of the axioms (which I would argue formal logic can never establish, as formal logic requires axioms to say anything), and the other relates to connecting actual truth of the axioms to actual truth of the theorems. So the two interesting questions are:
1) Can we ever know an axiom is actually true, or does actual truth always admit some uncertainty, as it initiates but does not itself flow from a formalizable reasoning process?
2) If actual truth is always somewhat "fuzzy" in regard to formal logic, can formal logic ever degrade the actual truth of an axiom in the process of connecting it to a theorem?

Note that I would claim we have already established that actual truth can go places that logical truth does not, so question #2 is asking if logical truth can go somewhere that actual truth has trouble.

It appears that you are trying to establish the legitimacy of a reasoning outside of "logic" but still include some sort of manipulation. Dangerous territory, because it is vaguely or informallly defined, and you seem to have opened up the discussion to allow statements like "This statement is a reasoning falsehood," without the formal constraints of logic.

As long as you don't try too much, you don't get into trouble that way. But, if you try to do something actually useful, you'll probably get into trouble. In other words, sure we can agree on a lot of reasonable things just by acclamation, but that is notoriously error prone, or misleading.

The fifth postulate is a good example.
No, it's not just undecidability. What Godel showed was that reasonably complex systems have undecideable true statements--if they don't, they are complete AND they're inconsistent.

The "truth" of mathematics is an entirely different thing from the "truth" of physics.

Mathematical truth is nothing more and nothing less than that which results from application of valid inference/logic to the fundamental axioms. There is no "absolute truth". Neither is there "approximate truth". There is no contingency on the "truth" of logic, since the game is simply to determine what follows from accepted logic. "Truth" is simply the result. It is not debatable outside of the rules of logic.

Physics is looking for something else -- a version of "truth" that is consistent with experiment. To the extent that physics is describable via a mathematical model, one can apply logic to deduce conclusions (aka predictions) as to how nature would behave if an experiment were to be performed. Generally those conclusions are correct, but occasionally they are contradicted by experiment. That is a reflection of the fact that physics is not an axiomatic discipline, but rather a field in which there is still need for research.

Physics proceeds as a series of successive approximations. New theories refine and supplant old theories, and the status of current theories is provisional -- they are viewed as the best available models but not necessarily as the last word. Physics admits, indeed thrives on, "approximate truth."

Mathematics is not science. Logic is the arbiter in mathematics. It does not rely on experiment. Physics is a science. Experiment is the arbiter of "truth". Logic is only a useful too. There is a symbiosis between the two subjects, but they are quite different subjects.

One can legitimately adopt the philosophy that physical law is written in mathematics and that the objective of physics is to discover those laws. That has been a fruitful approach for many of the most famous theoretical physicists -- Einstein, Dirac, Feynman, Weinberg among them. In this case the role of logic and mathematics is accentuated.

But it is not necessary to adopt that philosophy. One can equally well adopt a purely empirical perspective and reject the idea of the existence of any ultimate mathematical description of nature, remaining content with the latest approximations and their limitations. In this case logic and mathematics would seem to play a lesser role.

Actually, I'm not doing that, the human intellect has done that for millennia.
Yes, it is dangerous territory, and the dangers of that territory are what the thread is about. They are the dangers of connecting logical truth with the actual truth that stems from meaning.
Why wouldn't one want to do something useful? Of course we want to do something useful, and we do expect to get into trouble. That is the nature of the beast.

Yet perfectly essential, or mathematics is nought but a game like chess.

Quite so, hence the thread. For example, the statement 1+1 = 2 has a truth in mathematics, by which I mean it can be proven from the Peano axioms (and I'm calling that logical truth), and it has a truth in physics, by which I mean that we were taught to understand it when we were quite young and knew squat about the Peano axioms (and I'm calling that actual truth).

We disagree on the meaning of an "axiom" as it pertains to physics (where the term "postulate" would be more appropriate, but the distinction between axioms and postulates simply doesn't exist in logical truth, as logic treats them both the same and has no way to distinguish them). But that's not really relevant, let's not hijack the thread. So far what you have said is a kind of summary of why this thread exists.
Now we are getting into the heart of things. An "approximate truth" is hard to distinguish from what I'm calling actual truth, as it relies on meaning, and we all know how approximate meaning is.
This is a very tricky question in the philosophy of science. You're right, it's related to this thread-- what is the connection between logical truth and actual truth?
Yes, that is the issue of rationalism versus empiricism. But even the empiricists must use logical truth to help them manipulate their empirical truths, so the question of the thread is just as relevant for both camps.

Eudlid's geometry need not describe any part of the physical world. It is a purely abstract mathematical construct, and one that has been shown to be one example out of many consistent geometries. It is a fully consistent geometry, but not the only one, and others may be more representative of the universe in which we find ourselves.

However, Euclidean geometry is distinguished by its usefulness as a local model for the formulation of the more general theory of manifolds and Riemannian geometry.

I think you have things backward. Euclideana geometry does not require any intuition regarding space, but rather our usual intuition regarding space is built on Euclidean geometry. As Einstein showed us, that intuition is incorrect, and the Euclidean model is only valid locally.

Logic is largely useless without a means of determining when a departure from logic has taken place. This is why no approach involving logic would be complete without mastering the ability to spot logical fallacy, a disease that's heavily prevailent here on BAUT.

__________________
If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

WRT the OP, One is not the other, but occasionally the other may lead to the one, or it may not.

__________________
"What you think you thought you saw you did not see." Agent J, MiB - Manhatten Bureau

That may well be true, but for this thread we may imagine that logic itself is a very clear and unbending prescription, essentially symbolic logic. I think what you are refering too is a "fuzzier" form of logic that is much closer to rhetoric. The connection between symbolic logic and rhetorical logic, I would say (rhetorically), parallels the relationship between logical truth and experiential truth. Let us imagine that on this thread, only the force of solid logic will prevail, as I think has so far been the case.

Those all seem like fair statements, so open up the possibility of expounding on them. If they are not the same, what characterizes their differences, and under what circumstances can one lead to the other-- and when is the connection not as reliable?