On 21 Jun 2015, at 20:32, meekerdb wrote:
On 6/21/2015 8:50 AM, Bruno Marchal wrote:
On 19 Jun 2015, at 23:32, meekerdb wrote:
On 6/19/2015 10:57 AM, Bruno Marchal wrote:
On 19 Jun 2015, at 02:36, meekerdb wrote:
On 6/18/2015 4:11 PM, Bruce Kellett wrote:
meekerdb wrote:
On 6/18/2015 1:10 PM, John Clark wrote:
On Thu, Jun 18, 2015 at 1:51 PM, meekerdb <[email protected]
> This is gitting muddled. '2+2=4' is a tautology if the
symbols
are given their meaning by Peano's axioms or similar axiom
set and
rules of inference. If the symbols are interpreted as the
size of
specific physical sets, e.g. my example of fathers and
sons, it's
not a tautology.
In an equation, ant equation, isn't a tautology then it isn't
true.
An equation is just a sentence. A tautology is a declarative
sentence that's true in all possible worlds. 2+11=1 in worlds
where addition is defined mod 12. That's why an equation
alone can't be judged to be a tautology without the context of
its interpretation.
But your counterexamples are simply changing the meaning of the
terms in the equation. I agree that a tautology is true in all
possible worlds, because its truth depends only on the meaning
of the terms involved. If the meaning is invariant, the truth
value does not change. But this is not invariant under changes
in meaning.
"2+2=4" is a theorem in simple arithmetic, and a tautology
because of the way we define the terms. In a successor
definition of the integers:
1=s(0),
2=s(s(0)),
3=s(s(s(0))),
4=s(s(s(s(0)))),
2+2=4 can be proved as a theorem. But that relies on the above
definitions of "2", "4" etc. In modular arithmetic, and with
non-additive sets, these definitions do not apply.
Note, however, that this interpretation of 'tautology' differs
from the logical interpretation that Bruno refers to.
Bruce
I don't think it's different if you include the context. Then
it becomes "Given Peano's axioms 2+2=4". Isn't that the kind
of logical tautology Bruno talks about? Within that meaning of
terms it's a logical truism. I don't think it's necessary to
restrict logic to just manipulating "and", "or", and "not".
Bruno introduces modalities and manipulates them as though they
are true in all possible worlds. But is it logic that a world
is not accessible from itself?
As you say, it depends of the context. Yet, the arithmetical
reality kicks backs and imposed a well defined modal logic when
the modality is machine's believability(or assertability), for
simple reasoning machine capable of reasoning on themselves, as
is the case for PA and all its consistent effective extensions.
But why should we think of modal logic and the measure of true? I
still haven't heard why a world should not be accessible from
itself. Logic is intended to formalize and thus avoid errors in
inference, but it can't replace all reasoning.
Don't confuse Logic, the science, with some of its application.
Then in our case, computationalism ovites us to study machines and
computations, which are not logical notions, but needs non logical
assumption (like x + 0 = 0, or like Kxy = x).
Then we study what those machine can really believe ratioanlly and
non ratioannaly about themselves, and modal logic appears there by
themselves, because provable/believable, knowable, observable
simply *are* modalities.
Sure. And implies is a inference,
Implies entails inference, by the modus ponens; and inference entails
implication, by the deduction theorem (which is not always true for
modal logic, so we have to be cautious).
but that doesn't mean material implication is the right
formalization of it.
It is usually, but not in all case. But this asks for some caution
which have been taken.
You've assumed that Kripke's formalization IS the modality.
?
It is part of Solovay theorem that the provability logic admit a
Kripke semantics. That is true for a large class of modal logic. It is
enough they prove K and are closed for the necessitation rule (which
is a form of self-awareness).
You still haven't explained why the formalization denies that a
world is accessible from itself.
It is a consequence of Gödel's second theorem and kripke semantic. []p
-> p is not a theorem, indeed []f -> f is true, but not provable, so
by Kripke semantics there are some world not accessible by itself. I
can come back on this (but this has already been explained in detail
once).
Arithmetical truth is a well defined notion in (second order)
mathematics. It does not ask more than what is asked in analysis.
But all first order or second order *theories*, effective enough
that we can check the proofs, can only scratch that arithmetical
reality, which is yet intuitively well defined.
It is not "Given Peano axioms 2+2=4". It is because we believe
since Pythagorus, and probably before, that 2+2=4, that later we
came up with axiomatic theories capturing a drop in the ocean of
truth.
I didn't say that's why we believe 2+2=4; I said that's what makes
it a tautology, i.e. when you include a context within which is
provable.
What about Riemann hypothesis, or even the (apparently solved)
fermat theorem?
Today, we might still believe that both are provable in PA. Would
this made them into tautology?
Would you say that it is a tautology that even numbers have 24
times (the number of its odd divisors) clothes and odd numbers have
8 times (the set of all its divisors) clothes (with the cloth of a
natural number being a representation of the sum of four squared
integers)?
Well, as they involved non logical axioms, the expert in the field
call them theorems.
Every sentence of the form "axioms imply theorem using rules of
inference" is a tautology.
It depends what you mean by tautology. It is usually seen as a
metatheorem (which is not a logical tautology).
If the theory is reasonable enough, theorem-hood entails truth in
all interpretations of the theory,
But by "all interpretations" you mean all interpretations that make
the axioms true.
Yes. And we have completeness, so the reverse is true too: what is
true in all interpretation is a theorem too.
Bruno
Brent
which means that the statement is "true" independently of its many
possible meanings/interpretations/models. But we use the term
"valid" for that weaker sense of "truth".
Once a theory get the sigma_1 complete complexity threshold, it
becomes *essentially* undecidable. Not only it cannot prove all the
truth (notably about itself), but none of its consistent extensions
ever will.
Once you get the sigma_1 complete treshold, you are forever
incomplete and incompleteable.
RA illustrates well that treshold, because all of its subtheories
(non sigma_1 complete) are incomplete but all admits complete
consistent extensions. This is explained in a famous paper by
Tarski Mostowski and Robinson which has been printed by Dover.
http://www.amazon.com/Undecidable-Theories-Studies-Foundation-Mathematics/dp/0486477037
Bruno
Brent
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