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Self-referentiality (cf. Gödel) and Logic

ontological_realist said:
beero1000 said:
In classical logic, double negation elimination is a theorem.
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O.K. but would you please also give a direct answer to my question.
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I thought I did already. We can prove that propositional calculus is consistent as Godel's incompleteness theorems do not apply to axiomatic systems that are simple enough. Since double negation elimination is a rule of replacement in propositional calculus, there is no problem with Godel's theorems.

Some constructivist logics reject double negation elimination (essentially, the law of excluded middle) anyway, but that is a choice of axiomatic system and not a consistency issue.
 
ontological_realist said:
Does self-referentiality (cf. Gödel) disprove that A is not ~A ?
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Surely, A is not ~A!

Either you start from "A <=> ~~A" or from something else. The former assumes "A <=> ~~A" is obvious enough to be accepted without proof. The latter might be problematic if you get into systems that are not consistent. Self-referentiality certainly seems problematic in this respect. Logicians (all?) use meta-languages and I guess it's a way of avoiding self-referentiality. Our thoughts are not self-referential in a problematic way*. So I guess we should stick to systems that are not self-referential in a problematic way.

* Obviously, consciousness seems to be quintessentially self-referential, but not in a problematic way. It's not self-reference per se which is the problem but self-referential truth claims. and it's not really a fundamental issue. It's just a misunderstanding.
EB
 
I don't know if this is on topic, but a self referential example often quoted is :-

(1) "This statement is false".

It's said by some, that if true, then it is also false; and if false then it is also true.
However, it is neither true nor false. What it is, is not true. Not true, simply means it is in another category to true.
There exists both false, and not true in the category other than true. As with court cases, the verdict tends to be
guilty or not guilty. But there is more in the category of other than guilty, namely not guilty and innocent. The concept
of not guilty, is not identical to innocent.

Further to this, take the self referential example :-

(2) "This statement is true".

To my mind, that example is not true as well. The simple reason is that it is not a statement at all.
The phrase says nothing - what is it stating? Nix. There is no statement in it.

Neither of the examples above say :-

A = B
or
A = ~A
 
Gila Guerilla said:
I don't know if this is on topic, but a self referential example often quoted is :-

(1) "This statement is false".

It's said by some, that if true, then it is also false; and if false then it is also true.
However, it is neither true nor false. What it is, is not true. Not true, simply means it is in another category to true.
There exists both false, and not true in the category other than true. As with court cases, the verdict tends to be
guilty or not guilty. But there is more in the category of other than guilty, namely not guilty and innocent. The concept
of not guilty, is not identical to innocent.
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Problem solved if you accept to limit logic to affirmative statements that are either true or false, in which case "false" just means "not true".

Gila Guerilla said:
Further to this, take the self referential example :-

(2) "This statement is true".

To my mind, that example is not true as well. The simple reason is that it is not a statement at all.
The phrase says nothing - what is it stating? Nix. There is no statement in it.
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Well, it is a proper sentence, in the affirmative mode, and one that makes sense, although a proper context is laking. Which statement is true? So your point rather is that the statement is meaningless, i.e. it seems impossible to decide what it means, singularly. But, the conclusion is the same, it is neither true nor false, i.e. it's not a logical statement.

Gila Guerilla said:
Neither of the examples above say :-

A = B
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It's a statement. It says that A is equal to B. We just can't say whether it's true or not because a proper context is laking. Again it's meaningless unless you could tell what A and B as supposed to refer to. If A and B are statements (they could be numbers, objects etc.) and "=" is read as "equivalent" then it means something if A and B are assumed as either true or false. Usually, you'll come across that kind of statement in a book and the book will give you the proper context to interpret the statement. But all statement require some level of interpretation and therefore a proper context.

Gila Guerilla said:
or
A = ~A
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That's a bit different. Here we can accept this as representing a very large class of false statements like "It's raining = ~(It's raining)" etc. Of course we have to assume that A is either true or false but that's a prerequisite for reading the statement as logical in the first place. Logic does require cooperation between willing minds.
EB and ~EB
 
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