(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.

 

Yes, these x which (rationally) cannot be questioned (x*) can well-nigh be compared with platonistical x, thus eternally true non-"empirical" x, but of course without the eternity stamp, existing per se eternally (always) valid, but it is only the matter of assumptions, even if then unquestionable for a rational mind, a human rational mind shall be said, for another "rational" mind, then than the human's, maybe not sees x* as rational, yes, there maybe even be human minds which not sees x* as rational, especially then Up; It really would be interesting to hear a justification of how x with exactly (identically) the same properties can be different.

 

It is thus (rationally) about assumptions, there exists neither (given, eternal) platonistical x nor given (eternal) "empirical" x, of course corresponding to empirical x, because if "empirical" x not (completely correct) corresponds to empirical x, then of course it is a matter of purely abstract (only thought) thoughts; If x≠x, then x≠x (in accordance with Ip/Up), however much it wants to be claimed that (especially) approximations are (approximate) truths, they simply not are (objective) truths, but then something pure abstract (only thought).

 

That E defines the existence of eternal possibilities (which can be interpreted platonistically)** given T1 not changes anything of that said in the previous paragraph, since E is based on assumptions (rational, or not), especially T1, and thus not on anything categorically (eternally) given, by which E of course also is an assumption, not something categorically (eternally) given.

 

__________

*  Up defines Ip (x=x) in a very categorical way, especially the property concept is underlying valid. "Ip" can especially be defined weaker, less categorically, that x=x without specifying more than that (so that x=x then not is as specifically defined as in Ip). A stronger (even more categorical) definition of "Ip" is for example to define "Ip" the E-theory given (whit the E-theory as basis), thus especially define that x=x; x={mx}(≥mx) (equivalently can be defined for mv, and for E (restated): E=E; E=∞’mv).

 

** Even if everything possible given E, existing as eternal possibilities in E (given T1), one with E, manifested as mx or not, are eternal possibilities, "ideas", in E, which of course for example makes the idea that the Earth is round as eternal as that the Earth is flat, that the former is a rational idea and the latter an irrational idea according to "the empiri" (in broader sense, for example seen by an astronaut who has travelled around the Earth, one who has lived in one and the same place all life may have harder to embrace that idea, but may then have to believe the astronaut if they would meet) hasn't the slightest significance concerning the eternity stamp, given that these thoughts exists, which of course constitute proof of that they exists, which, given E(/T1, of course) directly also defines their eternal status; That something, x, never will be thought or manifested as mx other than defining a thought, not exclude the existence of x as possibility, can well be added.

 

 

About N again

 

Given N then non-x then is a unique specific x(=y), not =0, and this then always, which then commonly not is valid:

 

Commonly non-x just only defines that x not is valid, or rather that x not is in focus, but it is non-x that is in focus, even if x can, or rather is there in the background, through x in non-x, and non-x not defines anything specific, before non-x maybe is defined to define something specific, directly (as then for example N does), or indirectly, through a context:

 

Is a context defined, non-x more specifically maybe at least can begin to give itself, for example, restated, if E and xÎE are defined, in which non-x closest defines E-x, but non-x commonly naturally also defines all specific non-x=yÎE, y which also can be clusters(/sets) of y, as well as maybe xÏE, as well as 0 (or Nothing) if x≠0.

 

Or take non-true (x), in accordance with N, then false (x) is meant by non-true, which commonly naturally not necessarily is valid, closest non-true can mean decidable/undecidable (x), probable(/improbable) (x, with some probability), possible/impossible (x), etcetera, except of course everything else that not is true as a concept/word, such as mould (mould≠true as concepts, and consequently non-true=mould is valid (interpreted) in that way), Sweden or E; Not-true=false is an (irrational) definition, which further lowers Classical logic, especially in its truth table form.

 

 

Basic mathematics given E

 

Houses are built from the ground up, and the same shall (should) be valid for logic, trying to find axioms that fit some (that are seen as) results(/theorems) is commonly irrational. So if (rational) axioms do not lead to any (desired) result, then commonly the only alternative to not reject those "results", is to assume them ad hoc, as axioms simply. The foundation is thus the most important, theory is at least as loose as its foundation. "Commonly" because there may be exceptions, when it may be rational to fit the axioms to "results", an example of this is the "empirical" assumption that mx have attraction force, given the consequences of not assuming that (which at first sight is the most rational).

 

The geometrical basis

 

All continuous geometric shapes (where the pencil so to speak not is lifted) are defined by the curve (if p’=p, then either (a) p is defined, or that the curve ends in p, in the p in which the curve began to be "drawn"):

 

d(p,p’); p)=p]; p]=p) (given the continuity/homogeneity of E; Curves principally exists between all p's that not are identical, not Nothing, it is so to speak possible to draw a line between p and p’; p’≠p):

 

The smallest curve/distance:

 

dp=min[d(p,p’)]

 

The smallest area (triangle):

 

y=min[d(dp,p)]; pÏdp.

 

The smallest volume (tetrahedron):

 

v=min[d(y,p)]; pÏy.