That Gödel formally proves the existence of undecidable/independent x depends on N, and is with that of no importance, because that formalism (that builds on N), Classical logic, thus simpliciter is wrong/irrational. Because already (the assumption of) N (the Negation) principally defines this with undecidable/independent x, defines the existence of x which (platonistically) unprovable exists per se (equivalently empirically), N just only (platonistically) is valid (thus either (the unique) x or the (the unique) y, then per assumption of N, it is thus the matter of platonism per assumption), precisely what also is valid (is assumed to be valid, or then Classically logically is proven to be valid, of course provided N) for undecidable/independent x; Given N, N is valid for every z within X domain, if z so are provable within/by X or not, x or y is true for z, then given N. For example it is (then) valid that it is unprovable if mx have attraction force in the E-theory (the "empiri" is then in the E-theory taken to help answering this question), a question (z) which of course is within the E-theory’s domain/area of definition, a question which if N would have been valid in the E-theory of course (platonistically) would have had an answer, namely then either x or y, where x and y of course nearest define the two alternatives that mx either has attraction force or not (a not-case which then negated in accordance with Dl defines (brings back to) that mx possesses attraction force, which simpliciter is absurd (how can something so common as this not-case bring back to something that specific, or what does this not-case more specifically define?), clearly shows how absurd Classical logic is). Fundamental logically it is then about definition, plain and simple (only), if especially then mx has attraction force, or not, if one of these alternatives (empirically, genuinely, de facto) is true, and the other false, is irrelevant, yes, nothing that at all can be determined, but then only can be ASSUMED, defined something about (in accordance with the thinking, the experience, especially then a rationally thinking, which then (especially) this work seeks to give an idea of (what it is)), and in this then perhaps the "empiri" ((the) thinking which is assumed to refer, correspond to empiri) can give a hint:*

 

Counter-proof is in accordance with Kp to show that an x contradicts some for true held x in the theory in question.** If x unusually cannot be counter-proved, then x must be provable/derivable in the theory in question, if x now not wants to be assumed as an axiom (especially then perhaps on the basis of "empiri", as then what is valid for the (mx-)attraction force), in accordance with FT; If (non-axiomatic) x is assumed to belong to a theory without being able to be counter-proved or proved, then it is simpliciter the matter of irrational undecidable/independent x in accordance with FT.

 

__________

* For Gödel's Incompleteness theorems no difference is of course valid, can be noted, they are about, are assumptions, especially is assumed around the truthfulness of the axioms, and quite especially is assumed around the truthfulness of the derivations (“true” axioms/rules of inference not necessarily derives true (conclusions), but that (assumed) true axioms derives true is an axiom especially in Classical logic, and with that of course something (intuitively/unintuitively) assumed, defined true.

 

** That is thus sufficient for contradictoriness, need not necessarily be the question of an (absurd) p-superpositionality, which in accordance with N is a conventionally common view, that both x and y(=non-x) in N, "contradictorily", cannot be valid at the same time, the "Law of (non-)contradiction" ((x Ù y)’; N). The Fundamental logic thus defines much more commonly, "loosely", or rather actually very much more strictly. It is enough with the smallest deviation from something assumed true (x), then this deviation is false (x(=y)≠x is false (for x)).

 

Principia Mathematica

 

Principia Mathematica, at the pages 94-97, reproduced on the website Law of thought, defines these six "primitive propositions" of importance (which here are defined with = instead of ® (É), which changes nothing, because the intension of ®, just like with =, is that the left side can be exchanged for the right side, well, rather it is better with =, because ® loses the (weaker) meaning (≠[=]) to be able to implicate without it being the matter of implicative identity if [®]=[=]):

 

2) (x Ú x)=x(; x=(x Ú x)).

 

3) y=(x Ú y).

 

4) (x Ú y)=(y Ú x).

 

5) (x Ú (y Ú z)=(y Ú (x Ú z).

 

6) (y ® z)=((x Ú y) ® (x Ú 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 implicates z, which x of course commonly can, in which case the formula reads: (y ® z)=((x Ú 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 defines 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 Ú y):

 

y=(x Ú y); N.

 

4:

 

(x Ú x)=(x Ú 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 Ú y)=(y Ú x); N.

 

5:

 

Given 4:

 

(x Ú y)=(y Ú x):

 

(x Ú (y Ú y))=(y Ú (x Ú x)); 2:

 

(x Ú (y Ú y))=(y Ú (x Ú y)); N:

 

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

 

6:

 

(y ® y)=((y Ú y) ® (y Ú y)); 2:

 

(y ® y)=((x Ú y) ® (x Ú y)); N:

 

(y ® z)=((x Ú y) ® (x Ú 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 Ù x)’:

 

ü (x ® y)= (x Ù y’)’.

 

x=x:

 

y’=(x Ù x):

 

ü (x ® y)’=(x Ù y’).

 

(x ® y)=y:

 

(x ® y)=(y Ú y):

 

ü (x ® y)=(x’ Ú y).

 

x=x:

 

y’=(x ® x):

 

(y Ù y)’=(x ® y’):

 

ü (x Ù 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-logical 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))):

 

ü (x=x’)’.

 

N=N:

 

ü (x=y)=(y’=x’).

 

N=N:

 

(x=y)=((x ® x)=(y ® y)):

 

(x=y)=((x ® y)=(y ® y))(; N):

 

ü (x=y)=((x ® z)=(y ® z)); z=y.

 

N=N:

 

(x=y)=((x Ù x)=(y Ù y)):

 

(x=y)=((x Ù y)=(y Ù y)):

 

ü (x=y)=((x Ù z)=(y Ù z)); z=y.

 

N=x:

 

(x=y)=(x Ú x):

 

(x=y)=((x Ù x) Ú (x Ù x)):

 

(x=y)=((x Ù y) Ú (y Ù x)):

 

ü (x=y)=((x Ù y) Ú (x’ Ù y’)).

 

x=x:

 

(x Ù x)=y’:

 

(x Ù y)=(y Ú y)’:

 

(x Ù y)=(x’ Ú x)’:

 

ü (x Ù y)=(x’ Ú y’)’.

 

y=y:

 

(x Ù x)’=(y Ú y):

 

(x Ù y)’=(y Ú y)’:

 

(x Ù y)’=(x’ Ú x)’:

 

ü (x Ù y)’=(x’ Ú y’)’.

 

(x ® y)=(x ® y):

 

((x Ù x) ® y)=(x ® (y ® y):

 

ü ((x Ù y) ® z)=(x ® (y ® z); z=y.

 

(x ® y)=(x ® y):

 

(x ® (y ® y))=((x Ù x) ® y):

 

ü (x ® (y ® z))=((x Ù y) ® z); z=y.

 

Well, how it works should be clear after this (the other formulas are just as easy to prove), but takes one more formula because it defines both z and å (Double Composition Zalta calls it), which is a bit unusually to see:

 

(x ® y)=(x ® y):

 

((x ® y) Ù (x ® y))=((x Ù x) ® (y Ù y)):

 

ü ((x ® y) Ù (z ® å))=((x Ù z) ® (y Ù å)); z=x, å=y.

 

Zalta's text is fascinating in how he entangles the most, the simplest, the text is almost impenetrable if you don't know in advance what it is about, namely then development of N, given Tp. Zalta also defines the modal further development of Classical (N-)logic, so to speak draws a blanket of uncertainty over everything, which is completely meaningless, yes, nonsense. Rational logic adheres only to what it assumes to certainly be true (primarily then the Rational basis), and revise this analysis if it comes to the conclusion that what it previously assumed not holds, not is valid, not is rational. Assuming from the outset that analysis only is about possibilities (modality(/modalities)) only introduces uncertainty no one is happy about. Even if everything fundamentally is uncertain, but everything then is about assumptions, these assumptions per se not need to include uncertainty, only be possible. Ok, probability-theory is one thing, in some specific context where it fits, it by definition is about probabilities, but to generally assume that theory in itself is uncertain/modal, is pointless, because it always is possible to revise (well, rationally the Rational basis is not possible to revise, but otherwise so).

 

The Fundamental logic primarily only deals with general properties, thus whit properties valid for all x, the all (universal) quantifier (") is valid if it is about (properties for) all x, so it is simpliciter rationalized away, can be added apropos Zalta's text (which deals with " and $). The existence quantifier ($) may have its value in certain contexts, but has not had in this work, with which it of course hasn’t been any reason(/need) to drag it out.

 

Then (further) Zalta claims to prove existence of very basic concepts in chapter 10 (and onwards).*** No, commonly proofs always presupposes more fundamental concepts, which outmost always are about assumptions, with which the proofs (rationally) also are assumptions. Even Up, the most fundamental principle of all, is an assumption, although no rational person can deny Up, Up equally is an assumption which (undecidable) either is true or false, (undecidable) is valid in the World, or not. By which then of course everything that is derived with(/from) Up as basis also are assumptions, but given that it has Up as basis, and otherwise not hover out, there is in any case reason to talk about rational proofs (of course unlike irrational "proofs"). Zalta nowhere assumes Up, nor N is explicitly defined/expressed in Zalta's voluminous text, despite that whole his text completely rests on (presupposes) N. N which then is irrational, by which Zalta's other ("higher order") assumptions are needless to mention, it is enough to mention Zalta's N-assumption to state that Zalta's proofs in no way are (rational) proofs.

 

__________

* Commonly there is nothing that defines 3 to be true, that x’ ® y’ would implicate the right side, it is only valid if it is only (and only) x ® y, which naturally commonly not has to be valid (x’ can commonly very well implicate y). To also say a few words about 2, 2 presupposes that it is x ® y (which is valid in accordance with N, why 2 is valid in accordance with N), by which the left side for common validity shall be defined: x ® y ® z, and then not: x ® (y ® z), because if x not implicates y, then of course y not is valid (and of course neither z (implicated by y), if y now implicates z), unless for example å implicates y. So if it were the case that x ® (y ® z), but it is å that implicates y (which implicates z), then of course the right side not is valid. Classical logic simply (irrationally) defines too restricted:

 

A better way to convince of the irrationality of Classical logic is probably to show on this, that the formulas derived by Classical logic simply not holds, are irrational, commonly. Instead of directly starting with confutation of N, which I persisted with for a long time, which people just not seems to be able to understand/realize, so what N, they seems to want to say, Classical logic is not just N! Well, it thus is, in principle. But this with derived formulas then, where especially the "implication definition" ((x ® y)=(y Ú x’)) almost overclearly shows the irrationality of Classical logic (å can thus perhaps give y, in which case x may prevail (de facto, not only as possibility), but maybe give ä or nothing (x ® y is unfulfilled)), may seem, really wonders how long it will take before this irrationality commonly is realized?

 

** For example: (x ® y)=x=(x Ù x)=(x Ù y’), which is contradictory in Classical logical meaning (y’≠x (over-interpreted)), but not commonly, because commonly it can for example be valid that (x ® y)_o=(x((® z)) Ù y’((® å))); y’ (in the latter formula) can be =x, because it commonly (then) not is excluded that x can give ("produce") several different y, which if it occurs in the same moment of course presupposes a more complex ("bigger") x.

 

Or: (x ® y)=x=(x Ù x)=(x Ù y)(; N), in accordance to N it then Classically logically interpreted is valid that (x Ù y)’ (the Law of (non-)contradiction), even if it Classically logically actually not is contradictory, but just shows on N, that x and y superpositionally are valid (if x is a proposition, then also y is that (which both (platonistically) exists at one and the same time)), but it still not is a formula Classical logic defines.

 

Problems with contradictions Classically logically also is present regarding "De Morgan's laws": The first is defined by the condition in N that (x Ú y)(=(y’ Ú x’))=(x Ù y)’ (that the "Law of excluded middle" identically is the Law of (non-)contradiction (and vice versa)), the second: y=y ® x’=(x’ Ù y) ® (x Ú x)’=(x’ Ù x) ® (x Ú y)’=(x’ Ù y’)(=(y Ù x)), Classically logically interpreted contradicts the first law, but is intuitive, intuitively interpreted (if x and y are valid (symmetrically (reversed) interpreted), then intuitively x or y not is true, but x and y then cannot be valid in accordance with the Law of (non-)contradiction),^ why it Classically logically still is assumed/formulated.

 

*** Especially Zalta "proves" around the concept of property, which is a concept so fundamentally that nothing can be proven about it at all, it only exists or is, whatever "property" is called, designated, whatever word "property" is given. For either Nothing prevails, in which nothing prevails, Nothing defined as just only nothing, and consequently no properties prevails in Nothing either, Nothing is propertiesless, propertieslessness as it was then defined. Or then Nothing not prevails, and consequently also properties prevails, whatever they are called, x perhaps, x which of course not prevails in Nothing defined as x-lessness, but x then of course prevails in non-Nothing, or identically in non-[x-lessness], in the non-[x-less]. x, especially then called properties, which given that they not exists in Nothing, necessarily exists if Nothing not prevails, otherwise Nothing of course prevails:

 

The concept of property is completely given, given that something≠Nothing prevails (properties (whatever they are called) are simpliciter that which exists if Nothing not prevails).

 

It can be added that Zalta puts enormous effort into defining an artificial language that he (and many with him) believes is necessary to fit the N-logic. No, it is not the least necessary, any language will do, the important is only that it obeys the defined logic, especially then the N-logic, all its formulas, which fundamentally is the (N-logical) (x-)language. Which of course makes the language, whichever it now is, "artificial", if it obeys defined logical (x-)rules, but that is no stranger than learning the grammar of a language (whichever).

 

Rather this artificial language shows what N already shows, namely that it is a matter of subjectively controlling the thought, concerning N then to that y=x’ (given x), a y which Fundamental logically then not exists (ex ante), but N-logic then (platonistically) wants us to believe that it does. The artificial language does principally the same, through its strict/restricted construction, steers thinking towards certain (N-logical) goal. In and for itself the Fundamental logic also steers towards a certain (Fundamental logically) goal, but this in any case in accordance with rational thinking, especially then Up. And towards a more common goal than the tremendous restrictedness that the N-logic stands for, defines. Although given the E-theory (which then is a consequence of especially Up) it is not about total unrestrictedness, since the E-theory also defines extremely specific, albeit very much more freely than the multitude of formulas the N-logic defines, it is for example only to again look at the "implication definition" of N-logic: (x ® y)=(y Ú x’), which then excludes the commonly valid that for example z can implicate y and that x (if not implicating y now, for the moment (but then maybe in another moment provided that x ® y is a possibility)) can implicate å, or just only be (not implicate anything): (x ® y)_o=((z ® )y Ù x(( ® å))).

 

Or take all the axioms that Principia Logico-Metaphysica defines in Chapter 8, basically to be compared with Up in the Fundamental logic. An almost comical comparison. That Up is valid in the World should be something all rationals can agree upon, but Zalta's myriad of axioms? A myriad which furthermore, in fact (Zalta is nowhere near to define anything equivalent to E), defines the World ("E") more loosely than (primarily) Up does. And a myriad which then de facto has its first basis in the irrational N (since Zalta then defines/presupposes Classical logic), which of course commonly makes all these Zalta's axioms worthless, which he and everyone else who not knows that N is the bedrock of Classical logic of course not knows, as just unknowing this. But commonly they should argue (have argued) more for their axioms, not commonly more or less (ad hoc) just take them for granted (and this should of course not only be valid for classical logicians, but for all logicians). It might even have made them realize that N (and Tp) are the bedrock of Classical Logic, with which of course this work would not have had to mention Classical logic (it had already been rejected, or perhaps never defined/invented), yes, this work might already have been done, and I could have focused on other things.

 

^ This second De Morgan law shows parenthetically said clearly the importance of interpretation, because commonly if x or y not are valid, then it only is a possibility that x and y are true, just as well one or more (other) z≠0 can be valid (as possibilities), just as nothing can be valid (z=0), but given N, thus that it just only are about x and y, then of course the only possibility is that it is x and y that are valid (x Ù y; (x Ú y)’; N), then disregarded from the Law of (non-)contradiction ((x Ù y)’)).

 

 

A little more about the meaning of Fundamental logic

 

That nothing is determined before it is determined means so to speak that the mind/thought is trapped in its experience/thought(/thinking)/mind, impossibly can get outside of it. This with which the thought/mind as only possibility for more solid knowledge has to find thoughts which it believes in more than others. And the Fundamental logic then finds such a more fixed thought in primarily Up. Many argue that this more fixed thoughts exists in the "empirical" experience. For example of an oak. But, if Up not is assumed to be valid (or some other (approximately) equivalent principle which defines that x=x, that oak=oak),* then it not is certain that the oak is the oak, thus if it not is assumed that x=x, but it is then possible that x≠x. So (in particular) Up is a necessary presumption for the oak to be the oak (or more commonly then that x=x).

 

Some want to claim that an oak is an oak beyond all (thinkable) principles, but it is evident that the experience of an oak not is identical to an oak beyond the minds experience:

 

x≠y; x is the experience of y, and y that which (per se) exists beyond the experience (experienced as x).

 

If y=x, then y then is the minds experience (of y), which not is valid if y exists per se.

 

If x=y, then x (the minds experience of y) then is y, that which exists per se, which of course not is valid if y exists per se.

 

If y not exists per se, it immediately follows that y=x, given which it also is valid that x=y.

 

There exists a distance between x and y, if y exists per se. A distance which makes it impossible for the mind that experiences x to determine whether x has anything to do with any per se existing y. The mind can only ASSUME something concerning a relationship between x and y, if x corresponds more or less well to y, or not at all, y=x of course excluded, thus that x completely corresponds to y, x exactly defines, "depicts" y, which of course requires that y is x, which y as existence per se (≠x) of course not is.

 

So, it is thus about assumptions regarding "empirical" x, and just as much it is about assumptions regarding non-"empirical" x, it is evident that the latter is valid. Non-"empirical" x has however the advantage of being one with x, they are x, this "closeness", this non-distance, makes non-"empirical" x more "homely", more reliable, which becomes especially clear when the thought finds an x as Up, to which the (rational) thought sees no alternative, one can actually be said incredible feeling to find such an x, which (rationally) cannot be questioned, it is as close to an objective fact as it at all is possible to get. "Empirical" x cut a poor figure in comparison.