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
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,
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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