Re: universal quantification, because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

On 05/11/2024 02:05 PM, Ross Finlayson wrote:

On 05/11/2024 12:24 PM, Jim Burns wrote:

On 5/11/2024 11:47 AM, Ross Finlayson wrote:

On 05/11/2024 07:40 AM, Jim Burns wrote:

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In the logical, the purely logical,
the syntax "is" the semantics.

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If what makes logic impure is
to be about something,
then it would make some sense to say that
pure logic has no semantics
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...which leads, by default?
to syntax being the missing semantics, I guess?
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Sorry, I will not sign your petition.
Syntax and semantics are more different than
cabbages and kings.
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It seems to me that
the purest of ultra.pure logic is actually
_about_ claims,
analogous to geometry being _about_ points,
lines, plane.figures, and so on.
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It is an unbreakable law that
the sum of the squares of the two shorter sides
of a right.triangle is equal to
the square of the third and longest side.
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It is an unbreakable law that
a finite sequence of only not.first.false claims
holds only true claims.
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It is an unbreakable law that
Q preceded by P and P⇒Q is not.first.false.
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It's exactly about "not.ultimately.untrue" that
describes how there are "inductive impasses"
that belie their finite inputs.
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These are found in all the greatest seminal arguments
of objects of reason their dissonance yet harmony,
complementary duals, anything that's otherwise a
"paradox" of mathematical logic, which yet, is not.
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The "pure theory" is, "all the things, theoretically".
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Then in set theory there are ubiquitous ordinals,
for example, sharing the background, sharing the substrate,
of a continuum of objects, making the _bridges_,
the analytical _bridges_, the ponts.
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Where it's so that induction is unbreakable,
whether what _holds_ it, _is_ it, is the most
usual sort of example of comprehension and
quantification together.
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It's the most usual sort of example that
it _is not_, point to any argument about
Russell's "set" and any inductive set
which according to itself is the entire
world, or universe, of sets that don't
contain themselves, which you forget
is prohibited.
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So, the "conscientious" bit, is, even having
to always take into account any exceptions to
the rulial what are so rulial themselves,
regular, regularity, because "truth is regular".
Not: "regular is truth".
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It's not an unbreakable law that the Law of
Excluded Middle holds for all propositions,
because LEM or Tertium Non Datur TND, only
effect a reflection on a class of propositions,
not including those most pivotal and crucial,
of what would otherwise be logical antinomies,
which are not, because they are purely logical.
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Of course any sort of gathering of propositions
is its own little pure logic itself, yet, we're
interested here in the foundations and altogether,
besides.
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You speak of the meta-theory, and that there is one,
and we might call it pure logic, and it exists,
and we attain to it, because we're conscientious,
and, altogether thorough, as diligently as we can be,
mathematicians qua logicians qua theoreticians.
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Lock is lit, ..., and around it goes.
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Here's an example.
You encounter a river, it's either Styx or Lethe, I forget.
It's upon you to cross the river, or gorge, quite similar
to the recent episode recounted of the requirements
and consequences of crossing, or, not.
So, you see a guy across the river, no surprise, it's Zeno.
You imagine to consult him.
"HOW DO I GET TO THE OTHER SIDE OF THE RIVER?"
He hears your question.
"COME AGAIN?
"HOW DO I GET TO THE OTHER SIDE OF THE RIVER?"
He considers you for a moment.
"YOU ARE ON THE OTHER SIDE OF THE RIVER."
Then I imagine you might clarify,
"NO I MEAN TO YOUR SIDE OF THE RIVER."
Then you notice he's standing next to you and
says "go all the way across".
It's like two inductive analysts were contradicting
each other. One says "base case, subsequent
case, case closed", and the other says "base case,
subsequent case, case not closed".
You just pick one?