Does the liar paradox prove...

[Ripheus27]

You know what: let´s just say parconsistentialism and dialetheism are right and wrong.

But that's trivialism. Dialetheists aren't trivialists; they don't think that rejecting the LNC means you can say just anything and have it come out true and false. A dialetheist thinks a contradiction is true if it can be proven somehow to be so, not simply asserted to be. Since most self-contradictory sentences can't be proven after the dialetheist's fashion, the dialetheist doesn't have to countenance all such sentences.

Again, trying to reject the LNC has more implications for the validity of inferences than the soundness of premises. That is, the anti-LNCer will not accept, in all circumstances, reductio ad absurdum, nor will he or she be troubled by paradoxes (at least not as troubled as others tend to be). Deriving a contradiction from premises will not be looked upon as so problematic.

 

[quatona]

But that's trivialism. Dialetheists aren't trivialists; they don't think that rejecting the LNC means you can say just anything and have it come out true and false. A dialetheist thinks a contradiction is true if it can be proven somehow to be so, not simply asserted to be.

I don´t know what they mean by "proof".
"Proof", as we know it, rest squarely on the acceptance of the LNC.
Since they don´t consider the LNC as binding, they will be able to reject any "proof" (as we know it) as circular reasoning - and so am I (when they claim to prove a contradiction to be true) if accepting to play after their rules.
All I get is word salad. A contradiction can´t be true - that´s part of the very definition of "contradiction".
These guys want to explode logic and language but want to use it nonetheless and want me to accept their claims as meaningful.

 

[Ripheus27]

I don´t know what they mean by "proof".
"Proof", as we know it, rest squarely on the acceptance of the LNC.

Who are "we"? Is "acceptance of the LNC" what most mathematicians and logicians base their concept of proof on?

Since they don´t consider the LNC as binding, they will be able to reject any "proof" (as we know it) as circular reasoning - and so am I (when they claim to prove a contradiction to be true) if accepting to play after their rules.

That's not how it works. Don't you think it would be better to understand how paraconsistent logic works before making claims like this?

All I get is word salad. A contradiction can´t be true - that´s part of the very definition of "contradiction".

I don't think analytic philosophy would support such a definition. A contradiction would just be (a and not-a). Modality (the concept "can't") doesn't seem to appear in this formula.

These guys want to explode logic and language but want to use it nonetheless and want me to accept their claims as meaningful.

You're still assuming the the LNC is absolutely essential to logic, which assumption is in question, here. You seem to be saying that you know what the word "not" means, but that dialetheists, et. al. don't know this. Perhaps you're right, but you haven't given me enough evidence to conclude that you are.

 

[Ripheus27]

Here's one example of a system of logic that rejects "from a contradiction, you can infer anything":

The approaches taken for motivating the systems of paraconsistent logic which we have so far seen isolate inconsistency from consistent parts of the given theory. The aim is to retain as much classical machinery as possible in developing a system of paraconsistent logic which, nonetheless, avoids explosion when faced with a contradiction. One way to make this aim explicit is to extend the expressive power of our language by encoding the metatheoretical notions of consistency (and inconsistency) in the object language. The Logics of Formal Inconsistency (LFIs) are a family of paraconsistent logics that constitute consistent fragments of classical logic yet which reject explosion principle where a contradiction is present. The investigation of this family of logics was initiated by Newton da Costa in Brazil.

An effect of encoding consistency (and inconsistency) in the object language is that we can explicitly separate inconsistency from triviality. With a language rich enough to express inconsistency (and consistency), we can study inconsistent theories without assuming that they are necessarily trivial. This makes it explicit that the presence of a contradiction is a separate issue from the non-trivial nature of paraconsistent inferences.

The thought behind LFIs is that we should respect classical logic as much as possible. It is only when there is a contradiction that logic should deviate from it. This means that we can admit the validity of ECQ in the absence of contradictions. In order to do so, we encode ‘consistency’ into our object language by O. Then ⊢ is a consequence relation of an LFI iff

∃Γ∃A∃B(Γ, A, ¬A ⊬ B) and
∀Γ∀A∀B(Γ, OA, A, ¬A ⊢ B).
Let ⊢C be the classical consequence (or derivability) relation and O(&#915 express the consistency of the set of formulas Γ such that if OA and OB then O(A * B) where * is any two place logical connective. Then we can capture derivability in the consistent context in terms of the equivalence: ∀Γ∀B∃Δ(Γ ⊢C B iff O(&#916, Γ ⊢ B).

Now take the positive fragment of classical logic with modus ponens plus double negation elimination (¬¬A → A) as an axiom and some axioms governing O:

A → (A → (¬A → B))
(A ∧ B) → (A ∧ B)
(A ∧ B) → (A → B)
Then ⊢ provides da Costa’s system C1. If we let A1 abbreviate the formula ¬(A ∧ ¬A) and An+1 the formula (¬(An ∧ ¬An))1, then we obtain Ci for each natural number i greater than 1.

To obtain da Costa’s system Cω, instead of the positive fragment of classical logic, we start with positive intuitionist logic instead. Ci systems for finite i do not rule out (An ∧ ¬An ∧ An+1) from holding in a theory. By going up the hierarchy to ω, Cω rules out this possibility. Note, however, that Cω is not a LFC as it does not contain classical positive logic. [Graham Priest & Koji Tanaka, "Paraconsistent Logic," sec. 5.5]

 

[quatona]

Who are "we"? Is "acceptance of the LNC" what most mathematicians and logicians base their concept of proof on?

Yes, I´d go so far to say that it is essential to mathematics, logic and language.

That's not how it works. Don't you think it would be better to understand how paraconsistent logic works before making claims like this?

Sure. Was the introductory question "Does the liar paradox prove...?" misleading?

I don't think analytic philosophy would support such a definition. A contradiction would just be (a and not-a).

That´s impossible, by virtue of the semantics of the formal system language. If two statements can coexist without problem they simply can´t and won´t be called "A" and "not-A" in Aristotelean logic.

Now, maybe the entire point is: Oftentimes, what we perceive as contradictory (and signify as such) actually isn´t contradictory. I would wholeheartedly agree with that notion. However, in regards to systems based on Aristotelean logic, the conclusion would be: "They aren´t A and -A" (although we erroneously assumed them to be).


Modality (the concept "can't") doesn't seem to appear in this formula.[/quote]
Obviously not. So?

You're still assuming the the LNC is absolutely essential to logic, which assumption is in question, here.

All I am saying is: If we want to operate meaningfully with a formal system we need to keep to the rules and definitions. Of course, you can "explode" any formal system by violating its constituents.

You seem to be saying that you know what the word "not" means,

Yes. And for to have a meaningful conversation I am dependent on the participants being willing to accept the definition of the most basic signifiers of the language we are using.
Now, maybe all dialetheists et.al. want is establishing a parallel language based on different definitions. That´s ok. Then, however, those two languages have to be kept carefully apart.
You can´t superimpose the statements made in on language upon statements made in the other.
It would be like trying to "explode" the decimal system by pointing to the fact that "2=10" (in the binary system).
You yourself are permanently pointing this out when telling me I can´t superimpose Aristotelean logic on paralogic. However, you seem to forget about it when you assert that it´s possible for Paralogic to question Aristotelean logic.

I am not asserting that Paralogic cannot be a meaningful system within its own semantics. I am not asserting that it´s possible for both to coexist.
I am saying that a conclusion arrived at in one system don´t affect the validity of the other.

but that dialetheists, et. al. don't know this.

Don´t you think it would be better for them to understand the most basic signifiers of a formal system before operating with it or tackling its use?

 

[Ripheus27]

Okay, but then is Aristotelian logic the logic of natural language? There is significant evidence that it is not, wherefore some other logic (e.g. paraconsistent) might be what natural language encodes. That is, the exceptionless LNC expresses Aristotelian "not," but "not" in American English, for example, might be better expressed by an LNC with exceptions.

 

[quatona]

Okay, but then is Aristotelian logic the logic of natural language?

What do you mean by "natural language"?

There is significant evidence that it is not, wherefore some other logic (e.g. paraconsistent) might be what natural language encodes.

(emphasis added)
The bolded "not" - is it used in the dichotomic sense of the LNc or in the sense of "natural language" (whatever that may be)?

That is, the exceptionless LNC expresses Aristotelian "not," but "not" in American English, for example, might be better expressed by an LNC with exceptions.

I can´t think of a single instance where I regretted that the word "not" was LNC-based. When I wanted to make a non-dichotomic statement I have always found there to be other words for that.

 

[Ripheus27]

What do you mean by "natural language"?

Natural languages are opposed to constructed/formal ones.