What exactly is supposed to be "beyond" logic?

Many would disagree, including me.

My intuition is right much of the time, and that is good enough for me.

The fact that it isn't right all of the time doesn't take away from it's value.

We are often forced in life to make decisions with very incomplete information, and utilizing a combination of reason and intuition can help us make a better judgment.

A good deal of thought goes into religious belief, a good deal of free thought and rational thought.

I park my car in the garage, and I believe it is still there. Do I know it is still there? No. Someone may have broken in and stolen my car. Therefore, according to you, I can't know my car is the garage. I must only feel that my car is there.

I must go through life with lots of unknowing. Somehow I manage, and thrive.

Are beliefs either facts, or feelings, or is it a possiblity that they are both?

You believe my thinking if faulty, if it doesn't agree with yours. I just think my thinking is different than yours.

It just is.

You make a distinction without much of a difference, in my opinion. You are looking for a form of absolute knowledge, when such a thing is rarely available to us in life.

What's the problem?

What do we do about the unknowable????????

What do you do when you need information that you need and can't possibly get using the scientific tools available to you in this lifetime?

Anyone interested in scientific reasoning will recognize immediately that the most obvious matter beyond mere are the material facts disclosed by observation and experiment. If one were able to deduce such facts by logic, there would be no need for observation and experiment.

One such feature is that fact that just about every nontrivial theory expressible in the language of propositional calculus is undecidable, meaning that there is no general algorithm to decide whether a given sentence is true or false -- and in fact there's no general way to determine whether a typical theory leads to a contradiction or not.

Thus, an axiomatic assertion, such as the law of the excluded middle, has no computational or observable meaning in most cases and simply reduces to an ideological claim, which is nevertheless seldom discarded, because it has the convenient effect of simplifying certain complex grammatical forms.

An example is the assertion "If it's not true that everything fails to have property X, then something exists which has property X," which might be applied as follows: upon showing that the assumption "everything fails to have property X" leads to a contradiction, one realizes then that "it's not true that everything fails to have property X," and concludes that "something exists which has property X."

And sometimes, of course, the analysis is simply too complicated to do: a calculation, promising to give a needed answer but requiring a hundred years of computer time, is a mere fantasy if a decision is required in the next year or two.

I suppose you are free to take the view that the material world should simply be regarded as the 'input' to a logical proposition, if you so wish, but many of us regard the world as having more presence than a mere term in a proposition and notice it contradicts our ideations with some regularity.

You claim It would be more correct to say ... if you ran the computation for an infinite amount of time you'd have an answer, but you can't do this. Supposing, for a moment, that one could actually make sensible statements about running computations for an infinite amount of time,

I should object to your statement on the grounds that the incompleteness of Peano arithmetic is not analogous to the halting problem: there are perfectly well-formed sentences in Peano arithmetic that cannot be shown from the axioms to be either true or false -- and this problem cannot be solved by any constructive enlargement of the axiom set, if Peano arithmetic is consistent.

I suppose you will want now to say that this consistency could be determined by running computations for an infinite amount of time, but any practical person will recognize such a statement as pure ideology: it has no experimental content, and its meaning must therefore be purely a matter of verbal convention; in short, you want to assert the truth of some untestable claim.

You want to call If it's not true that everything fails to have property X, then something exists which has property X a silly axiom; in fact, you'll find some consequence of that assertion in any standard treatment of quantifiers.

I'll agree that it's pleasant when one can find a polynomial-time algorithm for a problem but as a purely practical matter "polynomial-time" doesn't really mean the same thing as "tractable." A number of problems which can supposedly be handled in polynomial time are still infeasible.

Jabber however much like you like about logic or computational complexity: still, the natural interpretation of a claim Such-and-such a method would solve this problem but we can't actually do anything like that, because it would take hundreds of years is simply Such-and-such a method seems to be useless for this problem

and also has some other beliefs A about unicorns

And humans are often required to make decisions in contexts where a logical approach is impossible.

Science cannot solve all of our problems:

Too literal a reliance on language leads to known paradoxes, and this limits the "knowledge" that can be obtained by mere linguistic gymnastics.

This leaves the question of what one does in important cases when careful logic or extensive computation is a too-expensive luxury or is unknown to yield a correct answer. Such cases seem to comprise much of almost everybody's daily life.

However, it seems impossible to pin down exactly what you mean by logic or computation -- you take a vague and expansive view, which appears mainly to involve continually redefining everything as a computation and then merely re-asserting your view that everything is a computation. Frankly, I prefer not to use words in such ill-defined fashion.

As far as I can tell, you never provided anything like the requested computational scheme. One natural interpretation would be that you realize there's no real prospect of doing that.

I think I myself would be inclined to try any technique, no matter how irrational, to restore what is missing there.