This is where we have problems, as expertly outlined already. Your use of the word "system" varies, and as a result you apply concepts to things that shouldn't have them applied. Let me try again, similar to what we did before: If a collection of axioms and rules of inference contains a contradiction, any statement can be proven true within that collection of statements and rules. Within the domain of axiomatic mathematics, there are multiple such collections in which different axioms are assumed true or false, and if we take all possible axioms as simultaneously true then the resulting collection is definitely inconsistent. However, many collections contain consistent axioms, and within those collections we can examine the consequences of believing those axioms to be true.Treatid wrote:Given a system that contains a contradiction; that system can prove anything (is inconsistent).

The system of axiomatic mathematics contains contradictions. Therefore the system of axiomatic mathematics is inconsistent.

It is true that all inconsistent collections of axioms can prove all other inconsistent collections of axioms true, but that doesn't add more axioms to the collection - this will be critical later.You might like to consider THE inconsistent system. An inconsistent system can prove anything. Proving a statement within a system means that statement is part of the system. So... All inconsistent systems contain every statement. As such, all inconsistent systems are equivalent. They all contain every possible statement. We can thus refer to The inconsistent system; which just happens to be the system of all possible statements.

Here be dragons. So you're saying that {x,y,z} are axiomatically true in A, and then creating B with the axioms {~x,y,z}. This is absolutely what math is about, and is a valid step. The shift from Euclidean to noneuclidean geometry is exactly this. What you seem to be missing is that the collection of axioms that make up A don't impact B, and vice versa. The supercollection {A&B} is an inconsistent collection.Description/emulation

Take an assumed consistent formal System A containing a number of statements that have been proven true with respect to the axioms. We take three of these statements and (preserving their truth value) use them as the axioms for System B. So - axioms x, y and z are true statements within the context of System A. And asserted true axioms of system B.

System B is a subset of System A. Anything we can prove in System B can also be proven in System A.

If we happen to prove that System B is inconsistent; then System A must also be inconsistent.

Suppose, instead, that we alter the truth value of one of the statements. Either we assert that one of the axioms is false (just as reasonable as asserting the axioms true), or we take the inverse of a statement (e.g. !x) and assert that axiom to be true. (We'll assume !x is true).

!x is obviously false with respect to System A. Were !x to appear as true in System A alongside x is true; then System A would contain a contradiction and therefore be inconsistent.

In this case, System B's axioms of !x, y and z cannot possibly come from System A.

Changing the truth state of axioms based on an existing formal logical system means you are discarding the formal logical system in favour of... what?

So - describing or emulating an axiomatic system means taking statements with a known context and using them as the basis for defining/describing/emulating another system.

Let's create a collection similar to what we did before, this time containing {x,~x,y,z}, and for the sake of argument let x,y,z be mutually independent, meaning no combination of them (true or false) can create contradictions. Your claim from before that this inconsistent collection contains all statements is problematic now - it really doesn't. There are 4 axioms. If you remove y or z, the inconsistency remains and you can still prove either/both true, but they are no longer axiomatically true. If you remove x or ~x, however, you now have a consistent collection again. All of the statements that you proved true in the supercollection may become indeterminate with respect to this new axiom collection, and fall out of the collection until you can re-prove them with the new axiom list. This is a simple path back to consistency.Context is important.Gwydion wrote:some things

Given an inconsistent system, our context is that inconsistent system. Within that inconsistent system we can prove that every statement is contradicted by itself. Remove any statement from System A and every other statement still allows us to invoke the PoE.

We know enough about the inconsistent system to know that within the context of that system, we can not only prove anything, but we can claim any statement means anything we want. A set of axioms taken from an inconsistent system could each have any or all conceivable meanings. This, after all, is the reason for axioms. We need a known starting point.

I'm not trying to say that a proof within an inconsistent axiomatic system must apply to every other axiomatic system. I'm pointing out that once a system is inconsistent everything is lost. Including any conceivable path back to some known set of axioms.

Your objection, then, needs to be extended further. Let's take another collection {x,y,~x,~y}, with x and y independent again. This axiomatic collection contains two independent contradictions, so removing any axiom leaves the collection inconsistent (WLOG let's take out ~y). However, while you can still prove ~y to be true within this collection, it is no longer an axiom. This means that when you remove either x or ~x, you are back to a consistent system - ~y and lots of other things fall away. The take-home idea here is that once you change the axioms of a collection, everything that isn't an axiom has to be reevaluated with respect to the new collection, because things that were previously true/false/indeterminate may not be any more.

This entire argument assumes the consequent - you're saying that because you assert "all axiomatic systems are inconsistent", any counterexample or argument is wrong because all axiomatic systems are inconsistent. You have not provided proof of inconsistency, just assertion of inconsistency, which means anything you want to demonstrate needs to be independent of that assertion.@LaserGuy: Two contradictory proofs within an axiomatic system prove inconsistency - not consistency. Indeed, given that the argument is that all axiomatic systems are inconsistent; I would expect you to be able to prove anything - including that a given axiomatic system is consistent. Proof of inconsistency trumps all other possible proofs.

And now we're back to the misuse of "system", so let me restate your summary once again: A collection of statements and rules of inference needs an alphabet and grammar in order to be expressed. Frequently we use natural language, because this is the way we communicate almost everything. Natural language has the ability to communicate different statements, and some of these statements are inconsistent with one another. Therefore, natural language has the ability to express contradictions. However, natural language does not have axioms, nor does it have rules of inference, so it is not a collection of the sort we have been discussing throughout my post, and indeed this thread. The principle of explosion states that if a collection of logical statements with certain rules of inference contains a contradiction, then any statement can be inferred from that collection. The principle of explosion does not apply to things which are not collections of logical statements or to things without the rules of inference which would allow that contradiction to "explode". Natural language, lacking these things, is in fact exempt from this principle, as are my computer and the paper I write on. The medium is distinct from the material.[b]Summary

A system cannot describe itself. In order to describe a system we need some other known system to describe a set of known axioms.

Hence we use natural language to describe our first axiomatic system and then use the formally defined systems to describe other formal systems.

For all of axiomatic mathematics; System A provides the context for understanding the set of axioms for System B.

Where System A is an instance of the inconsistent system; we cannot describe anything (or we cannot know what it is that we are describing) (The inconsistent system can prove any meaning of any statement).

Where System B is inconsistent, then System A is also inconsistent (See LaserGuy's last paragraph). If System A ever describes any instance of the inconsistent system, then every system described by System A is inconsistent (from the context of System A).

Change the truth value of a statement within System A and you have discarded System A. Whatever you assert beyond that point is arbitrary and has no relationship with mathematics.

Can we change the truth value of a statement in natural language and still preserve the known state of axioms described by natural language?

It doesn't matter.

If natural language is capable of describing both consistent and inconsistent systems, then natural language is inconsistent - and every system from that point on is also inconsistent.

"But natural language is exempt from all the rules...".

The Principle of Explosion applies if it can apply. Can we apply the PoE to natural language? Yes.