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.
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.
Peano arithmetic here is only an example of sound and correct Löbian
machine. The truth of 2+2=4 does not depend of the truth of if this or
that machine believes it or not. Yet with comp, the proposition "the
machine x believes y" becomes theorem of sigma_1 complete machine.
It is an ideal case, amenable, by comp, to mathematics. That ideal
case leads to an already very subtle theology, with some canonical
struggle between the different views the self can take. The machine's
soul is bipolar at the start, well octopolar.
Although PA only scratches the arithmetical reality, PA is already
quite clever and self-aware about its own abilities.
Bruno
Brent
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