4) (x y)=(y x).

 

5) (x (y z)=(y (x z).

 

6) (y z)=((x y) (x z)).

 

7) If x is an elementary proposition, x=y is an elementary proposition (=N).

 

These principles are by Principia Mathematica assumed to be true without proof, commonly none of them are evident true, except for 2, apart from the parenthesis sentence, which actually (the parenthesis) Classically logically is the most important part of this axiom, 2 which really is an axiom apart from the other sentences apart from 7. 3-5 are certainly not evident. 6 is valid provided that x not impli-cates z, which x of course commonly can, in which case the formula reads: (y z)=((x y) z); x z, and with which of course 6 is false. 3-6 are however true given 7 and 2 (2 which is in accordance with the more common Tp). 7 which defines N, although 7 only de-fines that x=y, that y=x is valid is implicitly assumed, simpliciter because Principia Mathematica (Classical logic) assumes Dl (the Law of double negation) to be valid, which thus presupposes an assumption of N (without (assumption of) N Dl not exists (not is valid)): 

 

3:

 

y=(y y):

 

y=(x y); N.

 

4:

 

(x x)=(x x); 2 and that x=x, which then is valid given N, which more specifically then defines that N=(x y),(x y),(y x),x,y=N(; N):

 

(x y)=(y x); N.

 

5:

 

Given 4:

 

(x y)=(y x):

 

(x (y y))=(y (x x)); 2:

 

(x (y y))=(y (x y)); N:

 

(x (y z))=(y (x z)); z=y; z=x,y; N.

 

6:

 

(y y)=((y y) (y y)); 2:

 

(y y)=((x y) (x y)); N:

 

(y z)=((x y) (x z)); z=y.

 

It is thus 7, or then N, which is the primary principle, and then then 2 (or more commonly Tp), 3-6 are theorems which follows from these two principles. This which consequently Principia Mathematica did not see, evident so, otherwise of course 3-6 would not have been set up as axioms ("primitive propositions", but then have been proved). And no one else has seen either, until this work, amazing to say the least. To further exemplify how N-logicians (Classical logicians) fundamentally view their N-logic, a look can be taken at the following contemporary (at the time of writing ongoing) work:

 

 

Principia Logico-Metaphysica

 

Which defines the following three "axioms" in chapter 8:

 

1) x=(y x).

 

Which of course directly follows from N (N=(y x),x=N; N).

 

2) (x (y z))=((x y) (x z)).

 

Already proven, but to take it again:

 

(x  y)=(x y); N:

 

(x (y y))=((x y) (x y)); Tp:

 

(x (y z))=((x y) (x z)); z=y.

 

3) (x y)=((x y) x).

 

It is a little fascinating that N-logicians find these sentences/formulas without (as it seems) taking direct help from N, but given N, in the context especially that x=y and y=x, and Tp, then 3 of course is a trivial theorem:*

 

(y x)=(y x):

 

(x y)=((y y) x):

 

(x y)= ((x y) x).

 

In chapter 9 a number of formulas are set up, all of which are easy to prove given N and Tp, to take some out of this heap, then N and Tp given:

 

(x y)=y:

 

(x y)=x:

 

(x y)=(x x):

 

(x y)= (x y).

 

x=x:

 

y=(x x):

 

(x y)=(x y).

 

(x y)=y:

 

(x y)=(y y):

 

(x y)=(x y).

 

x=x:

 

y=(x x):

 

(y y)=(x y):

 

(x y)=(x y)(; N).

 

y=x=(N) (y=N also is valid, so here N-logic is contradictory, which N-logic (of course) ignores, given that it assumes N (by which of course the consequences of this assumption (this validity) only have to be taken/accepted, which parenthetically means that many N-lo-gical formulas are sorted out, because they Classically logically are so obviously unintuitive (contradictory),** but then not the following, because it is intuitive, even if then contradicting the validity of N (x which derives contradictions are false (Kp)), which N-logicians simpliciter not sees, because they not have N explicitly in front of their eyes, yes, they simply not sees N, are unaware of N's meaning/-importance, despite then explicitly define N many times, then by assuming that x is a proposition if x is (and vice versa))):

 

(x=x).

 

N=N:

 

(x=y)=(y=x).

 

N=N:

 

(x=y)=((x x)=(y y)):

 

(x=y)=((x y)=(y y))(; N):

 

(x=y)=((x z)=(y z)); z=y.

 

N=N:

 

(x=y)=((x x)=(y y)):

 

(x=y)=((x y)=(y y)):

 

(x=y)=((x z)=(y z)); z=y.

 

N=x:

 

(x=y)=(x x):

 

(x=y)=((x x) (x x)):

 

(x=y)=((x y) (y x)):

 

(x=y)=((x y) (x y)).

 

x=x:

 

(x x)=y:

 

(x y)=(y y):

 

(x y)=(x x):

 

(x y)=(x y).

 

y=y:

 

(x x)=(y y):

 

(x y)=(y y):

 

(x y)=(x x):

 

(x y)=(x y).