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
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