Can a logically inconsistent universe exist?

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True, and the theorems generalize to first and second-order logic in general, and to a very broad class of formal systems.




Then propositional logic wouldn't be logic.




This doesn't amount to anything other than arguing about the definition of the word, logic. I'm talking about deductive logic, which is what I thought the original question was talking about.




Again, this seems to be a matter of definition. On the one hand we have the term as it's used by Rand, and on the other, we have the term as it's used within fields like mathematics and (formal) logic, in which axioms are taken as starting points for deductive logic, but it's not clear that they must be true (that mathematicians like to work with axioms that "seem right" is almost incidental).




That is correct.

I don't see the connection or how it relates to anything other than mathematics. Explain please?




If it deals with fantastical supernatural elements then no... it isn't logic, it's makebelieve. If it deals with the proper methods of building more complex knowledge on top of other knowledge in a hierarchical nature... then yes, it is logic. Only the supernatural will invalidate logic. Because the two can not be joined. Logic deals with reality. That is the point being made.




True. Because it is only by the perversion of the meaning of logic that one can properly go about plotting its demise. Straw man argument. Because you said you were having trouble with logic I thought I would be kind enough to provide a definition for you. Because if you treat concepts as approximations you can more easily arrive at errors in thinking.




Actually, I got the word "constrain" from merriam websters dictionary. And the definition of logic from "thefreedictionary.com" and it is true that I take a definition of axiom from Ayn Rand. But the definition of axiom is the same across all philosophy. I just took her wording because it is the most convenient and i have it on file. It is also defensible.

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In relation to the original poster, and your assumption. I'll quote the OP



It is indeed true that he starts off using mathematical proofs as the base of his understanding of proof. He also goes further to say that his question is no longer about maths but about reality in general. Once you do that you go away from the subcategory of logic used in the formulation of mathematical proofs and move into the realm of formal logical systems that deal with a person's cognitive apparatus when dealing with reality. As his question is related to a proof of the non-contradiction principle of reality we move to axioms. Because the law of identity is an axiom. And he also asked for proof. And so, in relation to the request for further information, and based on the apparent errors I saw in your posts (which is so far the most elegant and best defense of a logically consistent universe I have so far seen on this thread) I thought I would help you out.

I only posted here because you asked for help. I am not "having a go" at anything you have said. Just clarifying some points you MAY be having a problem with. My understanding of your problem (which you claim to have: " I need to sharpen up my background in formal logic, methinks. " - attribute: Escued) was that you were using approximate definitions for some of your concepts and mistaking a criticism of mathematical logical constraints to the formal logic systems that deal with reality.

Again I am not strictly saying you are wrong. I agree with most of what you say. Just clearing up some points that you claimed to be having problems with.

i think i was the one "plotting its demise", ill try and set up

we only have the data that is available to us, now i dont think there is a logical way of proving whether the data that is available to us is all the data we need to draw the correct conclusion [but i dont know]

in any case im saying there is a chance that reality runs somehow deeper than we have so far encountered or will ever encounter, that there could be something out there that could undercut everything we believe to be true

but it would be unreasonable to reject what we perceive, all i'm saying is that just because we have had thousands of years of black crows dose not mean that there isent a green one out there

im saying that i believe nothing is infallible, i believe at the very least that our current methods of logic could be inconsistent with the universe

Formal logic and mathematics are essentially one and the same. When I have referred to logic in this thread, I have meant "deductive logic" which is made most precise as a formal logical calculus. All of the formal calculi that correspond to our intuitive notions of what is "logical" are subject to a form of Gödel's theorem.




I don't accept this definition, then, and I don't think you could make the case that it's anything more than an esoteric one. Deductive logic is an analysis of what statements follow from premises (axioms, if you will), independently of whether those axioms actually are true. This is basically equivalent to the practice of modern mathematics.




Are you saying I have equivocated somewhere, or do you mean to imply that words have a true meaning rather than one they're assigned for the purpose of communication?




I didn't mean to argue in a way that misrepresents you, but where did I do this, exactly?

Incidentally, I referred to formal logic. I haven't noticed that you've said anything that has to do with formal logic. Incidentally, I was not actually asking for help so much as commenting that it had been a while since I'd looked over things like Gödel's theorems carefully to address all of the subtleties about how they apply to the relevant systems of deductive logic. I'm only saying it's been a while since I've looked through the grungy details of the mathematical logic involved.




I'm not sure what you mean by "treat concepts as approximations".




The definition of "axiom" is certainly not used in the same way across all philosophy. In formal logic and mathematics, the term axiom is generally used as I described. The axioms that mathematicians choose to work with often seem intuitively to directly correspond to reality, but this doesn't make them necessary truths. A famous example is Euclid's fifth axiom which is now regarded as false, although one can construct a coherent geometrical framework out of it.

Dictionaries address common usages, but they aren't always good when one is trying to be precise, and good ones almost always contain multiple definitions. Even so, dictionary writers often miss subtleties that are significant in things like philosophy, math and science. I have no idea what you thought I meant by "constrain". I was using the term in the sense of narrowing the set of possible truths (possible given what you know). E.g. in the sense that adopting Euclid's fifth axiom would constrain the set of possible geometries of space.




My interpretation was that he meant "Are all statements that are true about reality logically consistent in the sense that they don't imply any statements of the form (X)&(~X) ?". My most serious reply would be "If truth means anything, yes." There is no meaningful "if not" I can construct.




Formal logic and mathematical logic are basically synonyms. Once you're dealing with cognitive systems, it's a matter of psychology, biology, and meta-logical introspection. These (or large subsets of these) might fall entirely within the realm of the definition of logic you're using, I suppose. But as I understood it, the question ultimately came down to whether there can be two true statements (knowable or not) that contradict one another. All I can say is that as much as I know anything at all, I know that there cannot.

Not to be overly pedantic, but to expand on this: only Godel's incompleteness theorem (and some related results) require the formal system to express arithmetic. Other results like Godel's completeness theorem (entirely different) don't have any requirement on whether the system can express arithmetic or not.

And, Godel incompleteness only really works when the system can express basic arithmetic. A surprising number of quite-expressive first order logic systems are complete (in the sense that arithmetic is incomplete)

Thanks to the replies so far, I think I can refine my question.

Let's say someone proposes a grand unified theory of physics. It accounts for all current observations, it makes new predictions, which are confirmed empirically. The theory also predicts that, when the universe is twice as old as now, two mutually exclusive things should happen, A AND not A. As a thought experiment, assume that this theory is the true description of how the universe works.

Then we have this problem outlined by wolphin:

At this point, I have two different mental models for thinking about my question. If I say the laws of nature work like procedures in a computer programme, then I would expect no problem until the condition occurs under which the contradictory rules operate. I don't know what might happen then. Would all order and regularity in the universe disappear (including all the laws of nature described by the theory), because from then on anything can be true?

As an alternative, does the mere presence of a contradiction in the theory mean that anything can be true even before the condition occurs when both A and not A must happen? And would this destroy all order in the universe immediately, as soon as it tried to operate according to rules that contain a contradiction?

Stepping back from the thought experiment, I can split my question into two: Would the presence of a contradiction in the rules that govern the universe destroy all order in the universe, because anything can be true? If that is so, would order break down even before the contradiction applied to an event in the universe?

If the answer to both questions is yes, then from the fact that there is order in the universe (things happen according to recognizable rules) we could conclude that whatever rules describe the universe contain no contradictions.

There is also another way of looking at this topic. I am sure most scientists would reject out of hand any theory with internal contradictions. I would. I am asking whether it can be proven that this rejection is justified. If I can reason as in my previous paragraph, then rejection of logically inconsistent theories is justified.

Does this help?

It's not just that anything can be true, it's that everything must be true by logical implication given A&(~A). That is, all statements follow from this by logical implication. It's not just that they're ruled out.

I can't make physical sense out of a hypothetical question of this form. It's evident that not all statements are true, I would say.



Well, if a statement like A&(~A) is true at some point in time, since that implies everything, it would imply sentences like "A&(~A) was true five minutes ago." So yeah, I think it's pretty justified to reject logically contradictory theories out of hand as being "not true" (though they may well be excellent models within limits).

Thankee. I'll go read up on the proofs.

Here's one:

What happens if an irresistible force meets an immovable object?

nothing; in any universe that contained an irresistible force, there could not be an immovable object... and vice versa...