Inconsistent Logics

[Polymath257]

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

 

[ChristineM]

I used to be a fan of fuzzy logic, i kind of liked the Idea of true, false and various maybes

 

[Polymath257]

I used to be a fan of fuzzy logic, i kind of liked the Idea of true, false and various maybes

Yes, fuzzy logic essentially has *probabilities* instead of truth values. But the probabilities are all between 0 and 1 on the number line (so they are all comparable).

There are versions of logic with 4 possible values (actually, any power of 2) where you have True, False, and two other values that are not directly comparable. I have heard claims that some Buddhist philosophy accords well with such versions of logic.

 

[Revoltingest]

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

Any examples of logic leading to contradictions
that you can dumb down to my level?

 

[mikkel_the_dane]

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 will be honest - formal logic is not my strength.
But here is how I see it from a natural point of view with philosophy in regards to the hard problem of consciousness.
If consciousness is both physical and not physical, then it may be connected to a paraconsistent logic. But that is beyond me to show that with that kind of logic and also science.
Indeed it may be that a theory of everything is paraconsistent.

Just my 2 cents.

 

[Polymath257]

Any examples of logic leading to contradictions
that you can dumb down to my level?

Well, one old saw is the liars paradox: the person who says that they always lie. That quickly leads to a contradiction, but one that doesn't spread to other claims made by other people. The contradiction is tolerable.

I've seen non-standard logic used in analyzing quantum mechanical systems as well. I'm not convinced that is the best way to do it, but it allows for a particle to both be and not be in a location.

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

 

[Revoltingest]

Well, one old saw is the liars paradox: the person who says that they always lie. That quickly leads to a contradiction, but one that doesn't spread to other claims made by other people. The contradiction is tolerable.

I'd call that a contradictory premise.
Is it really a problem with logic?

I've seen non-standard logic used in analyzing quantum mechanical systems as well. I'm not convinced that is the best way to do it, but it allows for a particle to both be and not be in a location.

That seems appropriate for a probabilistic phenomenon.

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

Well, I have seen some negative theologies.
They don't seem helpful.
<snicker>

 

[Polymath257]

I'd call that a contradictory premise.
Is it really a problem with logic?

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.

 

[Dan From Smithville]

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?

I find it interesting, but I know that I do not fully understand the concepts you are expressing.

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

Unfortunately for me, it will require some education before I possess even a hint of understanding to make a prediction.

 

[Dan From Smithville]

Well, one old saw is the liars paradox: the person who says that they always lie. That quickly leads to a contradiction, but one that doesn't spread to other claims made by other people. The contradiction is tolerable.

I've seen non-standard logic used in analyzing quantum mechanical systems as well. I'm not convinced that is the best way to do it, but it allows for a particle to both be and not be in a location.

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

They should rename that Mudd's paradox or something like that. After the great Harcourt Fenton Mudd.

 

[Dan From Smithville]

Yes, fuzzy logic essentially has *probabilities* instead of truth values. But the probabilities are all between 0 and 1 on the number line (so they are all comparable).

There are versions of logic with 4 possible values (actually, any power of 2) where you have True, False, and two other values that are not directly comparable. I have heard claims that some Buddhist philosophy accords well with such versions of logic.

Stochastic?

 

[ChristineM]

Yes, fuzzy logic essentially has *probabilities* instead of truth values. But the probabilities are all between 0 and 1 on the number line (so they are all comparable).

There are versions of logic with 4 possible values (actually, any power of 2) where you have True, False, and two other values that are not directly comparable. I have heard claims that some Buddhist philosophy accords well with such versions of logic.

Fuzzy gives you logic 0, logic 1 and logic anywhere in between. The thought of logic 1.2 or 1.75 etc fascinated me.

 

[Dan From Smithville]

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

So in traditional logic, we cannot prove that someone is both a liar and telling the truth about being a liar at the same time, but there are forms of logic that can operate with that inconsistency?

 

[Dan From Smithville]

Fuzzy gives you logic 0, logic 1 and logic anywhere in between. The thought of logic 1.2 or 1.75 etc fascinated me.

I think the logic of .2 and .75 seems fascinating too. If I am understanding all of this. I may not.

 

[Polymath257]

So in traditional logic, we cannot prove that someone is both a liar and telling the truth about being a liar at the same time, but there are forms of logic that can operate with that inconsistency?

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.

 

[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.

I am part way there it would seem. Is there a way to explain paraconsistent logic on my level?

 

[Polymath257]

I think the logic of .2 and .75 seems fascinating too. If I am understanding all of this. I may not.

Fuzzy logic and fuzzy set theory go together nicely. So, we can ask if a certain collection of rocks is a 'pile'. For a small collection, the answer is close to no (probability zero), but as the collection grows, the probability also grows. So some collection is a 50/50 for being a 'pile' and larger collections are more 'pile-ish'.

For fuzzy logic, the same holds true for truth values. A statement may not be true or false, but, say, 25% true and 75% false.

 

[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.

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?

 

[Dan From Smithville]

Fuzzy logic and fuzzy set theory go together nicely. So, we can ask if a certain collection of rocks is a 'pile'. For a small collection, the answer is close to no (probability zero), but as the collection grows, the probability also grows. So some collection is a 50/50 for being a 'pile' and larger collections are more 'pile-ish'.

For fuzzy logic, the same holds true for truth values. A statement may not be true or false, but, say, 25% true and 75% false.

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

 

[Polymath257]

I am part way there it would seem. Is there a way to explain paraconsistent logic on my level?

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.