There are many Logics, based on different Axioms.
/ EN: 'Axiom' = PL: 'Aksjomat' /.
Axioms are statements that are not proven, but assumed as true, taken on faith.
/ EN: 'assumption' = PL: 'założenie' /.
Depending on the Axioms used, Theorems can be proven or disproven, and the whole Mathematical & Logical Apparatus can be developed.
Basing on Boolean Algebra, double negation evaluates to confirmation, but in some languages - polish for example - double negation does not mean confirmation, it does mean emphasis on negation, giving negation more power.
There are rules for negating Quantifiers as well.
/ EN: 'negation' = PL: 'zaprzeczenie' /,
/ EN: 'confirmation' = PL: 'potwierdzenie' /,
/ EN: 'emphasis' = PL: 'nacisk' /,
/ EN: 'quantifier' = PL: 'kwantyfikator' /.
/ według Algebry Boole'a i Teorii Kwantyfikatorów wylicza się do:
'istnieje taki moment w czasie kiedy zgodzę się na te warunki'. /
/ a w mowie potocznej znaczy: 'nie istnieje taki moment w czasie kiedy zgodzę się na te warunki'. /
Therefore, Speech of the Art, Literature, can have the Logic based on different Axioms than Boolean Algebra.
We can say that: 'Life is more than Boolean Algebra and Quantifiers' when we want to use casual, non-logical talks.
When Boolean Algebra and Quantifiers are useful then? It's useful in computer programming, or when we want to talk logically and precisely, or when we want to express our wishes logically and precisely. But when we opt for logical and precise speech, let's make sure first that other people we talk with understand our logic.
We can also define the addition operation / it's an Axiom too / differently as well.
We can have an exception:
1+1 = 2, 1+2 = 3, 2+1 = 3, 1+3 = 4, 3+1 = 4, 2+2 = 5, 1+4 = 5, 4+1 = 5, 2+3 = 5, 3+2 = 5, 1+5 = 6, 5+1 = 6, 2+4 = 6, ...
We can also redefine the addition operation differently on the more general, more universal scale:
n+2 in our redefined addition operation is n+2+1 in classical addition operation.
By doing so, by changing Axioms, we just have revolutionized the Mathematics. ;)
Many different theorems apply now, but at least we know that we can make expression 2+2 = 5 to be evaluated as true in a certain Context - even if this brings more or less desired effects in process ;).
We are free to assume any Axioms we want, examples can be multiplied infinitely.
Which Logic is 'better' than other, then?
... it depends on the assumed Criteria, which might be Axioms as well.
