Re: The set of necessary FISONs (axiomless natural deduction)

On 02/05/2025 10:25 AM, Jim Burns wrote:

On 2/5/2025 8:25 AM, Ross Finlayson wrote:

On 02/04/2025 08:26 PM, Ross Finlayson wrote:

On 02/04/2025 11:38 AM, Jim Burns wrote:

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[...]

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What it's all about is
"The Principle of Sufficient Reason".

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⎛ The principle of sufficient reason states that
⎝ everything must have a reason or a cause.
https://en.wikipedia.org/wiki/Principle_of_sufficient_reason
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What causes everything to have a cause?
Does that have a cause? Is the cause itself?
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I think that 'cause' is insufficiently described
For this discussion.
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Judea Pearl and his colleagues have done admirable work
in this area.
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There's that
the principle of sufficient reason is satisfying, and,
the principle of sufficient reason is satisfied.

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Does making an unsupported claim ("There's that...")
count as offering a reason?
May I offer unsupported claims as reasons, too?
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So, axiomatics,
or modern weak logicist positivism
or the nominalism or fictionalism
all about same,
have unfounded axioms that supposedly
thusly make for both
that anything that can be derived can be derived,
yet also of course
that anything that can be derived must be derived,
here that's model theory,
and a structuralist view,
and it's equi-interpretable with proof theory,
insofar as inter-subjectivity is established,
and equi-interpretability, in language.

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"That anything that can be derived can be derived"
is clearly true. Axiomatic, even. So what?
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"That anything that can be derived must be derived"
is clearly false,
if you mean what I mean by 'derive':
among other things,
a non.empty list of actions by finite beings (me and my ilk)
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Those which can be derived
are infinitely.many.
The resources available to derive with
are finite.
If the rule is
"That anything that can be derived must be derived",
then the rulemaker will be disappointed.
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"Unfounded axioms" sounds to me like
a key to making sense of what you mean by
"axiomless geometry" and its ilk.
Is it
not "no axioms", but "no unfounded axioms"?
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Hm. Have you heard of "first principles" and "final cause"?
In logicist positivism, it's either/or "an axiom system"
and "science", is the usual idea.
Anyways it's usually attributed to "idealism"
vis-a-vis "the analytical", and there's quite
a long story about it it's sort of nice to
have ideals.
Then, something like Hegel's "Being and Nothing",
if you read the Wissenschaft der Logik, is pretty
great, then there's Kant's Sublime, pretty usual,
these being the things that the analytical tradition
does not and cannot say much about, yet, Hegel and
Kant do, because they're idealists.
Then, "anti-Platos" like Wittgenstein, Nietszsche,
and Heidegger, say, sort of have that Gadamer arrives
for hermeneutics at "amicus Plato", and, the Tractatus
Logico-Philosophicus, which is really sort of a reading
of Leibniz' monadology or about the radical origination
of things, is quite, strongly platonic, and idealistic.
If you're going to have a theory at all,
it might as well be the good one.
Then, here Duns Scotus' super|natural is considered
pretty great, and Chrysippus of course provides the
modal against Plotinus and his fallacies of material
implication, making sure that Aristotle won't be fooled.