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]
<mailto:[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, but that doesn't mean material implication is the right
formalization of it. You've assumed that Kripke's formalization IS the modality. You
still haven't explained why the formalization denies that a world is accessible from itself.
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.
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.
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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