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, 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