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.


Well, don't take this too much seriously. My problem is that you need to do the math to evaluate how much seriously you can take this remark.

Note that in machines' theology, some theorem cannot be proved without the reduction to contradiction, so that it misses them. (Unlike intuitionism which can still get them by the use of the double negation).

Classical logic is the simplest logic to (re) discover the many non classical logics of the realities/dreams.



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