Inconsistent Logics

[Dan From Smithville]

Exactly. There are also other paradoxes along the same line that are allowed in the paraconsistent logics. For example, the Russell set, which was such a problem for naive set theory, is no problem if you allow for contradictions.

My understanding is undoubtedly a flaw of never having had a formal logic course and only assembling a sub-functional working knowledge of the subject as I have gone along. There are surely some basics that I have missed out on.

 

[Polymath257]

That makes sense. Something I can understand. Similar conditions exist in trying to assign taxonomic status to organisms.

Exactly, hence the potential value of fuzzy sets or fuzzy logic in such cases.

For paraconsistent, we could also allow hybrids that are 'both'.

 

[Polymath257]

I am trying to understand Russel set from this description I found on the Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/russell-paradox/

I am trying to wrap my head around this example provided on that site, but my head remains obstinate.

"Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves “R.” If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself."

What exactly does it mean by not a member of themselves?

OK, a set is a collection of objects. The things in that collection are called the members of the set. So, the set A={1,2,3} has 1,2,and 3 as members. But it does not have A as a member.

On the other hand, we might imagine a set that has itself as one of its members, so maybe B={1,2,3,B}.

The Russel set is the collection of all sets that do not have themselves as members. So, the set A would be a member of the Russel set but B would not.

The problem is that if R is the Russel set, we can ask if R has itself as a member. But, if it does, then by definition it does not have itself as a member. And conversely, if it does not have itself as a member, then by the definition of R, it has to be in R and thereby R has itself as a member.

It is sort of an extension of the liar paradox to set membership.

 

[Dan From Smithville]

Exactly, hence the potential value of fuzzy sets or fuzzy logic in such cases.

For paraconsistent, we could also allow hybrids that are 'both'.

It is understanding of the latter that seems to be critical to my grasp of this subject. It eludes me for the moment. Your metaphor of the hybrid is something different than a 25/75, 50/50 or 75/25 answer that might arise in fuzzy logic, but I am not quite there yet on seeing what that difference is.

 

[mikkel_the_dane]

One way is to think of there being three truth values: true, false, and both. A statement is 'accepted' if it evaluates to be either true or both in all models. The article in wikipedia for paraconsistent logic goes into how to define the logical operations from 'truth tables'.

But, for example, the negation of a statement that is 'both' will again be 'both' and will also be accepted.

So, in the liar paradox, the truth value of the original statement is 'both'.

There are also four-valued logics that have true, false, both, and neither as options.

Could you explain neither?

 

[Dan From Smithville]

OK, a set is a collection of objects. The things in that collection are called the members of the set. So, the set A={1,2,3} has 1,2,and 3 as members. But it does not have A as a member.

On the other hand, we might imagine a set that has itself as one of its members, so maybe B={1,2,3,B}.

The Russel set is the collection of all sets that do not have themselves as members. So, the set A would be a member of the Russel set but B would not.

The problem is that if R is the Russel set, we can ask if R has itself as a member. But, if it does, then by definition it does not have itself as a member. And conversely, if it does not have itself as a member, then by the definition of R, it has to be in R and thereby R has itself as a member.

It is sort of an extension of the liar paradox to set membership.

Thank you. I figured it would take an elementary approach for me to achieve something here.

So it must be both a member of R and not of R at the same time and this paradox cannot stand on that inconsistency?

 

[Dan From Smithville]

OK, a set is a collection of objects. The things in that collection are called the members of the set. So, the set A={1,2,3} has 1,2,and 3 as members. But it does not have A as a member.

On the other hand, we might imagine a set that has itself as one of its members, so maybe B={1,2,3,B}.

The Russel set is the collection of all sets that do not have themselves as members. So, the set A would be a member of the Russel set but B would not.

The problem is that if R is the Russel set, we can ask if R has itself as a member. But, if it does, then by definition it does not have itself as a member. And conversely, if it does not have itself as a member, then by the definition of R, it has to be in R and thereby R has itself as a member.

It is sort of an extension of the liar paradox to set membership.

But there exist logic structures in which it can stand? So the next step for me seems to be to understand those. I will have to spend some time on the links you provided to even ask reasonable questions or you will suffer the frustration of trying to explain to the ignorant that wants it all without effort.

 

[Polymath257]

Another paradox along these lines: say a word is antireferential (I forget the original word used) if it does not describe itself. So, for example, the word polysyllabic does describe itself, so is NOT antireferential. But monosyllabic does NOT describe itself, so monosyllabic *is* antireferential.

is the word 'antireferential' antireferential? It gets into the same sort of circular dilemma.

 

[Polymath257]

Thank you. I figured it would take an elementary approach for me to achieve something here.

So it must be both a member of R and not of R at the same time and this paradox cannot stand on that inconsistency?

Yep. it shows a problem with our intuitions concerning set formation. A good part of the early 20th century was spent figuring out how to avoid some of these inconsistencies in set theory.

 

[Polymath257]

But there exist logic structures in which it can stand? So the next step for me seems to be to understand those. I will have to spend some time on the links you provided to even ask reasonable questions or you will suffer the frustration of trying to explain to the ignorant that wants it all without effort.

Well, by allowing the 'both' possibility, we find that R both is and is not a member of itself.

 

[Polymath257]

Could you explain neither?

Well, again, this is formal logic. A specific truth table for each operation would have to be given. But the idea is that the truth value is indeterminate.

 

[mikkel_the_dane]

But there exist logic structures in which it can stand? So the next step for me seems to be to understand those. I will have to spend some time on the links you provided to even ask reasonable questions or you will suffer the frustration of trying to explain to the ignorant that wants it all without effort.

I am also trying. And I will be following.

So here is how far I got. Consider true(T) and false(F). Now accept as Poly said for the 4 outcomes for meta-true.
It is meta-true that T and non-F
It is meta-true that non-T and F
It is meta-true that T and F
It is meta-true that non-T and non-F

There are four outcomes possible.

Now I don't know if that is it? We will see.

Edit: Scratch that. Poly answered it.

 

[Polymath257]

One application I have seen is in databases, which can have inconsistent information, but still can be used intelligently.

https://pdf.sciencedirectassets.com...7fca4441be3b43e935e2cae392f7gxrqa&type=client

 

[Revoltingest]

Yes, it is contradictory. That is the whole point. But, in a paraconsistent logic, it would be allowed in the logical system as a contradictory statement and analyzed from there. Traditional logic would stop once it is seen to be contradictory.

Makes sense.
We see many apply "logic" to complex things with questionable
premises which must be weighted. So when posters make
political claims as The Truth, me eyes roll about.
We should start with premises of personal values & preferences,
& fuzzily reason from those, recognizing that our conclusions are
not (typically) factual. Wider application of this should make RF
discussions less testy.
Instead of...."You're immoral & wrong, you poopy head!"
It would be...."Here's how I see it....you poopy head."

 

[mikkel_the_dane]

Well, again, this is formal logic. A specific truth table for each operation would have to be given. But the idea is that the truth value is indeterminate.

Okay
Consciousnesses is physical and not non-physical
Consciousnesses is not physical and is non-physical
Consciousnesses is physical and non-physical
Consciousnesses is neither physical nor non-physical (because it is totally unknown what it is)

I am struggling with an example of neither as for indeterminate truth value.

 

[Polymath257]

Okay
Consciousnesses is physical and not non-physical
Consciousnesses is not physical and is non-physical
Consciousnesses is physical and non-physical
Consciousnesses is neither physical nor non-physical (because it is totally unknown what it is)

I am struggling with an example of neither as for indeterminate truth value.

Something that is nonsense or where ordinary truth value doesn't apply.

 

[mikkel_the_dane]

Something that is nonsense or where ordinary truth value doesn't apply.

I got it. It was about truth value.

So what about this example with fuzzy logic thrown in for a real life example.

Now it is going to be woke so here it is:
For gender, not sex nor sexuality if you asked someone the following: For the 2 genders male and female you can for each answer 100% to 0% and it doesn't have to be 100% combined. It can be both 0% or both 100% or any combination here off. And of course since it is not an oppressive question you are always allowed to answer that you can't do that because you don't accept gender as valid.

 

[Jayhawker Soule]

Another example would be some versions of 'negative theology'. Maybe having a system where God both exists and does not exist is more helpful.

What does 'negative theology' have to do with "a system where God both exists and does not exist"?

 

[Polymath257]

What does 'negative theology' have to do with "a system where God both exists and does not exist"?

God can not not exist but not exist.

 

[Rational Agnostic]

I have seen it claimed many times that there is only one possible logic. This post is, to some extent, to point out that this not the case. In particular, there are non-trivial logics that allow for inconsistencies. Such logics have even been used to do higher level mathematics.

So, in traditional logic, one of the principles is that of non-contradiction: that it is impossible to prove both a statement and its negation. A system in which such a contradiction is provable is called inconsistent and, in traditional logic, such a contradiction trivializes the system: from a contradiction every statement can be proved. This is called the explosion principle.

But, there are versions of logic that are 'contradiction tolerant' in the sense that the explosion principle does not hold and where contradictions are allowed without trivializing the system. Furthermore, it is possible to *prove* that not all statements are provable, in spite of contradictions.

Such versions of logic are called paraconsistent logic. Some models of such paraconsistent logic have three-valued logic (as opposed to the standard true-false). ONE way to view paraconsistent logic is that it focuses on the 'consequence relation' as opposed to the usual 'implication'.

In any case, there are now paraconsistent versions of set theory, number theory, and analysis (including calculus). In fact, it is possible to regard paraconsistent systems as extensions of the usual standard logical systems, but where the extension is contradiction tolerant.

So,

1. What do you think about the possibility of logic tolerating contradictions?

2. Is it possible that paraconsistent logic is useful in some religious discussions?

Some references:
Paraconsistent logic - Wikipedia
https://plato.stanford.edu/entries/logic-paraconsistent/
https://plato.stanford.edu/entries/mathematics-inconsistent/
Inconsistent Mathematics | Internet Encyclopedia of Philosophy

I see logic as a way of using language to make sense of the real world. So, since the "laws of logic" are dependent on definitions of words, and since words can only be defined in terms of other words, eventually, definitions in logic and math become circular and ultimately objectively meaningless as far as I can tell. But they are still USEFUL though, even if ultimately subjective.