Describe an Inconsistent System.

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Treatid
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Describe an Inconsistent System.

Postby Treatid » Sun Nov 29, 2015 12:07 am UTC

  • A consistent system cannot describe an inconsistent system.
  • If a system can describe an inconsistency, that system must, itself, be inconsistent.
  • The 'Principle of Explosion' must apply to any system capable of describing the 'Principle of Explosion'.
  • The 'Principle of Explosion' tells us that there is no such thing as a partially-inconsistent-system.
  • Given an inconsistency (contradiction) within a system, then every statement within that system can be contradicted (is inconsistent).
  • There is no possible subset of an inconsistent system that is consistent.
  • Every possible subset of an inconsistent system is also inconsistent.
  • If any superset-of-a-system is inconsistent then the system (set) is inconsistent

The usual consideration is that natural/informal languages are qualitatively different to formal systems; given that we know formal systems cannot bootstrap a known starting point, and natural/informal systems appear to provide suitably well known axioms - then natural/informal languages must be exempt from the rules of formal systems.

1. This requires us to believe that natural/informal languages are not a system or part thereof ( which, given that natural languages are part of the universe, has its own problems).
2. The Principle of Explosion does not specify the nature of the systems it applies to. The Principle of Explosion specifies the conditions required for it to apply. Specifically, if a 'thing' contains a single contradiction, that contradiction allows us to contradict every other statement within that 'thing'.
3. You cannot design an inconsistent system - unless you start with an inconsistency.


https://en.wikipedia.org/wiki/Principle_of_explosion

Edit: Corrected the title.
Last edited by Treatid on Tue Dec 01, 2015 12:00 am UTC, edited 1 time in total.

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Re: Des

Postby madaco » Sun Nov 29, 2015 1:44 am UTC

The following lines in your post are false:

1 (for the meaning of "describe" that I assume you mean), 2, not sure what you mean by "apply to", or quite what you mean by "describe", but probably 3, 6, 7.

So, I think either you have some fairly fundamental misunderstandings, or you are deliberately making false statements to make some sort of point or joke that I don't understand.

Consider the following systems:

System 1:
The sky is orange.

System 2:The sky is orange. The sky is not orange.

System 3: The sky is not orange.

System 1 is a subsystem of system 2. System 1 is consistent. System 2 is inconsistent. System 3 is consistent. System 3 is a subsystem of system 2.
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Re: Des

Postby Xanthir » Sun Nov 29, 2015 4:39 pm UTC

Treatid wrote:
  • A consistent system cannot describe an inconsistent system.
  • If a system can describe an inconsistency, that system must, itself, be inconsistent.
  • The 'Principle of Explosion' must apply to any system capable of describing the 'Principle of Explosion'.

You're using "describe" in three unrelated ways here, rendering these statements all non sequiturs.

In your first line, "describe" must be taken to mean "construct within itself". But then the first line is wrong, because you can totally simulate inconsistent systems within a consistent one. You're using the old system to *implement* the new system, but the new system constitutes a logically separate universe. For example, I can use a computer to write a program that simulates a broken logic. The *mechanics* of the broken logic are consistent, it's just the higher-level notion of truth that the broken logic implements (separate from the notion of truth that the underlying logic implements) is inconsistent.

In your second line, "describe" must be taken to mean "prove in the system". But then the second line is just a restatement of the Principle of Explosion.

Your third line is a bit of a muddle. If we take "describe" to mean the English act of description, then it's false - natural language is not a logical system, so the PoE doesn't apply to it. But we can't take it in any more formal manner either, as PoE is not a logical statement, it's a natural-language observation about logical systems. It just describes what happens when you have an inconsistency.

  • The 'Principle of Explosion' tells us that there is no such thing as a partially-inconsistent-system.
  • Given an inconsistency (contradiction) within a system, then every statement within that system can be contradicted (is inconsistent).

This is a restatement of the Principle of Explosion, yes. Using the one contradiction, you can use proof by contradiction to prove any other statement, including direct contradictions of other true statements.

  • There is no possible subset of an inconsistent system that is consistent.
  • Every possible subset of an inconsistent system is also inconsistent.
  • If any superset-of-a-system is inconsistent then the system (set) is inconsistent

All of these are restatements of each other, and wrong in basic, trivial-to-demonstrate ways.

Take any consistent system (for example, the system consisting solely of the axiom "A"), and add a contradicting axiom (such as "~A"). It's now an inconsistent system. The system "before" is a subset of the system "after", but was consistent, so all your lines are wrong.

madaco wrote:So, I think either you have some fairly fundamental misunderstandings, or you are deliberately making false statements to make some sort of point or joke that I don't understand.

It's the first, madaco. treatid is a crank with some very persistent misunderstandings about how truth works in formal systems.
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Re: Describe an inconsistent system

Postby Treatid » Mon Nov 30, 2015 11:59 pm UTC

Sorry for messing up the title. It should have been: "Describe an inconsistent system."

Yes, The statements provided are all related. I hoped it was clear that I was covering the same point from different angles.

I think I am using "describe" consistently. In mathematics, describing a system is the same as emulating/simulating that system (at least, that is what I intend when using the word "describe").

Xanthir wrote: because you can totally simulate inconsistent systems within a consistent one.

Well - this gets straight to the heart of things. How exactly do you do this? It seems to me that if you can construct an inconsistency within a system then you have invoked the Principle of Explosion and proven that your initial system wasn't consistent.

You're using the old system to *implement* the new system, but the new system constitutes a logically separate universe.

When System A emulates System B, the emulation of System B is wholly contained within System A. The logical separation that your refer to is not absolute. Two systems might be entirely distinct - until we use one system to describe (emulate/simulate) another system. Such emulation is only possible if System A is a superset (in terms of functionality) of System B. If system B has features that cannot be constructed in System A then System A cannot describe System B. (Hence the distinction between standard Turing Machines and Universal Turing Machines).

natural language is not a logical system, so the PoE doesn't apply to it.

Why does this mean that PoE doesn't apply?

This is, obviously, a critical point.

But we can't take it in any more formal manner either, as PoE is not a logical statement, it's a natural-language observation about logical systems. It just describes what happens when you have an inconsistency.

It is a statements about systems. You haven't provided a reason for it to only apply to logical systems. The PoE itself does not specify that.

Take any consistent system (for example, the system consisting solely of the axiom "A")

Stop right there.

Having a single axiom does not prove that a system is consistent (E.g. "x&!x").

It is normal practice to assume that a set of axioms is both true and consistent up until the point where they are proven inconsistent. However, an assumption is not the same as proof.

Proving consistency is hard. You are using assumptions to justify your conclusions. I am showing that those assumptions are not correct.

You have not proven that your initial axiom is consistent. Without that critical first step - the rest of your counter argument falls flat.

Alternatively: we have two paths that start from the same place and end at two contradictory results. Your path does not exclude my path. Offering an alternative does not invalidate the other.

madaco wrote:Consider the following systems:

System 1:The sky is orange.

System 2:The sky is orange. The sky is not orange.

System 3: The sky is not orange.

System 1 is a subsystem of system 2. System 1 is consistent. System 2 is inconsistent. System 3 is consistent. System 3 is a subsystem of system 2.

You are not proving your point - you are assuming it.

You assume that "System 1:The sky is orange." is consistent. This seems straightforward and rational. Indeed so obviously self-evident that to question it must be a sign of madness.

However, you haven't actually proven that System 1 is consistent.

On the other hand we have an argument that shows that every system that has an inconsistent superset is itself inconsistent.

You are correct that these two viewpoints are incompatible with each other. At least one of them must be wrong. What you haven't demonstrated is that your assumption of consistency for System 1 is absolute and inviolable such that it must be the favoured viewpoint.

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Re: Describe an Inconsistent System.

Postby Xanthir » Tue Dec 01, 2015 1:48 am UTC

You are not proving your point - you are assuming it.

You assume that "System 1:The sky is orange." is consistent. This seems straightforward and rational. Indeed so obviously self-evident that to question it must be a sign of madness.

Not madness; doing so just makes it clear that one has absolutely no idea what one is talking about. This is basic first-order logic, man.

Can you construct an inconsistency from System 1? Stated in symbolic form, it contains the single axiom "A". Go ahead, *try* and construct "A and ~A" from that. Just try.

You don't get to just assert that "it's inconsistent" is an argument with the same truth status. It's false, nothing more.
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Re: Describe an inconsistent system

Postby measure » Tue Dec 01, 2015 3:53 am UTC

Treatid wrote:When System A emulates System B, the emulation of System B is wholly contained within System A. The logical separation that your refer to is not absolute. Two systems might be entirely distinct - until we use one system to describe (emulate/simulate) another system. Such emulation is only possible if System A is a superset (in terms of functionality) of System B. If system B has features that cannot be constructed in System A then System A cannot describe System B. (Hence the distinction between standard Turing Machines and Universal Turing Machines).

Yes, the emulation of System B is contained within System A, but the things B treats as axiomatic/true/false/unknown are not necessarily the same things that A does. For example, consider madaco's System 1 (The sky is orange): inside of System 1 "the sky is orange" is true (it is an axiom). In the system used to describe System 1 however "the sky is orange" might be false or unknown, while "it is true that 'the sky is orange' is true in System 1" is true in the larger system.

Even if B contains a contradiction (such as P and ~P), A, in describing this contradiction, need only accept "it is true in B that P and ~P", and so A is protected from the explosion.

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Re: Describe an Inconsistent System.

Postby Cauchy » Tue Dec 01, 2015 3:41 pm UTC

Consider the system with one axiom, A (with A not equal to ~A), and no rules of inference. This system is consistent, because the only statement that can be proved is A. It's an axiom, so it's proved by default, and since there are no rules of inference, nothing else can be proved, and hence ~A cannot be proved. For any other statement, it cannot be proved, and hence it and its negation cannot both be proved. Thus the system is consistent.

Consider the system with two axioms, A and ~A, and no rules of inference. This system is not consistent, because both A and ~A can be proved.

The first system is a subset of the second system. Thus we have demonstrated a consistent system that is a subset of an inconsistent system.
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Re: Describe an Inconsistent System.

Postby Treatid » Wed Dec 02, 2015 9:34 am UTC

Let us take Cauchy's argument at face value. Cauchy has proven that a given system is consistent despite a superset being inconsistent.

At the same time, it has been shown that every subset of an inconsistent system is inconsistent.

Argument 1 leads to x.
Argument 2 leads to !x.

These arguments exist in parallel - one argument does not exclude the other.

It doesn't matter how many different ways we prove x; if we can prove !x at the same time then we have an inconsistency.

@measure: If you have to change properties of system A in order to emulate system B then you are no longer using system A to emulate system B. That is: a change to system A means that it isn't system A anymore. The true/false value of individual statements is an explicit part of a system - not something you can arbitrarily change and still claim to be using system A.

@Xanthir: As with Cauchy - we can take your argument at face value as being valid. What it isn't, is complete. There is an equally valid argument that comes to a diametrically opposed conclusion. Provided that both arguments follow the established rules of deduction then we cannot prefer one argument over the other - both must be taken together. This allows you to say that System 1 is consistent and me to say that System 1 is inconsistent. X&!x.

{The issue here is 'inconsistency'. "The sky is orange" is a fine statement by itself. It becomes a problem when we try to insist that it must be either true or false but not both. As soon as we introduce the concept of inconsistency then we can use the PoE to show that everything is inconsistent. i.e. - this is not a proof that language doesn't work - it is a proof that the concept of inconsistency is flawed.}

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Re: Describe an Inconsistent System.

Postby Gwydion » Wed Dec 02, 2015 2:07 pm UTC

Treatid wrote:Let us take Cauchy's argument at face value. Cauchy has proven that a given system is consistent despite a superset being inconsistent.

At the same time, it has been asserted by Treatid without proof that every subset of an inconsistent system is inconsistent.
Fixed that for you. Please justify your statement with the sort of rigor that you are asking of the rest of us.

I'm still struggling to see your point. Are you trying to say that the ability to write untrue statements makes a logical system inconsistent, because anything that can be written can be used as argument? Do you think the principle of explosion should apply to my computer system, on which I can type inconsistent statements? (I don't want it to explode.)

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Re: Describe an Inconsistent System.

Postby Treatid » Thu Dec 03, 2015 4:29 pm UTC

Given an inconsistent system, we know that we can prove that each statement within that system is both true and false (ref: the Principle of Explosion). As such, every statement within an inconsistent system is, itself, inconsistent.

There are no degrees of inconsistency. There is no hierarchy of inconsistency. The order in which we might discover inconsistency is irrelevant. Every part of an inconsistent system is equally as inconsistent as every other part.

Hence - every subset of an inconsistent system is also inconsistent.

That is all there is to it. Extremely simple and obvious.

The only reason anyone would have any difficulty grasping this is because the result is inconvenient.

...

'Inconvenient', in this case, means that any set of axioms that is a subset of an inconsistent system (all of them) must be inconsistent. Or(/and)... Inconsistency is a flawed concept.

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Re: Describe an Inconsistent System.

Postby Sizik » Thu Dec 03, 2015 4:57 pm UTC

If that is your definition of "inconsistent", then all systems are inconsistent.
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Re: Describe an Inconsistent System.

Postby Xanthir » Thu Dec 03, 2015 6:00 pm UTC

In the normal definition of "(in)consistent", tho, it's a word that applies to logical systems, not to statements. Statements are true or false, nothing more. Calling them "inconsistent" is a category error.

So...
Or(/and)... Inconsistency is a flawed concept.

Your notion of "inconsistency" is indeed flawed, as pointed out by myself and others. That has no bearing on the concept of "inconsistency" recognized by everyone else.
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Re: Describe an Inconsistent System.

Postby arbiteroftruth » Thu Dec 03, 2015 6:03 pm UTC

Treatid wrote:Given an inconsistent system, we know that we can prove that each statement within that system is both true and false (ref: the Principle of Explosion). As such, every statement within an inconsistent system is, itself, inconsistent.


False. Every statement within an inconsistent system is *contradicted* by another statement within that system, and that's what we mean when we say the *system* is inconsistent. In order for the *statement* to be inconsistent, it would have to contradict *itself*. But the statement does not contradict itself. Other parts of the system contradict it, but the statement does not contradict itself.

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Re: Describe an Inconsistent System.

Postby Gwydion » Thu Dec 03, 2015 7:26 pm UTC

Treatid wrote:Given an inconsistent system, we know that we can prove that each statement within that system is both true and false (ref: the Principle of Explosion). As such, every statement within an inconsistent system is, itself, inconsistent.

There are no degrees of inconsistency. There is no hierarchy of inconsistency. The order in which we might discover inconsistency is irrelevant. Every part of an inconsistent system is equally as inconsistent as every other part.

Hence - every subset of an inconsistent system is also inconsistent.
You are correct that within an inconsistent system, the principle of explosion shows that any statement can be proven both true and false. However, you are wrong to apply the idea of "consistency" to individual statements. A statement's consistency refers only to whether it contradicts itself - for example, the statement (A&~A) is inconsistent. However, a statement that doesn't contradict itself is internally consistent, meaning that it is not inconsistent except when combined with other statements. Groups of statements can be inconsistent despite all of the individual statements being internally consistent. For example, the system of three statements about real numbers { (A>B), (B>C), (C>A) } is inconsistent, because all 3 can never be simultaneously true. Any subset of those 3 statements, however, would be consistent for some values of A, B, and C.

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Re: Describe an Inconsistent System.

Postby Treatid » Fri Dec 04, 2015 2:14 pm UTC

I get the impression that people think the Principle of Explosion is weaker than it actually is.

Wikipedia wrote:As a demonstration of the principle, consider two contradictory statements – “All lemons are yellow” and "Not all lemons are yellow", and suppose (for the sake of argument) that both are simultaneously true. If that is the case, anything can be proven, e.g. "Santa Claus exists", by using the following argument:

  • We know that "All lemons are yellow" as it is defined to be true.
  • Therefore, the statement that (“All lemons are yellow" OR "Santa Claus exists”) must also be true, since the first part is true.
  • However, if "Not all lemons are yellow" (and this is also defined to be true), Santa Claus must exist – otherwise statement 2 would be false. It has thus been "proven" that Santa Claus exists. The same could be applied to any assertion, including the statement "Santa Claus does not exist".

Which is to say that using the Principle of Explosion on an inconsistent system we can prove anything/everything.

Given an inconsistent system we can prove that every subset of that system is self inconsistent.

Perhaps you are more comfortable thinking of subsets of a system as also being logical systems themselves. However, the proof is not dependent on your particular definition of 'inconsistent'. We simply assert that we are using the standard mathematical definition and the proof stands, no matter how irrational the result might appear.

So:

For any axiomatic system that is believed to be consistent we can prove that the system is inconsistent (since it is a subset of an inconsistent system) (This includes all the possible single axiom systems).


@Sizik: Yes - that is the point. Every axiomatic system is inconsistent. Thank you for understanding the argument (Even if you don't yet agree with it).

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Re: Describe an Inconsistent System.

Postby arbiteroftruth » Fri Dec 04, 2015 3:58 pm UTC

Treatid, if I understand you correctly, you're saying that because of the principle of explosion, an inconsistent system can prove absolutely anything, including the assertion that "every system is inconsistent", and you think this proves that every system is in fact inconsistent.

The thing is that statement itself is still only true in the context of the inconsistent system. In a different system, that assertion would be false. An inconsistent system has no more influence over actual truth than a madman shouting in the street. Just because the inconsistent system says that "every system is inconsistent" doesn't make it actually true.

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Re: Describe an Inconsistent System.

Postby Gwydion » Fri Dec 04, 2015 4:16 pm UTC

Treatid wrote:I get the impression that people think the Principle of Explosion is weaker than it actually is... using the Principle of Explosion on an inconsistent system we can prove anything/everything.

Given an inconsistent system we can prove that every subset of that system is self inconsistent.

Perhaps you are more comfortable thinking of subsets of a system as also being logical systems themselves. However, the proof is not dependent on your particular definition of 'inconsistent'. We simply assert that we are using the standard mathematical definition and the proof stands, no matter how irrational the result might appear.
You're missing some words there. Using the Principle of Explosion within an inconsistent system, one can "prove" anything and everything within that system. One can't create a "logical" system which is inconsistent, and use it to prove things in other systems. Think of it this way - every logical statement A within a system is actually saying "if {every axiom in this system is true} then A." The PoE states that in an inconsistent system, for all possible statements A, the larger if-then statement remains true because the antecedent is false - some of its axioms are contradictory.

The PoE is an incredibly powerful logical tool but you're not using it appropriately. The idea behind it is that if one defines a system of logical rules and statements, but that system is not consistent, the system is useless because literally everything follows logically from it, including contradictions of every rule and statement in the system. Therefore, defining such a system is not useful, and one must be careful in choosing those rules and statements accordingly. The existence of the Principle of Explosion does not invalidate all of math and logic, as you seem to suggest that it should.

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Re: Describe an Inconsistent System.

Postby Twistar » Fri Dec 04, 2015 4:49 pm UTC

Just a warning. We've beaten around this bush with Treatid before. http://forums.xkcd.com/viewtopic.php?f=17&t=107895. He refuses to come to a technical understanding of any aspects of formal logic and will continue to use words in whatever non-technical way he has contrived so that he can continue to support his claim that all of mathematics is wrong.

"a consistent system cannot describe an inconsistent system"

Literally the only words that are used in a technically correct sense here are "a" and "an". In my opinion, it's not worth it to go further in this discussion unless Treatid gets an understanding of and starts using the technical jargon of formal logic correctly.

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Re: Describe an Inconsistent System.

Postby Treatid » Sat Dec 05, 2015 1:49 pm UTC

Part 1.

Any proof within an axiomatic system only applies to that axiomatic system. On that we are in agreement.

Inconsistent systems tend to be discarded quickly because the ability to prove anything and its inverse isn't very constructive. However, this leaves the nature of inconsistency less examined than it might be - and that does prove devastating to all of axiomatic mathematics.

So, within an inconsistent system (and only relevant to that inconsistent system) we can prove that every subset is inconsistent.

A subset of a system is part of that system. Mathematics takes the view that we can take a subset of a system and consider the subset in isolation from the total system - thus avoiding the problems inherent in being part of an inconsistent system.

That is, we have a set of statements. This set of statements is considered a system in its own right. This same set of statements is also a subset of an inconsistent system. The exact same set of objects have two different contexts in which our perception of the objects are radically different.

While those statements are a subset of an inconsistent system - those statements are inconsistent. So... mathematics must assert that a given set of statements are not a subset of an inconsistent system - except that they are a subset of an inconsistent system. Obviously there must be two distinct meanings for 'subset'.

Part 2.

"Describe an inconsistent system."

For these purposes; 'describe' means 'emulate'.

System A emulates System B.

With regard to consistency we can build a truth table (ahem, a truth list):

  • A is consistent, B is consistent.
  • A is consistent, B is inconsistent.
  • A is inconsistent, B is consistent.
  • A is inconsistent, B is inconsistent

A is consistent, B is consistent.
This is fine.

A is consistent, B is inconsistent.

This is a problem.

In order for A to emulate B, A must construct an inconsistency within itself. If A constructs an inconsistency within itself then A is inconsistent.

A is inconsistent
If A is inconsistent then anything emulated within A is also inconsistent.

Part 3.

If we have some system A (e.g. natural language) that is capable of describing/emulating an inconsistent system; then A is inconsistent.

"But natural language is exempt from the rules of logical systems because... reasons...".

The Principle of Explosion does not claim to apply only to formal logical systems. While there is a formal logical expression of the PoE, the natural language example I posted from the wikipedia page is applied to an instance of natural language.

Given that we describe (emulate) both consistent (or so it was thought) and inconsistent systems using natural language; natural language is inconsistent and every system described by natural language is also inconsistent.

{That last sentence is giving to give you all conniptions. Think of it as a reductio ad absurdum rather than trying to take it too literally. Natural language works fine - it is 'inconsistency' and related concepts that are the problem. Axiomatic mathematics cannot be built upon natural language in the way that is currently attempted.}

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Re: Describe an Inconsistent System.

Postby Flumble » Sat Dec 05, 2015 2:15 pm UTC

Treatid wrote:So, within an inconsistent system (and only relevant to that inconsistent system) we can prove that every subset is inconsistent.

You can only do that because of the PoE. And you can similarly prove the opposite because of it.
Any proof within an inconsistent system is meaningless.

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Re: Describe an Inconsistent System.

Postby Twistar » Sat Dec 05, 2015 6:31 pm UTC

Treatid, I'll engage in this conversation. But here's how I'm going to engage. I'll read your whole posts. But when I respond, I'm only going to respond up until the point where you stop using the accepted definitions for words, or you say something that shows a blatant shortcoming in your understanding of formal logic. Yes, you might be onto something with your ideas. I'm not denying that. But if you can't even play ball and use the terminology established in the field it is not worth anyone's time to talk to you.

With that, here goes.

Treatid wrote:Part 1.

Any proof within an axiomatic system only applies to that axiomatic system. On that we are in agreement.

Inconsistent systems tend to be discarded quickly because the ability to prove anything and its inverse isn't very constructive. However, this leaves the nature of inconsistency less examined than it might be - and that does prove devastating to all of axiomatic mathematics.

So, within an inconsistent system (and only relevant to that inconsistent system) we can prove that every subset is inconsistent.

The middle block of text doesn't really make sense. Maybe YOU haven't examine the nature of inconsistency thoroughly but that doesn't mean other people don't understand it.

And the final block of text is where you show a blatant misunderstanding of the terminology of formal logic. You use the terms "inconsistent", "system" and "subset" incorrectly.

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Re: Describe an Inconsistent System.

Postby Gwydion » Mon Dec 07, 2015 1:30 am UTC

Treatid wrote:Part 1.

Any proof within an axiomatic system only applies to that axiomatic system. On that we are in agreement.

Inconsistent systems tend to be discarded quickly because the ability to prove anything and its inverse isn't very constructive. However, this leaves the nature of inconsistency less examined than it might be - and that does prove devastating to all of axiomatic mathematics.

So, within an inconsistent system (and only relevant to that inconsistent system) we can prove that every subset is inconsistent.
Correct, and within that same system we can prove that every subset is consistent, and any other statement you want. The conclusion here should be that any information reasoned from an inconsistent basis is useless, and can not be later used to reason anything else - whether that information is true or not doesn't matter.
A subset of a system is part of that system. Mathematics takes the view that we can take a subset of a system and consider the subset in isolation from the total system - thus avoiding the problems inherent in being part of an inconsistent system.

That is, we have a set of statements. This set of statements is considered a system in its own right. This same set of statements is also a subset of an inconsistent system. The exact same set of objects have two different contexts in which our perception of the objects are radically different.

While those statements are a subset of an inconsistent system - those statements are inconsistent. So... mathematics must assert that a given set of statements are not a subset of an inconsistent system - except that they are a subset of an inconsistent system. Obviously there must be two distinct meanings for 'subset'.
When you remove an axiom from a set (creating a subset), some statements that were previously true/false may remain the same but others may become undetermined. Therefore, it is not valid to assume any true statement in the superset will remain true in any/all subsets. This applies directly to the concept of the PoE - once you remove the inconsistency from your axioms, you can no longer "prove" all statements true in the same way.

I'll restate your last argument then: We have a set of statements which is a system in its own right, in addition to being a subset of a larger system that is inconsistent. Within the superset, we can prove every sentence, so any conclusions drawn about this subset within the superset are meaningless. If this subset is consistent, then we can still make logically meaningful inferences within it; if the subset is also inconsistent, then we are back where we started.
Part 2-3.
I don't understand what you're trying to say with this. I guess it's just an extension of the first part, but I don't understand what it means to "describe" or "emulate" a system as you're asking. I think you're looping back to the previous threads now, when you claimed that because language is simply a widely accepted social convention and nonabsolute, nothing is anything and everything is nothing. I'm not sure why anything from these parts follows from what you say in Part 1, which isn't a logically valid inference anyway.

I refer back to my previous question - is my computer inconsistent (and vulnerable to exploding) because I can type inconsistent statements on it? If the answer is no, I suspect it is because in the English language, the word "system" (like many others, including "set", "describe", and "proof") has multiple meanings which are not interchangeable. Referring to the English language as a "system" is correct, in that it is a collection of smaller parts that can be put together to form something greater than their sum. Referring to the English language as a formal system of logic which "contains" all of mathematics is wrong - it isn't one.

One last point - your entire argument boils down to "nothing is consistent, everything is meaningless," which renders your conclusion similarly meaningless. It really is a reductio ad absurdum, but when you do that you prove that one or more of your starting points was false. Why do you think axiomatic mathematics, or natural language for that matter, are the false starting points rather than your own statements?

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Re: Describe an Inconsistent System.

Postby LaserGuy » Mon Dec 07, 2015 9:05 pm UTC

Treatid wrote:Part 1.

Any proof within an axiomatic system only applies to that axiomatic system. On that we are in agreement.

Inconsistent systems tend to be discarded quickly because the ability to prove anything and its inverse isn't very constructive. However, this leaves the nature of inconsistency less examined than it might be - and that does prove devastating to all of axiomatic mathematics.

So, within an inconsistent system (and only relevant to that inconsistent system) we can prove that every subset is inconsistent.


This is trivially false.

Consider the system A, which contains only one statements: {a}. This system is consistent. Consider the system B, containing the statement {~a}. This system is consistent. Now we create a system C that contains the statements {a}, and {~a}, respectively. Then we have a situation where:
A: {a} is consistent and is a subset of C
B: {~a} is consistent and is a subset of C
C: {a, ~a} is inconsistent

In fact, you have the conditional in this argument backwards: If the subset of a system is inconsistent, then the system is inconsistent. However, if the system is inconsistent, that does not imply that all subsystems are inconsistent--the subsystems are the more restricted type. This is obvious, because any system can be turned into a subsystem simply by taking its union with some other system (as noted in the proof above).

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Re: Describe an Inconsistent System.

Postby Xanthir » Mon Dec 07, 2015 10:08 pm UTC

Treatid keeps making the mistake of thinking that "inconsistency" is a property of statements, and that conclusions in one logical system automatically apply to other logical systems. They need to fix their understanding on those two concepts before anything else will make sense.
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Re: Describe an inconsistent system

Postby madaco » Tue Dec 08, 2015 5:45 am UTC

Treatid wrote:
madaco wrote:Consider the following systems:

System 1:The sky is orange.

System 2:The sky is orange. The sky is not orange.

System 3: The sky is not orange.

System 1 is a subsystem of system 2. System 1 is consistent. System 2 is inconsistent. System 3 is consistent. System 3 is a subsystem of system 2.

You are not proving your point - you are assuming it.

You assume that "System 1:The sky is orange." is consistent. This seems straightforward and rational. Indeed so obviously self-evident that to question it must be a sign of madness.

However, you haven't actually proven that System 1 is consistent.

On the other hand we have an argument that shows that every system that has an inconsistent superset is itself inconsistent.

You are correct that these two viewpoints are incompatible with each other. At least one of them must be wrong. What you haven't demonstrated is that your assumption of consistency for System 1 is absolute and inviolable such that it must be the favoured viewpoint.


So, do you claim that System 1 is inconsistent? Or only that I have not proven to you that it is consistent?

If you do claim that it is inconsistent, what do you mean by this claim?

So, pretending that I have never heard the word "inconsistent" before, how would you explain it to me, in the sense that you mean when you say that System 1 is inconsistent? (Assuming that you do mean to say that system 1 is inconsistent)

Here is another system:

System 4:

(System 4 contains no axioms/statements. This statement isn't part of it, its just describing it. The alphabet for System 4 has 0 symbols, the only well formed formula in System 4 is the one of length 0. There are zero axioms of System 4. The set of inference rules for System 4 has 0 inference rules.)

Do you claim that System 4 is consistent, or that it is inconsistent, or that the question of whether System 4 is consistent or not is flawed?

Also, what do you mean by emulate?

If I draw a map of Chicago, does that map emulate Chicago in the same sense of "emulate" that you mean?

I don't think emulate is a standard term in this topic, so I do not understand what you mean by it. Could you explain what you mean? You seem to think that the term emulate makes it easier to understand what you mean, but I think it actually makes it harder.

You say that if something "emulates" something else which is inconsistent, that the first thing is also inconsistent. I, again, do not understand what you mean when you say "emulate", nor do I know what you mean by "inconsistent", so, I don't know how this differs from "System A is unhappy. System B looks at system A, so System B is unhappy." (with "is unhappy" standing in for "is inconsistent" as you use it, and "looks at" standing in for "emulates")

...

Are you trying to make a point related to Wittgenstein or Kant? Some sort of "there may be mathematical truths, but if there are it is impossible for us to know them" sort of thing? (somewhere between a "noumenal world" thing and a "whereof one cannot speak" thing?)
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Re: Describe an Inconsistent System.

Postby Treatid » Thu Dec 10, 2015 5:34 pm UTC

Context is important

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.

Axiomatic mathematics argues that inconsistency doesn't apply to the whole system. Specifically, a subset Z of an inconsistent system Y can be considered as a wholly distinct system, and thereby Z can be considered (potentially) consistent. (please read response to Gwydion, below)

{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.}

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.

Gwydion wrote:When you remove an axiom from a set (creating a subset), some statements that were previously true/false may remain the same but others may become undetermined. Therefore, it is not valid to assume any true statement in the superset will remain true in any/all subsets. This applies directly to the concept of the PoE - once you remove the inconsistency from your axioms, you can no longer "prove" all statements true in the same way.

I'll restate your last argument then: We have a set of statements which is a system in its own right, in addition to being a subset of a larger system that is inconsistent. Within the superset, we can prove every sentence, so any conclusions drawn about this subset within the superset are meaningless. If this subset is consistent, then we can still make logically meaningful inferences within it; if the subset is also inconsistent, then we are back where we started.

{Thank you for your input. You are clearly trying to understand what I have misunderstood and bring this mistake to my attention (you are being constructive in your criticism).}

Context is important.

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.

@Madaco: System 1 is inconsistent if whatever describes System 1 is inconsistent. The phrase "The sky is orange" is fine by itself. The problem arises when we try to insist that the phrase is definitely either true or false but not both and not some inbetween (or off to the side) state.

I would suggest that System 4 is a null system. Non-existent. As such, I am hesitant to try and define properties of 'nothing'.

Regarding Map/territory: If the map is sufficiently detailed then it may not be possible to distinguish between the map and the territory. This is what mathematics strives for - descriptions that can be treated as if they were the object itself.

Hopefully a close reading of the main section above will clarify what I mean by description/emulation with respect to an axiomatic system - and hopefully that coincides with the standard mathematical use of description.

Regarding (mathematical) philosophy: Nothing so subtle. My aim is to show that any system that contains the concept of inconsistency must itself be inconsistent.

@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.

@Flumble:
I sympathise with your viewpoint. However, throwing out a result because you don't like it isn't very rigorous mathematics. The fact that axiomatic mathematics contains singularities that give absurd results should, by itself, be a warning that there are deeper problems within axiomatic mathematics (No matter that it appears possible to isolate the singularities).

Moreover, knowing that we can prove anything with the inconsistent system, and that the inconsistent system contains all possible statements allows us to prove that starting from the set of all possible statements there is no possible way to construct a consistent system (Brief summary: start with the inconsistent system. Create a path to a supposedly consistent system by (e.g.) removing an axiom/statement at a time. Apply the known rules of the inconsistent system (not the supposed rules of the destination system - that system isn't the context we start with)).

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.

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Re: Describe an Inconsistent System.

Postby Xanthir » Thu Dec 10, 2015 6:22 pm UTC

Treatid wrote:Context is important

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.

No it doesn't. A logical system is a set of axioms and inference rules. An individual system can contain contradictions, is subject to the PoE, etc.

If by "axiomatic mathematics" you mean "the universe of logical systems that have something to do with math", then that's not a logical system. If by that you mean "the axioms and reasoning typically used in mathematical reasoning", there's also no contradictions there (that we are aware of). (But math is incomplete.)

Axiomatic mathematics argues that inconsistency doesn't apply to the whole system. Specifically, a subset Z of an inconsistent system Y can be considered as a wholly distinct system, and thereby Z can be considered (potentially) consistent. (please read response to Gwydion, below)

No, you're misreading again. Inconsistency *does* apply to the entire system. But the set of axioms (+ inference rules) you use constitutes the system; subsetting the axioms of an existing system is not making a "subset of the system", it's making a new system. The new system may or may not be inconsistent, based on what axioms you chose to keep.

Again, consider the system with a single axiom "A" and the standard FOPL inference rules. This system is not inconsistent. If you wish to argue that it is, show a contradiction. Otherwise you're just making bad arguments with inconsistent language and confusing yourself.

Now the system with two axioms "A" and "~A" IS inconsistent - you just need to use the "and" inference rules to produce "A and ~A", which is a contradiction. The first system contains a subset of the axioms of this second system, but it's an independent system with no actual relationship to this one.

Again, if you think the first one is inconsistent by virtue of having a subset of the second one's axioms, SHOW IT. Show us the contradiction you can produce in the first system. Inconsistency isn't some nebulous quantity you have to infer, it's simply a term we apply to system that can prove a contradictory statement. So prove it. Produce a contradictory statement, or realize that you've thoroughly confused yourself and you need to go back and study the basics of logic. (And that centuries of theorists aren't idiots or liars.)

{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.}

This is correct.

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.

True.

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).

Axioms are true by definition. They're the true statements in your system. You can't "change their truth value", because that means you're not doing ordinary logic, and not using words in the way everyone else uses them; you'll confuse yourself. You can, tho, replace an axiom with its negation.

"True" or "false" is a structural quality of a logical system. There are axioms, and there are statements you can validly infer from those axioms, and there are statements you can't validly infer from those axioms. We call the first two "true" and the second "false". It doesn't matter what the statements are - if you say "this axiom is false", you're actually making an English-language argument that "you're using the wrong system for this task, and will get bad results when you map your results back to real concepts; the correct system has the negation of that axiom instead".

!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.

Yes, adding a contradictory axiom will make the system inconsistent.

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

Yup. System B's axioms are not a subset of System A's. You literally just said that. That means B is no longer relevant to the "set/subset" point you keep trying to make.

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?

You can't change the truth state of axioms, they're true by definition. If you swap an axiom with its negation, you're making a new, independent system, which can prove different 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.

No you can't. Here's an inconsistent system: the axioms "A" and "~A", along with standard FOPL inference rules. You can prove this system is inconsistent in one line. Now, remove either of the axioms and prove an inconsistency in what's left.

I think you're assuming that you can excise the axiom, but keep all the statements you proved with that axiom? That's not how this works. The set of statements provable by a set of axioms is contingent on those axioms; if you change the axioms, the set of provable statements changes as well.

But seriously, if you disagree with me, do the task from two paragraphs ago. Or do the task from earlier in this post. They're both using toy systems that should be trivial to work with, and you assert that it's possible to show a contradiction in them. Do so. Put up or shut up, basically.
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Re: Describe an Inconsistent System.

Postby Twistar » Thu Dec 10, 2015 6:33 pm UTC

edit: Last time this topic came up I proposed a hard ban on the word "system" in this thread. I propose we use that ban again. Treatid's argument only stands because of ambiguous usage of the term "system" and "inconsistent". If we're not going to be rigorous this is a waste of time.


Treatid wrote:Context is important

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.

You're using the word system wrong.

When you say "the system of axiomatic mathematics" what you SHOULD be saying (if you want to be rigorous which it appears you do) is: "the theory derived from the Zermelo Fraenkel axioms and expressed in the langauge of first order logic".

In other words, the "system" to which you are referring is a theory, in the technical sense of the word that I have linked above.

HOWEVER, you aren't using the word "system" the way you should be using it. Instead, you're using the word "system" to refer to the language of first order logic, and you are pointing out the fact that within the language of first order logic you can write down a theory which is inconsistent. That is true. Once you have defined the language of first order logic you can always write down an inconsistent theory. HOWEVER, you can also write down theories that are consistent. This of course gets into your misusage of the term inconsistent. inconsistent or consistent is a term that applies to theories, NOT to the language of first order logic.

And look, this all basic logic stuff that you REFUSE to understand because you know it will undermine your big point you're trying to make. Instead, you just continue to make category errors and criticize others for not being rigorous.

If you want to play ball you have to learn the technical jargon

Let me translate your argument into the correct technical jargon so that you or others can clearly see the flaws in your argument. And again, I'm only replying the the first two sentences of your last post because they ALREADY reveal a blatant misunderstanding of formal logic so it is not worth my or anyone else's time to go past them, especially if we're trying to have a rigorous discussion.

Given a system that contains a contradiction; that system can prove anything (is inconsistent).

translated into technical jargon reads:
Given a formal language within which can be written a theory that contains a contradiction; that formal language can only express theories which are provably inconsistent.


However, the translated statement is NOT a true statement. Here is a statement which is RELATED to the translated statement, and which is true, but is DIFFERENT than the translated statement.

Given a formal language within which can be written a theory that contains a contradiction; within that particular theory any statement (as well at it's inverse) can be proven to be true.


This last sentence is doing nothing more than stating the consequence of the principle of explosion. However, notice how it differs from the translated statement above in how it refers to languages and theories differently.

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Re: Describe an Inconsistent System.

Postby Cauchy » Thu Dec 10, 2015 7:10 pm UTC

Logical systems are more self-contained than you're making them out to be, Treatid. A system comes equipped with:

1) Its alphabet, that is, the symbols that can be used to make its well-formed formulas (the logical sentences that could or could not be provable)
2) Its grammar, which is the collection of well-formed formulas, that is, the collection of sentences that the system could apply to
3) Its axioms, that is, the collection of well-formed formulas that are taken as proved
4) Its inference rules, that is, the mechanisms by which we determine which other statements are provable besides the axioms

When we say that an inconsistent system can prove anything by the Principle of Explosion, we mean that every well-formed formula in the system has a proof. This is not "anything" anything, it's only the collection of well-formed formulas, because the system can't actually "talk about" anything else. Even if Peano Arithmetic were inconsistent, it wouldn't be able to prove that Santa Claus is real, because "Santa Claus is real" is not a well-formed formula in Peano Arithmetic.

So your mention of "the inconsistent system" is not correct.

I'm not clear on what you mean by emulating here. The entire use of the term subsystem is a little shaky, I think, and a quick google search didn't obviously return a definition. I'd personally say that for B to be a subsystem of A, they have to share the same alphabet and grammar, and every axiom and rule of inference of B has to be an axiom or rule of inference of A. However, I could also see that you might want to expand the term to include cases where every axiom of B is merely provable in A. In either case, your second version of System B is not a subsystem of System A, except in the case that A is inconsistent.

So it sounds like when you talk about description/emulation, you mean that the described/emulated system has as all of its axioms provable statements of the larger system, so that the provability of any statement in the described system implies its provability in the larger system. You should note that this is almost always NOT what is meant when anyone else is using the term.


Your use of the word "context" is rather fuzzy. You talk about the "context of a system", but I'm not sure what that means? Within a system in which the principle of explosion applies (there needs to be the right alphabet and grammar to support negation, obviously, and even then there need to be the right types of inference rules for this to happen, it's not an inherent property of contradictions), every statement has a proof. I don't know what it means to "remove a statement from System A", but if you remove an axiom, it's quite possible sometimes to resolve the issue. After all, it could be the case that every time a statement and its negation are both proved in System A, the proof of at least one of them relies on a certain axiom. Then, by removing that axiom, the contradictions would go away.

Logical systems don't (typically) talk about meaning, and certainly none of the most well-known ones do, so your claim that "within the context of [an inconsistent] system . . we can claim any statement means anything we want" doesn't really make sense.


To summarize, I'd say that you're using terms in non-standard ways and a lot of your claims don't seem to really make sense as either "true" or "false", so it's very hard to communicate effectively with you.

With regards to your summary, I'd say:

Here you're conflating your previous definition of "description" with the natural language one. When we describe first-order logic, for instance, in natural language, we are not saying that the axioms of first-order logic are true in some mystical way. When we say that "first-order logic proves 'P or not P' for every well-formed formula P", this doesn't mean that "P or not P" is magically proved in reality. It means exactly what it says, that first-order logic proves "P or not P" for every well-formed formula P, without any mention of whether we think "P or not P" is true or not, or whether it's provable outside of first-order logic. This, loosely, is what we mean when we (that is, the rest of us) talk about describing a system. There's certainly a more formal description of this idea, though I don't have it on hand. This description has the side benefit of "protecting us" from contradictions in the system we're describing. After all, for a system A and a wff P, the negation of "A proves P" is not "A proves not P", it's "A doesn't prove P". So if A proves both P and not P, this is not a contradiction in the system we're using to describe A.

This blows up the rest of your summary, because it all relies on your use of "describes".

Finally, when you say "System A provides the context for understanding the set of axioms for System B", I really don't know what you mean. What does it mean to "understand" a set of axioms? What does it mean for a system to "provide a context for understanding"? You don't seem to be using the natural language definition for "understand", so I'm rather lost.
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Re: Describe an Inconsistent System.

Postby Twistar » Thu Dec 10, 2015 8:05 pm UTC

Cauchy wrote:Logical systems are more self-contained than you're making them out to be, Treatid. A system comes equipped with:

1) Its alphabet, that is, the symbols that can be used to make its well-formed formulas (the logical sentences that could or could not be provable)
2) Its grammar, which is the collection of well-formed formulas, that is, the collection of sentences that the system could apply to
3) Its axioms, that is, the collection of well-formed formulas that are taken as proved
4) Its inference rules, that is, the mechanisms by which we determine which other statements are provable besides the axioms


You are describing a Theory. Let's just call it that instead of a system because Treatid uses the word 'system' to mean various things depending on the desired result. edit: Or maybe the word system is being used consistently, I'm not sure, but in any case it's used in a way that isn't helpful for the discussion since Treatid can't/won't define it rigorously.

Again jargon.

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Re: Describe an Inconsistent System.

Postby Xanthir » Thu Dec 10, 2015 8:19 pm UTC

Cauchy wrote:When we say that an inconsistent system can prove anything by the Principle of Explosion, we mean that every well-formed formula in the system has a proof. This is not "anything" anything, it's only the collection of well-formed formulas, because the system can't actually "talk about" anything else. Even if Peano Arithmetic were inconsistent, it wouldn't be able to prove that Santa Claus is real, because "Santa Claus is real" is not a well-formed formula in Peano Arithmetic.

Only because you can't spell it with the alphabet in use, but that's not really a significant limitation. If you allow it to be represented symbolically, you can certainly prove it in a hypothetical inconsistent PA.
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Re: Describe an Inconsistent System.

Postby Cauchy » Thu Dec 10, 2015 8:58 pm UTC

Twistar wrote:
Cauchy wrote:Logical systems are more self-contained than you're making them out to be, Treatid. A system comes equipped with:

1) Its alphabet, that is, the symbols that can be used to make its well-formed formulas (the logical sentences that could or could not be provable)
2) Its grammar, which is the collection of well-formed formulas, that is, the collection of sentences that the system could apply to
3) Its axioms, that is, the collection of well-formed formulas that are taken as proved
4) Its inference rules, that is, the mechanisms by which we determine which other statements are provable besides the axioms


You are describing a Theory. Let's just call it that instead of a system because Treatid uses the word 'system' to mean various things depending on the desired result. edit: Or maybe the word system is being used consistently, I'm not sure, but in any case it's used in a way that isn't helpful for the discussion since Treatid can't/won't define it rigorously.

Again jargon.


I typed "logical system" into Wikipedia and used the definition I got, but yes, I agree that the terminology is quite confusing. "Theory" isn't much better because of how loaded the word is in natural language, but I'd be fine using it. "Formal system" is probably the best choice, actually, since that's what my search redirected me to.

Xanthir wrote:
Cauchy wrote:When we say that an inconsistent system can prove anything by the Principle of Explosion, we mean that every well-formed formula in the system has a proof. This is not "anything" anything, it's only the collection of well-formed formulas, because the system can't actually "talk about" anything else. Even if Peano Arithmetic were inconsistent, it wouldn't be able to prove that Santa Claus is real, because "Santa Claus is real" is not a well-formed formula in Peano Arithmetic.

Only because you can't spell it with the alphabet in use, but that's not really a significant limitation. If you allow it to be represented symbolically, you can certainly prove it in a hypothetical inconsistent PA.


And how would you represent it symbolically? You have to start interpreting the statements, which is exactly the can of worms we're not opening.
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Re: Describe an Inconsistent System.

Postby Xanthir » Thu Dec 10, 2015 9:17 pm UTC

You're allowed to say that "A" means "Santa is real". Standard FOPL stuff, no predicates or anything.
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Re: Describe an Inconsistent System.

Postby Cauchy » Thu Dec 10, 2015 9:47 pm UTC

Xanthir wrote:You're allowed to say that "A" means "Santa is real". Standard FOPL stuff, no predicates or anything.


This is certainly a model of first-order logic, which we're trying to divorce from the formal system itself in this discussion.
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Re: Describe an Inconsistent System.

Postby Xanthir » Fri Dec 11, 2015 12:49 am UTC

We don't really need to generalize that much. Treatid's problem is basic enough that we can use any particular simple model.
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Re: Describe an Inconsistent System.

Postby Cauchy » Fri Dec 11, 2015 3:42 am UTC

I'm not sure that playing fast and loose with definitions is the best course of action right now, given that looseness of definitions is basically the entire problem. So I'm very reticent to conflate first-order logic with one of its models.
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Re: Describe an Inconsistent System.

Postby gmalivuk » Fri Dec 11, 2015 3:44 am UTC

There's a difference between simplifying and ignoring some things and playing "fast and loose".
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Re: Describe an Inconsistent System.

Postby Twistar » Fri Dec 11, 2015 4:17 am UTC

It's not clear to me what any of you mean right now when you say "model".

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Re: Describe an Inconsistent System.

Postby Cauchy » Fri Dec 11, 2015 4:18 am UTC

I don't see too much difference between saying an inconsistent Peano arithmetic proves Santa Claus is real, and Treatid's talking about axiomatic mathematics as containing contradictions. If we're going to give him trouble for conflating two levels, then we should certainly hold ourselves to a higher standard. I mean, how sure are you that Treatid will be able to easily understand when it's okay to conflate a theory with its model, and when the distinction needs to be kept, such that the conflation here won't cause problems?

Edit:

Twistar wrote:It's not clear to me what any of you mean right now when you say "model".


https://en.wikipedia.org/wiki/Model_theory

Loosely speaking, a model is a mathematical structure that... well, models a theory. That is to say, it has meanings for the symbols in such a way that the provable statements in the theory are true in the model. A good example of this is in abstract algebra: the theory of groups is the collection of statements that can be proved from the group axioms, and a model of the theory is any particular group, like Z/2Z. The provable statements in group theory correspond to true facts about Z/2Z (though the true facts about Z/2Z don't all correspond to provable statements in group theory). A similar thing happens in first-order logic, where a model determines the meanings of the predicates and functions.
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Re: Describe an Inconsistent System.

Postby gmalivuk » Fri Dec 11, 2015 12:44 pm UTC

If I have a student who doesn't understand the difference between a noun and a verb, I don't spend hundreds of words (most of which the student also doesn't understand, most likely) detailing the distinction between gerunds and present participles.
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