Re: Minimal Logics in the 2020's: A Meteoric Rise

I minimal logic, if you define:
    ~A := A => f
You can already prove, thats just modus ponens:
    A, ~A |- f
Or if you want you can also prove, i.e. rephrase
it, with approproate introducton of a conjunction &
by Curry Howard inference rules for "products":
    |- ~(A & ~A)
Thats the law of non-contradiction:
https://en.wikipedia.org/wiki/Law_of_noncontradiction
Its hard wired into minimal logic.
olcott schrieb:

On 7/4/2024 5:03 PM, Mild Shock wrote:

Could be a wake-up call this many participants
already in the commitee, that the whole logic
world was asleep for many years:
>
Non-Classical Logics. Theory and Applications XI,
5-8 September 2024, Lodz (Poland)
https://easychair.org/cfp/NCL24
>
Why is Minimal Logic at the core of many things?
Because it is the logic of Curry-Howard isomoprhism
for symple types:
>
----------------
Γ ∪ { A } ⊢ A
>
Γ ∪ { A } ⊢ B
----------------
Γ ⊢ A → B
>
Γ ⊢ A → B           Δ ⊢ A
----------------------------
Γ ∪ Δ ⊢ B
>
And funny things can happen, especially when people
hallucinate duality or think symmetry is given, for
example in newer inventions such as λμ-calculus,
>
but then omg ~~p => p is nevertheless not provable,
because they forgot an inference rule. LoL
>
Recommended reading so far:
>
Propositional Logics Related to Heyting’s and Johansson’s
February 2008 - Krister Segerberg
https://www.researchgate.net/publication/228036664
>
The Logic of Church and Curry
Jonathan P. Seldin - 2009
https://www.sciencedirect.com/handbook/handbook-of-the-history-of-logic/vol/5/suppl/C >
>
Meanwhile I am going back to my tinkering with my
Prolog system, which even provides a more primitive
logic than minimal logic, pure Prolog is minimal
>
logic without embedded implication.

 Prolog logic such that provable means true and false
means unprovable is the key foundation of correct reasoning.
 It simply implicitly rejects expressions that would otherwise
result in mathematical incompleteness as not truth bearers
within its system. This is the way that correct reasoning
actually works.
 When expressions of language are self-contradictory such
that X and ~X cannot be proven within the system Prolog
rejects X. Mathematical logic would conclude that the
system is incomplete.