## Truth and Consistency

### Truth and Consistency

For a formalist mathematical system where truth is analytical (truth is based on their logical form), would you say that X being consistent means X is true (i.e. truth = consistency)? Or would consistency be a necessary/sufficient condition of truth (i.e. everything that's true must be consistent, but not everything that's consistent is true)? In the case of the latter, what's an example of something that's consistent but not true? I don't like the thought that they're equal in this case, but if the truth is based on their logical form, and that logical form is consistent with the given axioms, I can't see an alternative.

### Re: Truth and Consistency

Truth is derived solely from consistency where an inconsistent statement such as 2 + 2 = 5 still necessitates 2 and 5 as existing as true in the respect they exist. 2 and 5 show consistency through the replication of the counted phenomena which form them, yet it is also true that 2+2=5 is a contradiction. On one hand truth exists through consistency, on the other hand the truth of 2+2=5 is true as inconsistent.

Any contradiction, as inconsistent, is derived from truth values which are true as existing. This existence is consistency. This consistency is replication. 2+2=5 is true as a contradiction, in other words it is true to state a contradiction occurs. 2+2=5 repeats as a consistent contradiction.

One could state that truth value is necessitated by consistency, with there being multiple grades of consistency, but this does not negate the fact with the absence of consistency that consistency does not exist. One can have an inconsistent statement with multiple sub consistencies available thus necessitating that while consistency is necessary for truth an absence of consistency still allows for consistency to exist.

Any contradiction, as inconsistent, is derived from truth values which are true as existing. This existence is consistency. This consistency is replication. 2+2=5 is true as a contradiction, in other words it is true to state a contradiction occurs. 2+2=5 repeats as a consistent contradiction.

One could state that truth value is necessitated by consistency, with there being multiple grades of consistency, but this does not negate the fact with the absence of consistency that consistency does not exist. One can have an inconsistent statement with multiple sub consistencies available thus necessitating that while consistency is necessary for truth an absence of consistency still allows for consistency to exist.

- Arising_uk
**Posts:**12303**Joined:**Wed Oct 17, 2007 2:31 am

### Re: Truth and Consistency

No idea about a 'a formalist mathematical system' but in Logic truth is not based upon the logical form but on whether what the symbol stands for is actually true and this only applies to the contingent statements as the tautologies are necessarily true and the contradictions necessarily false.bmsdil wrote: ↑Fri May 08, 2020 11:06 pmFor a formalist mathematical system where truth is analytical (truth is based on their logical form), would you say that X being consistent means X is true (i.e. truth = consistency)? Or would consistency be a necessary/sufficient condition of truth (i.e. everything that's true must be consistent, but not everything that's consistent is true)? In the case of the latter, what's an example of something that's consistent but not true? I don't like the thought that they're equal in this case, but if the truth is based on their logical form, and that logical form is consistent with the given axioms, I can't see an alternative.

### Re: Truth and Consistency

I wouldn't conflate consistency with truth. There are two challenges here.

1. No mathematical system can ever prove its own consistency, and in fact, by Godel's incompleteness theorem, any system that can prove its own consistency is necessarily inconsistent. If consistency was truth you can't prove it.

Consistency is nothing more than the absence of (known) contradictions and if you stick to first order logic, there is a correspondence between semantic truth and syntactic provability: https://en.wikipedia.org/wiki/G%C3%B6de ... ss_theorem

2. A mathematical system cannot be both consistent AND complete. Basically - this is a choice. You can optimise for consistency; or for completeness but not both.

Where this leaves you is a world where IF the system is consistent, then there are necessarily true expressions that are provable from the axioms, and there are true expressions that aren't provable.

I wouldn't even go that far.

Para-consistent logics allow for dealing with contradictions.

Dialetheism allow for true contradictions.

Well, what do you mean by "true".

According to Tarski truth can't be captured in arithmetic: https://en.wikipedia.org/wiki/Tarski's_ ... ty_theorem

If truth was logical form, then the Structuralists were right. But it's not.

Logic/mathematics is reductionist/analytic in nature. Truth is holistic/synthetic.

### Re: Truth and Consistency

Consistency is the replication of a phenomena. 2 is consistent as the repetition of a single phenomena. A cow is consistent as the replication of specific forms (legs, horns, etc.) with the cow replicating across time. A contradiction is also replicatio in nature, 2+2=5 replicates the same inconsistency thus necessitating consistency in inconsistency.Skepdick wrote: ↑Tue May 12, 2020 8:01 amI wouldn't conflate consistency with truth. There are two challenges here.

1. No mathematical system can ever prove its own consistency, and in fact, by Godel's incompleteness theorem, any system that can prove its own consistency is necessarily inconsistent. If consistency was truth you can't prove it.

Consistency is nothing more than the absence of (known) contradictions and if you stick to first order logic, there is a correspondence between semantic truth and syntactic provability: https://en.wikipedia.org/wiki/G%C3%B6de ... ss_theorem

2. A mathematical system cannot be both consistent AND complete. Basically - this is a choice. You can optimise for consistency; or for completeness but not both.

Where this leaves you is a world where IF the system is consistent, then there are necessarily true expressions that are provable from the axioms, and there are true expressions that aren't provable.

I wouldn't even go that far.

Para-consistent logics allow for dealing with contradictions.

Dialetheism allow for true contradictions.

Well, what do you mean by "true".

According to Tarski truth can't be captured in arithmetic: https://en.wikipedia.org/wiki/Tarski's_ ... ty_theorem

If truth was logical form, then the Structuralists were right. But it's not.

Logic/mathematics is reductionist/analytic in nature. Truth is holistic/synthetic.

### Re: Truth and Consistency

This is a prime example of Quine's criticism of analyticity - it's circular.Arising_uk wrote: ↑Tue May 12, 2020 2:54 amNo idea about a 'a formalist mathematical system' but in Logic truth is not based upon the logical form but on whether what the symbol stands for is actually true and this only applies to the contingent statements.

Any contingent statement X can be turned into a yes/no question: "Is X true?"

This equates the contingent (structurally and semantically) to a decision problem.

One algorithm decides X is true, another decides X is false and the usual two-party consensus problems emerge.

### Re: Truth and Consistency

One is a synonym of the other, thus any definition derived from one is derived from another therefore necessitating the definitions as synthetic. As abstractions they reflect empirical phenomena and vice versa. Words, as phenomenon, show the same replicative nature as the empirical phenomena they are grounded upon. One word, "consistency", replicates into a newer form, "replication". The words, "consistency" and "replication" follow in the same form and function the definitions they describe. These definitions are self referential.

### Re: Truth and Consistency

The volume of words is premised upon one thing expressed in a variety of ways. One phenomena as replicating does so as an adaptation to void. For example "x" has one meaning. This meaning is empty in and of itself thus it progresses to "y" where the relationship of "x" and "y" form eachother. "Y" and "x" are both empty on their own terms thus progressed to "z".

This emptiness is grounded in the circularity of the phenomenon where "x is x", "y is y", "x is y" and "y is x". Each term is an empty loop, and as an empty loop is determined by what phenomenon they progress to. This progression is the the process of definition where one word Inverts to another symmetrical word allowing for the repetition of the original word.

### Re: Truth and Consistency

So? Why do you need to express the same thing in a "variety" of ways?

Why can't you express one thing in one way?

The entire notion of a synonym is that you have multiple words which mean the same thing.

If "consistency" and "replication" mean exactly and precisely the same thing, why bother having two words?

It's a rhetorical question. The fact that you do use both words, and that there is a clear nuance between the two uses is precisely the argument against semantic reductionism. You choose your words for a reason.

### Re: Truth and Consistency

Skepdick wrote: ↑Wed May 13, 2020 3:57 pmSo? Why do you need to express the same thing in a "variety" of ways?

Why can't you express one thing in one way?

All expression is inherently empty in itself. What gives definition to expression is its progress from one term into a new one. This progression allows for a relation of parts which enables a greater clarity. Each term is effectively renewed through a new one.

The entire notion of a synonym is that you have multiple words which mean the same thing.

If "consistency" and "replication" mean exactly and precisely the same thing, why bother having two words?

It's a rhetorical question. The fact that you do use both words, and that there is a clear nuance between the two uses is precisely the argument against semantic reductionism. You choose your words for a reason.

Each word is defined through the context in which it is used. With the expansion of words comes an expansion of context where the original word becomes self referencing in a newer and greater variation. It is this variation which allows for the same word to progress into a newer and higher context. Context expansion is the grounding of definition. The one words now exists under a variety of definitions thus expanding the definition of the original word.

Last edited by Eodnhoj7 on Fri May 15, 2020 5:51 pm, edited 1 time in total.

- Arising_uk
**Posts:**12303**Joined:**Wed Oct 17, 2007 2:31 am

### Re: Truth and Consistency

Don't see how? It's been a very long time since I read Two Dogmas but from what I remember is that he objected to the analytic/synthetic distinction with respect to the idea that some propositions were necessarily true or tautologies due to the meaning of the words or terms involved, basically concerned with what a synonym is. He did not disagree that the logical tautologies and contractions were respectively necessarily true and false as it didn't matter what the words or terms were in them. My point was that whilst I don't know what the poster meant by a " formalist mathematical system where truth is analytical (truth is based on their logical form)" in Logic the only things that are true or false in this way are the tautologies and contradictions, all else are the contingent propositions and many of those are exactly the ones that answer to the question are they true or false and the answer will not be found in their logical form but, as you say, by some other decision process, the best of which we have found so far, in my opinion, has been the repeatable experimental scientific methods of the natural philosophers but you could just run with 'Because 'God' made it so' or such like if you wished. Given Quine was an empiricist analytic philosopher I reckon he agreed as from his view point metaphysics was dead.Skepdick wrote: This is a prime example of Quine's criticism of analyticity - it's circular.

Any contingent statement X can be turned into a yes/no question: "Is X true?"

See above.This equates the contingent (structurally and semantically) to a decision problem.

One algorithm decides X is true, another decides X is false and the usual two-party consensus problems emerge.

### Re: Truth and Consistency

So, it depends on how you conceptualise the notion of a "contingent statement". It's neither true nor false on its own (e.g neither a tautology nor a contradiction) so for the purposes of discussion its truth-value is undetermined.Arising_uk wrote: ↑Wed May 13, 2020 5:52 pmDon't see how? It's been a very long time since I read Two Dogmas but from what I remember is that he objected to the analytic/synthetic distinction with respect to the idea that some propositions were necessarily true or tautologies due to the meaning of the words or terms involved, basically concerned with what a synonym is. He did not disagree that the logical tautologies and contractions were respectively necessarily true and false as it didn't matter what the words or terms were in them.

Here's one definition: A contingent statement is a statement whose truth value depends upon the truth value of its component substatements.

To pin the truth-value of contingent statements on its sub-components is reductionism in practice.

You've taken the problem "How do I determine the truth-value of a statement?" and turned it into a "How do I determine the truth-value of a sub-statement?" which doesn't get you any closer to addressing the general form: Ho do I determine truth-value?

Analytical means "you can arrive at truth by reduction".Arising_uk wrote: ↑Wed May 13, 2020 5:52 pmMy point was that whilst I don't know what the poster meant by a " formalist mathematical system where truth is analytical (truth is based on their logical form)"

In other words: if you keep reducing complex problems to simpler problems you will (eventually) arrive at some normal/simple form which you axiomatically accept as true.

You will arrive at some Mathematical identity where you accept that the left-hand side is equal to the right-hand side.

If you treat equality as material equivalence, then Logically you will either arrive at: (True ⇔ True) or (False ⇔ False)

A contradiction is defined as ¬(P ∧ ¬P) ⇔ TrueArising_uk wrote: ↑Wed May 13, 2020 5:52 pmin Logic the only things that are true or false in this way are the tautologies and contradictions

e.g the negation of a contradiction is materially equivalent to a tautology.

That's a structural statement, not a semantic one, because it doesn't tell you the truth-values of P or ¬P - those are yet-to-be evaluated.

The "P" in "¬(P ∧ ¬P) ⇔ True" stands for Proposition.Arising_uk wrote: ↑Wed May 13, 2020 5:52 pm, all else are the contingent propositions and many of those are exactly the ones that answer to the question are they true or false and the answer will not be found in their logical form but, as you say, by some other decision process

Logical propositions are exactly the same kind of objects as mathematical types and computer programs.

Structurally (analytically) there is absolutely no difference up to isomorphism.

And that's fine - logic produces consequences/entailment which still need to be empirically validated .Arising_uk wrote: ↑Wed May 13, 2020 5:52 pm, the best of which we have found so far, in my opinion, has been the repeatable experimental scientific methods of the natural philosophers but you could just run with 'Because 'God' made it so' or such like if you wished.

He did analytic philosophy - doesn't mean he could be reduced to a philosopher. There was more to Quine than philosophy.Arising_uk wrote: ↑Wed May 13, 2020 5:52 pmGiven Quine was an empiricist analytic philosopher I reckon he agreed as from his view point metaphysics was dead.

One of his (many) points is that the line between science and philosophy is not as clear cut as many insist.

But then ... is any line clear? Do lines even exist or are they just metaphysical constructs?

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