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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: Minimal Logics in the 2020's: A Meteoric Rise
Date: Thu, 4 Jul 2024 20:46:17 -0500
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On 7/4/2024 8:26 PM, Mild Shock wrote:
> 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.
> 

When provable means true and false means unprovable
then (Γ ⊢ X) means X is true in Γ.
then (Γ ⊢ ~X) means X is conventional false  in Γ.
the (Γ ⊬ X) ∧ (Γ ⊬ ~X) X is not a truth bearer in Γ.

> 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.
>>
>>
> 

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer