And then 2:
(x ® y)=(x ® y)(; N=(x ® y)=N; N, so (x ® y)=(x ® y)):
(x ® (y ® y))=((x ® y) ® (x ® y)); Tp:
2) (x ® (y ® z))=((x ® y) ® (x ® z)); z=y.
z, or any other x (ex ante) ≠x,y, which given N (ex post, after assumption of N) of course is x or y:
z=x,y.
Especially this is "hidden" or simply not is seen in Nlogic, because it complicates, so that then this that z=x,y is kept hidden. With which z is believed to be something other than x or y, which of course defines something much more complex than if it only is about x and y, which it then only is given N, Nlogic (Classical logic) commonalize (overinterpret) thus in a treacherously way, because it doesn't understand better (hopefully, otherwise it is of course a matter of fraud).
To also take the two "hypothetical syllogisms" at https://en.wikipedia.org/wiki/Hypothetical_syllogism#As_a_metatheorem that Łukasiewicz presupposes in his proof of Dl, for the sake of enlightenment:
(y ® y)=(((x ® y) ® (x ® y)):
HS1) (y ® z)=(((x ® y) ® (x ® z)); z=y.
(x ® y)=((x ® y) ® (x ® y)):
(x ® y)=((y ® y) ® (x ® y))(; N):
HS2) (x ® y)=((y ® z) ® (x ® z)); z=y.
That Lpformulas, to lastly also take two distributive formulas:
x=(x Ù x)=(x Ù y)=(x Ù (y Ú y))=(x Ù (y Ú z)); z=y.
x=(x Ú x)=((x Ù x) Ú (x Ù x))=((x Ù y) Ú (x Ù y))=((x Ù y) Ú (x Ù z)); z=y.
So:
(x Ù (y Ú z))=((x Ù y) Ú (x Ù z)).
x=(x Ú x)=(x Ú y)=(x Ú (y Ù y))=(x Ú (y Ù z)); z=y.
x=(x Ù x)=((x Ú x) Ù (x Ú x))=((x Ú y) Ù (x Ú y))=((x Ú y) Ù (x Ú z)); z=y.
So:
(x Ú (y Ù z))=((x Ú y) Ù (x Ú z)).
All this (which then follows from primarily N) then Łukasiewicz presupposes before he even so to speak begins to prove Dl, with which the whole of course appears to be very complicated, and in its own way it of course is, but then selfcreated complexity, because directly starting from N it is thus easy as a pie to prove Dl.
Anyway, the foregoing shows that extremely much can be defined based on/given N, but N is thus (rationally) completely false, with which these derivations are equally false, however much they may seem to define something that makes sense. But if any of these formulas are to be used in logical analysis, then they have to be argued for per se, each formula (ex ante) be ascertained if it is relevant to assume in some context, or not. To assume them valid on the basis of N is (rationally) simpliciter not possible, but they then must so to say stand on their own legs, (ex ante) be argued for to do so, be rational (intuitive) in the context in which they perhaps are assumed (to be valid); Such "simple" principle as Lp for example is extremely complicated to see (analyse) the validity of in specific contexts, for example if [x+z=y+z]=[x=y], because directly interpreted this sentence is naturally not valid, the left term is de facto not identically the right term, unless y=x, in which case the relation trivially is valid (given Up/Ip), but what if y≠x ‒ the more normal regarding this with x and y. If y=x, then it mostly is meaningless to define "y", mostly just confusing, if now then y=x, in which case it then most rationally is to stick to x(=y=x), the only exception is when rewriting identities, especially mathematically a common method of arriving at conclusions, simply then by rewriting identities (with help of the principles which are assumed to fit for that) ‒ is the [x=y]relation in some meaning remaining if z is added to x and y respectively? For example, is father+mother=child+mother identically father=child? No, hardly, but it really needs to be analysed if this Lpprinciple can be seen as valid/rational in some context.
By way of conclusion concerning Classical logic, it defines the following "Truth table" in accordance with N, to be compared with the rational table in the introduction section:
x y true false false true
This table which then shall be interpreted in accordance with N, which primarily is irrational in two ways: the N*perspective, that (unique) x ® (unique) y, then contradicting Ia and Ib, and the (x,y≠0)perspective, that there always is a unique true y if x is false, then contradicting that there can be completely false x; The assumption that x,y≠0 can Classical logic be argued to make because N is difficult to justify if x or y can be 0: x’=0 or 0’=x; x≠0, for which x’(=nonx) defines 0, and which x defines (is defined by) 0’(=non0)? Secondarily there is an "eye"perspective underlying N, that the "eye" finds, sees the relevant nonx among all the irrelevant nonx, which Classically logically usually is defined: If x is a proposition, then also x’ is a proposition, for example defined on page 68 of Language Proof and Logic, or page 97 of Principia Mathematica (â1.7), which only is absurd. There is no such connection/relation between different x, neither mentally (in that case it is about some prejudice) nor "platonistic", that it only (eternal) is so, which just only is even more absurd than that there is a mental connection between different words, concepts, that there in the language are eternally given connections between different words, concepts. Which not prevents especially Kurt Gödel (19061978), with his incompleteness theorems (contradicting FT), from still being a platonist, commonly in the meaning that theories X can exist equivalent empirically, thus per se (beyond especially the human's consciousness), and certainly, if the existence of platonistical X is thought, it is a fairly given thought that these X just only may define (undecidable/independent) x, without them being either provable or disprovable, without being axioms. But rationally, again (more rigorously of course in accordance with FT), this is only nonsense, it is the consciousness that defines X, nothing is determined/defined before it is determined/defined, see further the section Concerning FT in Addition II.
Given the Etheory the only rational alternative for 0=[no x(≠0)] is void/room ({mv}), with which it for example (superclonically, contradicting Up’) can be defined that xx=0, with the evident interpretation that if x is excluded from itself only empty space remains. For 0 to not cause problems in analysis, the best is to assume 0 to be idempotent, so that the right side in following example still is 0:
∞(xx)=∞0=0.
Because if 0 for example is assumed to be p (principally, as part of empty space p's are empty space), then of course ∞0=dp, something unintuitively arises; And of course even more so if 0>p, for example a volume, nonidempotent volume, with which of course x0>0; x>1.
With this it is into mathematics (again), which it especially given the Etheory is easy to slip into, because it is evident that the Etheory defines a fundament for development of mathematics, especially geometry, and the Etheory also defines (uses) certain basic mathematical concepts. But not much, because this work primarily (of course) not is mathematics (then it simpliciter not would be able to define what it does). More developed mathematics is so to speak a later issue, a problematic issue, which already has been discussed, especially Lp's unreliability shows. Lp which indiscriminately is used in mathematics. For example for proving the equivalent to Dl in mathematics, roughly as follows, given this work:
A) xx=0:
I) (xx)=0; Lp, []=[1]:
xx=0; (xx)=xx (distributive principle), 0=0, which is valid given that 0 is idempotent:
xx=0; xx=xx (commutative principle):
II) x=x; A(, Up).
Because it is the matter of manipulation of two superclones (of course contradicting Up’, but without the assumption of superclones there is no mathematics; Mathematics specifically assumes the existence of superclones through the assumption of the Axiom of extensionality), where then x superclonically is excluded from itself, the assumption of distributivity and commutativity is (intuitively) no problem (because (especially as) it is about summation (of (purely abstract) superclones) to 0), it is however in no way intuitive that exclusion of xx ((xx)) is the same as xx,* which it then given Lp is (for A to be identically (to) I). Neither is the result, II, intuitive, intuitively exclusion (x) of an exclusion (x) simply is a (tautological, pleonastic) double rejection (throwing away) of the exclusion, but primarily then given Lp, it brings back to x, just as if N had been assumed (x=x).** N which definitely not is assumed to be generally valid in (pure) mathematics, in which it not is one if x is plus or minus (in applied mathematics it can be one, for example for length differences):
II can also be proved without Lp (and other complexities), because in accordance with A (A:s intension) it also is valid that:
A’) x+x=0.
So if x=x, it insubstituted in A gives:
xx=0:
x=(+)x; A’.
But in any case is Lp mathematically extremely important, so not mathematically wrong to prove II whit the help of Lp, and in a way lucky that Lp given this later proof of x=x leads to just that, not leads to, proves something else.
Well, Lp leads, as already been touched upon, to strange conclusions (although Lp in its specific formulation in this case of course brings in negation, which of course N also is about), albeit of course, concerning II, a practical conclusion, whose practical validity simply has to be tested, and in that II has turned out to satisfaction, obviously, otherwise II (of course) would not (mathematically) be used. And given the latter proof, given A’, it (rationally) simply must be valid, completely independent of Lp.
All in all, perhaps the most important this work teaches is that no x is given (determined before it is determined), but it is all about assumptions, definitions, interpretation of reality which results in assumptions, primarily in first x, basic x, "axioms" (".." because rationally truly fundamental "axioms" are almost given, given true, especially then Up, but can they, especially then Up, be disproved, then of course they not are given, but disproving especially Up is (rationally) completely impossible, because every proof, whichever, must presuppose Up (or equivalently) simpliciter for the proof to be the proof (x=x; x=[the proof]), not being something else (x≠x)), and secondarily especially important is about Iiinterpretations/derivations based on these basic x, or perhaps concerns derivations given/provided some assumed principle of derivation (rule of inference, such as then for example Lp), theorems:
basicx ® theorem.
Another "empirical" possibility is that basicx sees, interprets as the (causal) basis for something "empirically" outinterpreted:
e) basicx ® hypothesis.
If an Iiinterpretation ("continuous logic") is difficult to find out in this, it has to be content with that, especially (pragmatically) if it is a practical implication. But naturally, the best is to try to find an Iiinterpretation, even if it not always is possible, take for example that that thrusted mx "empirically" seems to move reasonably in the thrusting mx "jump"direction, which it then is no Iiinterpretation in/for.***
Only e may rationally be left unexplained, if it is about purely abstract theory, it can rationally not be left unexplained, in accordance with FT.
__________ * (xx)=xx (implicatively identically, just like (xx)=) for example, but this then with no deeper meaning than that the right term is in the left term), but xx is not given =(xx) (exactly as x=x’ not is given, but then x’=x (because x is in the left term)).
** Given that x’=x, which without further ado can be stated to be what Classical logicians means that N defines, and it also is intuitive in one case, namely if it is defined that x’=Ex, thus that x’ is All exclusive (except) x, in which case excluding x (x) intuitively defines x’: x=x’=Ex, which defines that E=0’’; x±0’’=x, where 0’’ intuitively closest is 0*: 0’’=0*, which can be proved if Lp is assumed (which is done in section E), but Lp is then unreliable, so it has to suffice whit this intuition, which defines 0* to be a duality, both the largest (E, which 0* then can be interpreted as, as positionless) and the smallest (0’’, which it is evident that 0* is, interpreted as nonextension, only), which adds nothing to x, which argues for 0 to be defined to be (idempotent) void (empty space) ≠E, for the sake of distinction.
