Is there a refutation for Paraconsistent Logic/Dialetheism

[Ben_Sinner]

So are there any catholic philosophers on here that have anything to say on this?

 

[Ben_Sinner]

For example, the sentence “the sky is blue” can be true or false. But the sky being blue isn’t true or false. Again, this is as far as philosophy is concerned.

I’m afraid I don’t understand what you mean here. Could you explain more?

 

[Rhubarb]

I’m afraid I don’t understand what you mean here. Could you explain more?

In logic, truth-value is a property of sentences, which are parts of language. The sky being blue is a fact, a sentence about which can be true or false. The ‘truth’ involves the analyzing of the sentence, not the facts that are at hand. All these weirdo logics are doing is providing a certain way to look at sentences. At least, that’s how I understand it. Like I’ve said, theories of truth are a huge huge topic in logic, language, metaphysics and epistemology.

 

[Tomdstone]

The ‘truth’ involves the analyzing of the sentence, not the facts that are at hand.

Well then, according to that, is the following sentence true:
John Kerry is the president of the United States. ?

 

[Rhubarb]

Well then, according to that, is the following sentence true:
John Kerry is the president of the United States. ?

No it doesn’t. John Kerry is not the president of the United States, therefore the sentence you shared is false.

I mean, we could go deep down the rabbit hole about this. There are theories about truth-makers, and, what constitutes a sentence being true. Semantics and pragmatics is full of this stuff. Just, from the basis of logic, truth is a property of sentences. Generally speaking, the sentences are true when what the sentence is about ‘obtains’.

I mean, it’s really a pedantic distinction, but philosophy is a discipline of pedantry. We certainly use the word ‘true’ colloquially to describe facts in the world. But in the logical, technical way, truth and falsity is a property of sentences.

 

[Rhubarb]

No it doesn’t. John Kerry is not the president of the United States, therefore the sentence you shared is false.

I mean, we could go deep down the rabbit hole about this. There are theories about truth-makers, and, what constitutes a sentence being true. Semantics and pragmatics is full of this stuff. Just, from the basis of logic, truth is a property of sentences. Generally speaking, the sentences are true when what the sentence is about ‘obtains’.

I mean, it’s really a pedantic distinction, but philosophy is a discipline of pedantry. We certainly use the word ‘true’ colloquially to describe facts in the world. But in the logical, technical way, truth and falsity is a property of sentences.

 

[SimmieKay]

A refutation may be necessary because this sounds like a dangerous form of logic. “Not finitely many-valued” sounds like it is saying here is not one singular truth, there are many possibilities that are undetermined or whatever. This seems like a conflict with what Catholicism’s teaching regarding “truth”. Something is either true or false, if infinite possibilities are left open, it would seem we couldn’t really count on knowing anything to be true or false. Seems kind of scary.

It actually seems kind of borderline occultic as well.

Having read some of Graham Priest’s book (In Contradiction), I think one of the major motivations for these kinds of logical systems is trying to find a fresh approach to Gödel’s two incompleteness theorems. The first says that a theory powerful enough to express arithmetic cannot be both consistent and complete. The second says that any sufficiently powerful theory which can prove its own consistency must be inconsistent. This was important because it constituted a negative answer to Hilbert’s second problem. Paraconsistent logic says completeness can be saved at the cost of consistency, but inconsistency is not as a bad as everyone thought, because it can be contained.

So one can approach paraconsistency in such a way as to do little or no harm to ordinary thought - paraconsistent logic allows you to believe that A&~A is true for some A, but doesn’t require you to believe that for every A or even most A. If you limit its scope to statements related to Gödel’s theorems, the harm done to thought is little or none. So I don’t see why such a use of paraconistency should necessarily be verboten for Catholics.

On the other hand, paraconsistency has a forerunner in the Indian and Buddhist logical traditions, especially in the work of the Mahayana Buddhist philosopher Nagarjuna. Nagarjuna tries to use this kind of logic to prove certain Buddhist doctrines. Now, I think this use of paraconsistency is incompatible with Catholic teaching.

I would say that paraconsistent logic is a tool, and some ways of using the tool (to respond to Gödel’s theorems) should be acceptable in Catholicism, and other ways of using the tool (to argue for the doctrines of Mahayana Buddhism) should not be.

Simon

 

[Ben_Sinner]

So one can approach paraconsistency in such a way as to do little or no harm to ordinary thought - paraconsistent logic allows you to believe that A&~A is true for some A, but doesn’t require you to believe that for every A or even most A. If you limit its scope to statements related to Gödel’s theorems, the harm done to thought is little or none. So I don’t see why such a use of paraconistency should necessarily be verboten for Catholics.

The thing is, one of the “some A” this theory proposes is “There are no absolutes, which is an absolute”. This is a problem for Catholic doctrine that teaches that there are absolutes (God being one of them).

 

[Tomdstone]

• paraconsistent logic allows you to believe that A&~A is true for some A,

Can you give an example?

 

[SimmieKay]

The thing is, one of the “some A” this theory proposes is “There are no absolutes, which is an absolute”. This is a problem for Catholic doctrine that teaches that there are absolutes (God being one of them).

Paraconsistent logic permits us to believe that some contradictions are true, but it does not require us to believe that any particular contradiction is true. “There are no absolutes, which is an absolute” is an example of a proposition which paraconsistent logic permits us to believe, but does not require us to believe. So, paraconsistent logic permits one to believe things contrary to Catholic doctrine - yet classical logic permits you to do the same thing. Unless paraconsistent logic requires one to believe something contrary to Catholic doctrine - and I don’t believe it does - there is no problem.

Simon

 

[SimmieKay]

Can you give an example?

A classic example is the liar paradox - “This sentence is false”. Many adherents of paraconsistent logic would say that sentence expresses a proposition which is true and false simultaneously. Those who reject paraconsistent logic propose other analyses, for example, that “This sentence is false” does not correspond to any actual proposition, and as such, is incapable of being true or false.

Gödel discovered that any logical system powerful enough to perform arithmetic could be used to talk about itself. He did this by converting every logical formula to a unique integer, using a technique he devised known as Gödel numbering. Having done so, he proved that if he added to the logical system an axiom “This logical system is consistent”, he could use that axiom to prove the opposite “This logical system is not consistent”. So effectively he constructed the equivalent of “This sentence is false” in mathematical logic. (I’m simplifying things somewhat, and probably muddling them a bit too - I studied Gödel’s theorems in university, but that was 15 years ago now, though I hope I’ve got the gist of it right.)

Simon