On 10/23/2013 9:34 AM, Bruno Marchal wrote:
On 23 Oct 2013, at 17:39, Craig Weinberg wrote:
http://en.wikipedia.org/wiki/Dialetheism
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.
Brent
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.
Bruno
http://iridia.ulb.ac.be/~marchal/
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