On 24 Oct 2013, at 00:15, meekerdb wrote:

On 10/23/2013 9:34 AM, Bruno Marchal wrote:

On 23 Oct 2013, at 17:39, Craig Weinberg wrote:


Dialetheism is the view that some statements can be both true and false simultaneously. More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called "true contradictions", or dialetheia.

Dialetheism is not a system of formal logic; instead, it is a thesis about truth, that influences the construction of a formal logic, often based on pre-existing systems. Introducing dialetheism has various consequences, depending on the theory into which it is introduced. For example, in traditional systems of logic (e.g., classical logic and intuitionistic logic), every statement becomes true if a contradiction is true; this means that such systems become trivial when dialetheism is included as an axiom. Other logical systems do not explode in this manner when contradictions are introduced; such contradiction-tolerant systems are known as paraconsistent logics.

Graham Priest defines dialetheism as the view that there are true contradictions. JC Beall is another advocate; his position differs from Priest's in advocating constructive (methodological) deflationism regarding the truth predicate.
Dialetheism resolves certain paradoxes

The Liar's paradox and Russell's paradox deal with self- contradictory statements in classical logic and naïve set theory, respectively. Contradictions are problematic in these theories because they cause the theories to explode—if a contradiction is true, then every proposition is true. The classical way to solve this problem is to ban contradictory statements, to revise the axioms of the logic so that self-contradictory statements do not appear. Dialetheists, on the other hand, respond to this problem by accepting the contradictions as true. Dialetheism allows for the unrestricted axiom of comprehension in set theory, claiming that any resulting contradiction is a theorem.

It occurs to me that MWI is a way of substantiating dialetheism as a physical reality...in order to avoid having to internalize the possibility of dialetheism metaphysically.

No problem with that. Like Everett restore 3p-determinacy, comp restore also non-dialetheism, metaphysically, but does not (and cannot) disallow it it in some machine's mind.

G* says it; D(Bp & B~p), or <>([]p & []~p). read: it is consistent that p is believed and that ~p is believed, by the Löbian machine.
The machine cannot know that, note.

Sure. That's because logic assumes that if p<=>q then q can be substituted for p. Hence if you believe the morning star is a goddess and the evening star is a planet, you may believe a contradiction - but not if you know it.

That is a bit unclear to me. Substitution of equivalent if always dangerous in modal contexts. The reason is perhaps more prosaic, which is that a machine who believe in its inconsistency believes in some infinite ("non-standard) number(s), she agrees that 0 is not Gödel number of a proof of f, nor are 1, 2, 3, ... , but yet she believes in some number representing a proof of f.

Humans have a big non monotonical logic layers, making them able to say "I was wrong", and able to revise previews opinions. Evolution might exploit truth and relative lies too. That leads to complex questions.

Correcteness is when you forget all the lies, and nothing more. If you survive that, you get Löbian by necessity, and your physics will not change, normally (with comp).

No doubt that human actual theologies are more complex than the theology of the correct universal machine, platonist, and believing not in much more than the universal base (number, or combinator, or ...).

But PA, ZF, are only sort of "Escherichia Coli" of the person. They get personhood by the intensional nuances of the "provability" predicate. Detrivializing their physics and theology (the simplest one as it might be, but it is already quite rich).

Look how much information we already get in the UDA, where a person is defined by just the accessible memory (the diary "entangled" though their accompaniment in the annihilations and reconstitutions).

In the arithmetical version, a person is defined by a universal number with enough introspection and induction ability. PA and ZF are "well known" typical example. And incompleteness allows to define a notion of knowledge associate to them, and a notion of observation.

We all have a Löbian part, as believer in PA's axioms, for example. I think that that part is already conscious when we assume consciousness is invariant for the genuine universal digital substitution. The universal machine defines a canonical universal person, and the Löbian one, which knows, in some weaker sense that the Theaetetus' one, that they are (Turing) universal.



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