How consistent are the various logical systems?

[Kosh3]

how interrelated are the various logical systems that we have come up with.

How much do they vary from one another.

The problem I have is - if the universe is comprised of several or infinite "logics", then we can create several or an infinite amount of logical systems to describe the unvierse - and in such a case we have to do away with objective truth.

Since you could reason an unlimited number of answers from a single question, validly, using various logical systems - you have an unlimited amount of answers. None more true than any other. And then you could not say that there is objective truth.

[Volker.Doormann]

Originally posted by Kosh3
How consistent are the various logical systems?
So called 'logical systems' are never consistent, because they are imperfect phantomic creations of the physical mind. The problem I have is - if the universe is comprised of several or infinite "logics", then we can create several or an infinite amount of logical systems to describe the universe - and in such a case we have to do away with objective truth It is your problem, if you make use of imperfect systems. The point is, not to create and follow imperfect systems, but to recognize the truth and to recognize the false in the false. Nature, and also the spiritual nature does not need to be defined from imperfect minds, nature must be recognized, as it is. You can ignore truth, but truth will ignore you too.

Volker

[Clutch]

Originally posted by Kosh3
how interrelated are the various logical systems that we have come up with.

How much do they vary from one another. Well, they vary anywhere from a little to a lot. (Pretty useful answer, I know.)

It's hard to say how big the difference is without some criterion for bigness. Intuitionistic logic differs from classical logic in not including Excluded Middle (p or not-p) as an axiom. Hence it also gives up other tools: Double Negation Elimination (if not-not-p, then p), for example. It can't be modelled via a Boolean algebra, characteristic of classical logic (well, maybe it can, but not in any straightforward sense). Instead, its usual algebraic representation is made with a Heyting algebra. And its concept of negation must be understood modally, as representing the constructability of a reductio ad absurdum. You can build up the differences to be pretty big, in short, but on the other hand classical and intuitionistic logic are co-consistent: each is consistent only if the other is.

The slightly misleading presupposition of your question, in any case, is that we can have a notion of consistency independent of the logics we're comparing. It's the old problem of choosing a framework for the evaluation of two systems of thought: how do you do it without covertly assuming the truth or falsity of one or both systems?

[Kosh3]

Clutch: yea I assume that we are looking at different logics from the point of view of our own subjective innate logical system, since that is the only one we can ever actually look from. It could of course be right or wrong, if its wrong we're screwed - so mitas well say its right. Give ourselves that little curtisy for the purpose of being productive in the enquiry.

I found the list of differences you gave there quite helpful - got any more?

I dont know much specifically about the various logical systems, but find that the consequences of the logical systems is quite important philosophically.

[Peter Kirby]

One feature of some logics is that one can be represented with another.

For example, suppose that one had a logical system where the values for a statement referred to the knowledge of the speaker--"Yes, I know that is true" (Y) or "No, I know that is false" (N) or "I don't know, I'm unsure" (U).

This could be represented in binary logic by making a description of two properties: First, whether the speaker knows the answer (K); and Second, what the answer is (T).

So the mapping of one system to another would be:

K0, T0 => U. [I don't know, and it is false.]
K0, T1 => U. [I don't know, and it is true.]
K1, T0 => N. [I know, and it is false.]
K1, T1 => Y. [I know, and it is true.]

The other way around:

U => K0, T(0|1). [I don't know; it is true or false.]
N => K1, T0. [I know that it is false.]
Y => K1, T1. [I know that it is true.]

How to represent "fuzzy logic" (or, to use a more descriptive term, "continuous certainty logic" where, e.g., probability ranges from 0 to 1) in terms of binary logic and set theory is left as an exercise for the reader. Or for the computer programmer, as computers are machines that operate on binary circuits yet can and do manipulate fuzzy logic systems.

best,
Peter Kirby