“Up to a strong form of equivalence, the only sublogics of second-order logic given by first-order, restricted second-order quantifiers are first-order logic, monadic second-order logic, permutational logic, and full second-order logic”
So, to someone that’s fixed on first-order logic, these are options from which they can chose in order to keep the provability they love with the flexibility afforded by 2nd order logic.
Good paper. I think that’s what draws me to focusing on monadic second order logic as a _minimum_ basis rather than first order.
First order can’t use itself to prove itself. But monadic 2nd order logic can.
And – I just found this from that same site:
You gotta go up to 2nd order. That’s how you do it.
The paradox is imposed by the constraints of the described system. To me that shows it’s breaking point, the point where “it’s time to move to a more advanced level”.
But logics aren’t native to my thinking. My logic journey STARTED with 2nd order logic, in the form of Microsoft BASIC on my Tandy Color Computer 2 in 1983.
Sure, I learned boolean too. But I saw boolean as “within” a greater system which was afforded to me by programming.
It’s only as an adult – probably my early 40s – that I learned that “lambda calculus” is the “mother calculus” behind my early BASIC, and the logic that equates to it is 2nd order.
So for me, Godel seems silly in 2019 as multiple solutions came out.
I think what I’m mostly working towards is “How Can I Describe My Thinking?”
It’s not looking for “best answer for everything” per se, although that would be fun.
But I want to understand myself. Once I understand myself, I can explain myself.
Exactly! So, this is why I’m at “Ontologies”. I never cared for most “Master Ontologies” or Taxinomies as divisions are too often arbitrary.
Now I’m thinking: So if it’s arbitrary, I’ll arbitrate. So, what’s MY ontology?
That was the brilliance of Church’s lambda calculus. False=True, if you want.
Conditionally true can work for me. Truths can arise from within the distinctions.
What about actors and actions? Choices?
If I choose to fold or cut reality in some way, what validates my choice?
You said: “I have been unable to isolate any truth from distinction”.
Breaking that further down, is “distinction” a thing-in-itself or is an “action of distinguishing” performed by an “actor”.
I’m using actor as “one who is doing”. Do you do?