Logic & Axioms.

'My Logic is based on different Axioms than Yours'.


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.

/ EN: 'negation' = PL: 'zaprzeczenie' /,
/ EN: 'confirmation' = PL: 'potwierdzenie' /,
/ EN: 'emphasis' = PL: 'nacisk' /.


/ PL: 'nigdy nie zgodzę się na te warunki'. /

Therefore, Speech of the Art, Literature, can have the Logic based on different Axioms than Boolean Algebra.


We can also define the addition operation / it's an Axiom too / differently as well.

We can have an exception:

for example:

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:

for example:

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.

The Enigma Cipher of WW2 & the Turing Machine.

Polish and British mathematicians were among the best of people who cracked the Adolph Hitler's cipher named Enigma, it happened during the World War 2nd.

Enigma breaking was hard, and the mathematicians were hunted by Germany's spies.

Enigma evolved, so parts of the cipher were to be cracked again and again. It was not about making an automaton once, and letting it work for the rest of WW2 ... but statistically it worked so the effort was continued.

Doing maths when time flew and lives were at stake.... so stressful. The Germany's spies added to the dangers & to the stress too.

Polish mathematicians had one of few of first computers ... it was nicknamed: 'Bomb', for it was so big invention. It increased efficiency of the enigma cipher cracking.

And there was Alan Turing's effort of course. He was a British scientist who laid foundation-theory of computer's construction.

His thinking is still present in computer sciences of modern days.

(We were taught about Languages & Automatons in Warsaw University when i was studying computer sciences. Turing's Machine was a part of this lecture).

> [ https://en.wikipedia.org/wiki/Turing_machine ].

The Turing Machine is programmed in a similar way to programming the Register Machine.

> [ https://en.wikipedia.org/wiki/Register_machine ].


--
Sources:

1. 'Cubits & Shrodinger's Cat. From Turing Machine to Quantum Computers' by John Gribbin.
(Polish Edition).

2. The internet (wikipedia & the ważniak mostly) and my own thinking.

3. My (unfinished because of health problems) education at Warsaw's University (Mathematics, Informatics & Mechanics Faculty).