Wittgenstein and Gödel

This will be somewhat different version of Wittgenstein and Gödel pdf, e. g. footnotes are in main text.

Some remark for my notation. Underlines, boldings and spaces of the paragraphs of Tractatus, as I call the sentences of the book, are all (along) mine. If italics is mine I say it. – Also it is to notice that when I quote paragraphs, often I do not represent them in its entirety. Sometimes also certain parts of various paragraphs are compound. If so, of course I mark it.

Tractatus Logico-Philosophicus → http://people.umass.edu/klement/tlp/tlp.pdf

Wittgenstein and Gödel

I wonder that Ludwig Wittgenstein’s ”fundamental thought” in Tractatus / ”.. My fundamental thought is that .. the locig of facts cannot be represented” (4.0312 , 4.128 , 6.4321) can be seen to anticipate mu-tatis mutandis Kurt Gödel’s incompleteness theorems in 1931. Notably the first (even if Wittgenstein’s ”thought” imply also second theorem). – And, why not, also to anticipate Alan Turing’s paper in 1936 / negative answer to Entscheidungsproblem.

I mean that ”the facts” are precisely ”pictures of the facts” (1* , 1.1 , 1.11 , 1.13 , 2.06 , 2.063 , 2.141 , 3.12 + 3.14 , 4.01 , 4.06, italics is mine. What * means, look Tractatus.).

”The picture is fact”. (2.06 , 2.063 , 2.141 , 3.12 + 3.14 , 4.01 , 4.06)

”.. the proposition .. in its projective relation to the world .. is a fact”. (1* , 1.1 , 1.11 , 1.13 , 2.06 , 2.063
, 2.141 , 3.12* + 3.14* , 4.01 , 4.06, certain parts* of paragraphs 3.12 and 3.14 are compound.)

”The facts” can be read, well studied throughout ((2.06) → 4.06 , 4.1), and further (4.21 , 4.25 , 5 , 5.01).

”The existence and non-existence of states of affairs is reality. (We call the existence of states of af-fairs a positive fact, and their non-existence a negative fact.)”. (1.1 , 1.11 , 1.13 , 2.06 , 2.063)

Propositions can be true or false only by being pictures of the reality”. (1.1 , 1.11 , 1.13 , 2.06 , 2.063 , 4.06 , 4.1 , 4.25 , 5 , 5.01, italics is mine.)

”Reality” can be substituted in 4.06 by ”.. existence and non-existence of states of affairs”, or ”.. atomic facts” (2.06), that is ”positive facts and negative facts”, or shortly by ”facts” (1.1 , 1.11 , 1.13 , 2.06 , 2.063). Still reading we receive ”.. pictures .. of the truth and falsehood of the truth-arguments of propositions” (2.06 , 4.06 , 4.1 , 4.25 , 5 , 5.01, especially paragraphs 2.06, 4.1 , 4.25 and 5.01 are compound.).

Thus ”the facts” are all (3 , 4 , 4.001) the well ordered propositions, which can be proved true or false. This, what ”the facts” are in (4.0312), is the basis of my interpretation.

That ”the facts” are ”pictures of the facts” [((2.06) → 4.06 , 4.1) → 4.21 , 4.25 , 5 , 5.01] originates from ”the aim of the” Tractatus ”.. to draw a limit to .. the expression of thoughts .. in language .. from within ..” (Preface / (4.112 , 4.114)). Specifically ”the thoughts” are (precisely) ”pictures of the facts” (2.06 , 3 , 4 , 4.06) as well.

The logical picture of the facts is the thought”. (2.06 , 3 , 4 , 4.06, underline runs below.)

”A thought is a proposition with a sense”. (2.06 , 3 , 4 , 4.06 , 4.1 , ! 4.2 , 4.25 , 4.3 , 4.4 , 4.41 , 4.431 , 5.01, italics is mine.)

A sense” in in its proof-theoretical meaning (4.2 , 4.4 , 4.431). Paragraph 4.2 says, including paragraphs (2.203 , 3.02), what is needed to set out, decipher the meaning of well ordered propositions as ”.. pic-tures of the reality” (4.06), which can be proved true or false. – If one do not consider aforesaid para-graphs plus some other (essential) paragraphs (2.11 , 2.201 , 2.202 , 3.318 , 4.1 , 4.25 , 4.3 , 4.41 , 5 , 5.01) he or she cannot make sense of ”.. the logic of facts ..” (4.0312). Also ”the aim of the” book will remain obscure.

But why Wittgenstein’s aim is what is ?

In my book because of ! Russell’s paradox (though the aim makes sense also without it). Wittgenstein
comprehends Russell’s paradox straightforward, so that ! there is not any more a single well ordered,
consistent and provably true or false proposition .. but ”.. rauschen und brausen ..” (Tractatus-motto)
prevail. Thus there is ”.. a logical necessity ..” (6.375) –

”We make to ourselves pictures of facts .. pictures of the reality”. ((2.06) → 2.1 , 2.12 , 2.201 , 4.01 , 4.06
→ (4.1 , 4.25 , 5 , 5.01, paragraphs 2.1 and 4.06 are compound.))

The foremost task in Tractatus is how to make consistent and provably true or false propositions. The thing of 4.06 is ”.. can be true or false ..”, with provability (4.21 , 4.25 , 5 , 5.01). Then ask, Wittgenstein’s aim in Tractatus, can all of these well ordered propositions be said, ”.. be put into words ..” (4.112 , 4.114 , 6.552)? – One can say that Wittgenstein’s aim falls inherently due Russell’s paradox. If a logician has to wipe the slate clean, no well ordered propositions yet, last when the recipe of valid propositions is at hand and completed there is (consequently) ! the question whether the totality of well ordered propo-sitions can be decided. – A question alike .. Entscheidungsproblem.

To fulfill, or rather to try to fulfill his aim (and to show that it is not possible) Wittgenstein needs a theo-ry of ”truth”. It’s the correspondence theory of ”truth” (2.21 , 2.221 , 2.222 , 2.223 , 4.031 , 4.0311) ; not ac-cidentally. The consistency of correspondence theory must to be proved* (2.06 , 2.11* , 2.201 , 2.203* , 3.02* , 3.332 , 3.333 , 4.1 , 4.2 , 4.25 , 4.3 , 4.41 , 5 , 5.01). If it can be done, i. e. ! (4.0312 , 4.128 , 6.4321), then indirect – vis à vis – ”.. Russell’s paradox vanishes” (3.333).

It is due to the correspondence theory, its logic (1.11 , 1.12 , 2.021 , 2.024 , 3.203 , 3.3 , 4.031 , 4.061) that ”the facts” to be made in universal-language – the other component (with correspondence theory) of the ”substance of the world” (2.0124 , 2.021) – are ”.. pictures of the reality” (4.06) as well ordered, pro-vably true or false propositions. Due to the fact of the logic of the correspondence theory that univer-sal language must be in the beginning independent of reality*, i. e. no relation, connection to it*, it will be so as (4.06) says. It express, shows (that) relation between universal-language and reality (stand). And since independence presupposes well ordered propositions to be ”.. pictures of the reality” (4.06) to define ”reality” is necessary in order so say ”pictures of ! ”what””. Not to mention, that”reality” as a basic consept of the correspondence theory need to be (necessaryly) defined.

So, what is my wondering? – It is at first that we can put in (4.0312) in the place of / ”the facts” / follo-wing: ”.. My fundamental thought is that .. the logic of / the well ordered propositions that can be pro-ved true or false / cannot be represented” (4.0312).

Next I interpret ”.. cannot be represented” (4.0312) to mean that, *there is no (general) method or algo-rithm to deduce ”.. beforehand .. on one occasion ..” (5.47) all the well ordered propositions that can be proved true or false*. – Well ordered propositions of the correspondence theory.

*This* is mutatis mutandis the heart of Kurt Gödel’s first incompleteness theorem.

I can put ”my wondering” also so, that the relation, connection between universal-language and reality can be shown, but not said (2.18 , 4.12 , 4.121 , 4.1212) since to say that relation, connection would mean, as its corollary, the same as to to say all well ordered propositions (which derives a contradiction).

Altogether I think that Tractatus can be seen, at least when it concerns the picture theory, as an com-plex and wonderful analysis of the logic ((2.024 , 4.061) → (1.11 , 1.12 , 2 , 2.014 , 2.021 , 2.06 , 2.063 , 2.1 , 2.11 , 2.12 , 2.201 , 2.202 , 2.203 , 3 , 3.02 , 3.202 , 3.203 , 3.22 , 3.23 , 3.3 , 3.332 , 3.333 , 4 , 4.01 , 4.06 , 4.1 , 4.2 , 4.21 , 4.23 , 4.25 , 4.3 , 4.4 , 4.41 , 4.431 , 5 , 5.01) of the correspondence theory of ”truth” (2.21 , 2.221 , 2.222 , 2.223 , 4.031 , 4.0311), which derives the undecidability of ”.. pictures of the reality” (2.06). (I think that these 44 paragraphs of Tractatus plus some 20 others, alongside Wittgenstein’s aim, are enough to anybody to get my interpretation, wondering of the book.)

”The logical forms are anumerical. ..”. (4.0312 → ((2.18 , 2.21 , 4.12 , 4.121) → 4.128) → 6.4321) The totality of well ordered propositions ”.. cannot be said .. put into words ..” (4.1212 + 6.552, certain parts are put together.)

The logical forms ..” (4.128), expressions of the relation, connection between universal-language and reality, are with minimal reading, interpretation, propositions that can be true or false (2.06 , 2.17 , 2.18 , 2.21 , 4.06 , 4.12 , 4.121), that is the well ordered propositions that can be proved true or false (4.1 , 4.25 , 5 , 5.01).

I see that Wittgenstein’s ”fundamental thought” (4.0312 , 4.128 , 6.4321) is a consistency-result of the correspondence theory, and implies, indirect show ontological realism*. ”Das Mystische” in Tractatus (6.44* , 6.45 , 6.552) manifests necessarily existing ”true”, but yet ”inexpressible”, and so yet unprovab-le propositions. Them implies ontological realism, (in my book) as an inherent component of corres-pondence theory. Imply? – It have to be, for the sake of ! consistency, so that the outside of universal-language existing ”reality” can not be directly said ! in universal-language. – If could, correspondence theory would be a sort of ! Solipsism since ”reality” would be nothing but a computional task to be do-ne in universal-language itself, by enumerating all possible well-ordered propositions which can be made in universal-language .. if that ”enumerating” would be possible. That ”enumerating” is not pos-sible, indicates (in my book) the validity of correspondence theory.

Solipsism also if not ”.. make ourselves pictures of facts .. pictures of the reality .. from within ..” (2.1 + 4.06 + 4.114, certain parts are compound) in universal-language*, but it* and its* grammatical correct propositions, declarative sentences, only. These already existing propositions that ”.. already have a sense ..” (2.021 , 2.0211 , 2.0212 , 3.203 , 3.3 , 4.064) might be true or false, but none of them provably, i. e. not yet”.. with a sense” (4) in its proof-theoretical meaning (3.202 , 3.23 , 3.3* , 4.03 , 4.2 , 4.21 , 4.23* , 4.25 , 5 , 5.01) as expressions of ! all ”the truth-possibilities of the elementary propositions .. the truth-arguments of propositions” (4.3 + 4.41 + 5.01, some parts are compound.); and the question of ”truth” by correspondence theory would be a dead-end. – (If we study the question of realism versus solip-sism Wittgenstein has a point in paragraph 5.64, anyhow on the contrary from a point of view of cor-respondence theory / realism (but not of solipsism) → (4.064 , 5.473 , 5.4732 , 5.4733 , 5.5563.))

The key question of the picture theory, as a tool, to implement or apply correspondence theory in uni-versal-language – where ”everything” must be in principle possible to express – is the question of the relation between universal-language and reality. That (question) because of the logic of corresponden-ce theory (2.024 , 4.061). Correspondence theory’s consistency presupposes that universal-language, also that ”to be done” (2.06 , 2.1 , 4.06) well ordered language, is independent of reality Therefore con-sequently independent of ”reality”, as ”it” has to be defined, conceived in universal-language itself ! to ”.. make to ourselves pictures of facts” (1.1 , 2.06 , 2.063 , 2.1 , 2.141 , 3.12 + 3.14 , 4.06), well ordered pro-positions (which, because of Russell’s paradox, not yet exist).

”Reality” can be defined in universal-language without falling into circular argument because of univer-sal-language is still independent of reality, i.e. yet no ”to be shown” connection, relation to reality. Thus there is not yet a single proposition that can provably assert its truth-value. Since independence is due to the correspondence theory, to define ”reality” is possible by applying correspondence theory itself in universal-language. Nonetheless it’s exactly correspondence theory, whose analysis, that is relation ! ”wherto”* asks to conceive ”reality”*. – This self-reference is a benefit; less makers, bottom lines.

Therefore, actually the core of understanding picture theory as a proof-theory is that Wittgenstein says in paragraphs (2.022 , 2.023 , 2.16 , 2.161 , 2.17 , 2.18 , 4.03): ”.. The proposition only asserts something, in so far it is a picture” (2.06 , 2.18 , 4.022 , 4.03 , 4.06 , 4.1 , 4.2 , 4.21 , 4.25 , 4.3 , 5 , 5.01). What a ”.. propo-sition .. asserts ..” (4.03) is its ”sense” (2.221 , 4.031) to agree or disagree with reality (2.222), and to ”as-sert” its truth or falsehood, by correspondence theory, a proposition has to be ”a picture” (4.022 , 4.03 , 4.06). Wittgenstein’s picture theory is already an expression of applying correspondence theory to or in universal-language (3.5), to get first a recipe of the totality of valid propositions and then second to try to decide them all.

What can, must be done, ”.. the task ..” (6.4321) is that ”.. Reality must .. be completely descriped by the proposition ..” (4.023) – entailing (as such) to define ”reality” in universal-language – in order to get the logic of (all) well ordered propositions to be done, completely as well. What can and must be done is that the well ordered, well formed, propositions of correspondence theory must ”.. have in common with reality .. the logical form .. of reality” (2.17 , 2.18); so to ! be able to say, assert their truth or falsity (2.21 , 2.22 , 2.221 , 2.222 , 2.223 , 4.03 , 4.031). Not until ”.. the logical form of represetantion ..” (2.17 , 2.18 , 2.2 , 2.22) makes the truth-values (2.222) ! provably possible; so consequently ”the sense” of a propo-sition is independent ”.. of its truth or falsehood ..” (2.22). Else truth-values* would precede that (what) makes them* just possible (4.061 , 4.062). One of the points, things of Wittgenstein’s ”substance of the world”-doctrine is that there is ”.. already .. a sense ..” (4.064) of a proposition so that altogether to be possible to assert ”.. the agreement or disagreement of its sense with reality ..” (2.021 , 2.0211 , ! 2.0212 , 2.222 , 3.203 , 3.3 , 4.031 , 4.064 , 4.2211). Therefore a proposition* that ”.. already .. has such and such a sense ..” (4.031 + 4.064, certain parts are compund.), still independent of reality, must – a s ”.. a picture of reality .. communicate a n e w sense with old words ..” (4.01 + 4.03, some parts are compound.), that is represent its* existing sense** so that it** has ”.. the logical form .. of reality ..” (2.18).

Indeed, ”.. the logical form .. of reality .. can be shown .. completely ..” (2.18 + 4.023 + 4.1212 , 5.4711, so-me parts of paragraphs 2.18 , 4.023 and 4.1212 are compound.) because of ”it” is ! contained in well or-dered propositions (3.13 , 3.3 , 3.31 , 3.311 , 3.318 , 4.2 , 4.431)

”.. In the proposition .. its sense is not yet contained, but the possibility of expressing it .. In the proposi-tion the form of its sense is contained, but not its content”. (3.13 , 4.2)

What ”.. is contained ..” (3.13) in a well ordered proposition is – ”.. the possibilities of the existence and non-existence of the atomic facts” (2.06 , 2.1 , 2.11 , 2.201 , 2.202 , 2.203 , 3.02 , 4.1 , 4.2 , 4.21 , 4.25 , 4.3 , 4.41 , 5 , 501). But what ”.. is not yet contained ..” (3.13) in a well ordered proposition is its ”.. agreement and disagreement with the possibilities of the existence and non-existence of the atomic facts” (4.2, italics is mine.), which is ! that what ”.. cannot be said” (4.1212 , 6.522).

”The sense of a proposition is its agreement and disagreement with the possibilities of the existence and non-existence of atomic facts”. (3.13 , 4.2)

In princible it seems that it’s possible to express ”the sense” (4.2) of the ”.. pictures of the reality” (4.06) precisely due the fact what is contained in a well ordered proposition. Anyway to fullfil that possibility can’t be done (3.332 , 3.333). On the contrary of what it superficially may seem, it’s just due to ”contai-ning” that ”the totality” of well ordered, provably true or false, propositions ”.. cannot be said .. put into words ..” ((3.332 , 3.333 → (4.001 , 4.12 , 4.121 , 4.1212 , 4.128 , 4.4 , 6.522) → 7, parts of paragraphs 4.1212 , 6.522 are compound.). That paragraph 4.4 says, i. e. what is needed to do in order to express a well or-dered proposition, can’t be done (to avoid contradiction).

But why Wittgenstein’s aim fails ?What is the essential reason, premise (to his aim failing) ?

Answer is linked to the meaning, subject matter of (the consept of) ”independence” in the context of correspondence theory.

As said correspondence theory’s consistency entails that universal-language, the other component of Wittgenstein’s ”substance of the world”-doctrine (2.021 , 3.203 , 3.3 , 4.031 , 4.0311 , 4.2211), is indepen-dent of reality (1,12 , 2.024 , 2.06 , 4.061).

Else a proposition is also ! an expression of that ! reality what it, or its sense (2.21 , 2.221 , 2.222 , 4.031 , 4.0311) agrees or disagrees.

Not-independence would mean in the case of ”truth” a circular argument, and in the case of ”falsity” ! outright contradiction, i. e. ”falsehood” of a proposition* would ! express ! that ”truth” wich its* sense disagree.

Substance is what exists independently of what is the case”. (1* , 1.1 , 1.11 , 1.12* , 1.13 , 2.021 , 2.0211 , 2.0212 , 2.024* , 2.06 , 2.063 , 3.203 , 3.3 , 4.061 , 4.064 , 4.2211, italics is mine.)

It is easy to realise by paragraphs 2.021* , 2.0211 , 2.0212 , 3.203 , 3.3 that language, universal-language (2.0124), is the other part of Wittgenstein’s ”substance of the world”*-doctrine. In that sense even the basic part that correspondence theory as the other part presupposes ”something” to be applied (not to mention that correspondence theory is said and determined in language). And then further with para-graphs 3.202 , 3.23 , 4.23, combined with paragraphs 4.2 , 4.21 , 4.25 , 4.3 , 4.41 , 5.01, it is indeed pos-sible to infer the proof-theoretical meaning* of ”.. the simple signs ..” (3.202 , 3.203 , 3.23 , 4.23 , 5.01) which* is that ”the simple signs” (3.23) are ! exressions the truth-possibilities, BOTH ”truth” AND ”fal-sity”, of well ordered propositions. – This proof-theoretical meaning of Wittgenstein’s ”substance of the world”-doctrine is ”its” essential content. – And since these truth-possibilities ”.. cannot be said”. (3.221 , 4.1212 , 4.1272 , 5.55 , 5.5571), it is to conclude (4.0312 , 4.128 , 6.4321).

But not until paragraph 4.061 –

If one does not observe that propositions have a sense independent of facts, one can easily believe that true and false are two relations between signs and things with equal rights. One could then, for example say that ”p” signifies in the true way what ”¬ p” signifies in the false way , etc.”. (1* , 1.1 , 1.11 , 1.12 , 1.13 , 2.021 , 2.0211 , 2.0212 , 2.024 , 2.06 , 2.063 , 3.203 , 3.3 , 4.061 , 4.064, italics is mine.)

says the core of ”independence”. It is the fact that to be concistent correspondence theory presuppo-ses ”.. that propositions have a sense independent of the facts ..” (4.061). – Else, if not ”independencethen *”.. ”p” signifies in the true way what ¬ p” signifies in the false way ..”* (4.061), *which* is ! (in my book) a contradiction. ”Every proposition must already have a sense ..” (2.021 , 2.0211 , 2.0212 , 3.203 , 3.3 , 4.064) ! simply for the sake of there is something ”what” agrees or disagrees with reality, and if ”a sense” of a proposition is straight in touch with reality ”it” immediately either agrees or disagrees with reality, and so ”a sense” of a proposition signifies ! depending on the truth value – which both values are possible – of a contingent proposition. Thus each proposition have already ! all possible truth va-lues, i. e. each proposition is, would be a real mess of contradictions.

Anyway ”independence” is not an easy case (in the context of correspondence theory either).

First independence means external relation between universal-language and reality. Correspondence theory may now be consistent, but at the cost of that it is not possible to compare a proposition, or its sense with reality. The sense of a proposition do not reach to reality, to compare (it with reality). It is to say that the sense of a proposition and reality cannot be shown on the same occasion, at the same ti-me. There must exist some other relation than external between universal-language and reality. It’s by ”.. a logical necessity ..” (6.375) internal. – Otherwise, if no other relation, to prove truth or falsity of a proposition is impossible and → (2.222 , 2.223) .

It is especially the question of relation, internal relation to ”what”, that is ”picture of ”what”” that necis-sitates to define ”reality”. Thus, as an analysis of the correspondence theory, Tractatus begins logically by defining ”world” / ”reality” (1* , 1.1 , 1.11 , 1.12 , 1.13 , 2 , 2.06 , 2.063); irrespectice of Wittgenstein’s in-tentions.).

The internal relation must exist inside universal-language. Inside that universal-language ! itself where the analysis of correspondence theory’s validity will be made, vis-a-vis, self-referential, in that univer-sal-language ! as well defined ”reality” (which represents outside of u-language existing reality).It is a self-referential analysis of inside universal-language existing relation between universal-language it-self vis a vis in the same universal-languge itself defined ”reality”. – Universal-language must show, express its own valid relation, connection to reality (2.06 , 2.12 , 4.023). – There is no other place. The internal realation exists inside universal-language, and it is made by applying correspondence theory, the essential side of Wittgenstein’s ”substance of the world”-doctrine (2.021), to universal-language. The logic of correspondence theory*, the necessary premise, even axiom of ”independence” (2.024 , 4.061) to its* consistency (2.024 , 4.061), makes to internal relation.

In the end of the road – on the last step of ! not at all senseless Wittgenstein’s ladder (6.54) – of ”the internal relation” (2.014 , 2.021 , 2.06 , 2.1 , 211 , 2.201 , 2.202 , 2.203 , 3.02 , 3.202 , 3.203 , 3.22 , 3.23 , 3.3 , 3.332 , 3.333 , 4.1 , 4.2 , 4.21 , 4.23 , 4.25 , 4.3 , 4.4 , 4.41 , 4.431 , 5 , 5.01 , 5.55 , 5.5571 , 7) we are going to see ! ”.. the world rightly” (6.54) by meeting Wittgenstein’s ”fundamental thought” (4.0312 , 4.128 , 6.4321) as undecidability of ”pictures of the reality”; a l l ”.. pictures of the reality .. cannot be .. put into words .. beforehand .. on one occasion ..” (4.06 + 4.1212 +5.47 + 6.522, some parts of para-graphs are put together.)

What first was a necessary condition for correspondence theory’s consistency transforms at the last line to the undecidability of ”.. pictures of the reality” (2.18 , 4.06 , 4.128). It is obvious, axiomatic that independence as a necessary condition of correspondence theory’s consistency must stand further when there is a relation, connection between universal-language and reality, containing in universal-language.

To say shortly, tersely the reason why Wittgenstein’s aim fails, it is to say that the well ordered propo-sitions of correspondence theory as ”.. pictures of the reality” (4.06) contain the possibility of (provabi-lity of) their both ”truth” and ”falsity”. This ””both-and”-possibility” can be read by comparing carefully only (2.06 , 2.11 , 2.201 , 2.202 , 2.203 , 3.02 , 4.1 , 4.2 , 4.25 , 4.3 , 4.41 , 5.01). In the paragraph 4.2 ”.. the possibilities ..” are exactly the possibilities* of both ”truth” and ”falsehood” of ”.. the truth-arguments of propositions” (5.01) that* are contained in well ordered propositions. E r g o : (Anymore) ”No proposi-tion .. cannot be contained in itself .. because .. already contains the prototype of its own argument ..” (3.332 + 3.333, parts of paragraphs are compoud. Italics is mine.), i. e. contains ! exactly the possibili-ties of (provability of) its own both ”truth” and ”falsity”. – IF a well ordered proposition could ”.. be con-tained in itself ..” (3.332) THEN .. contradiction. Contradiction since now a(n invalid) proposition* would ! a g r e e ! with #the possibility of (provability of) its both ”truth” and ”falsehood”# .. #WHICH# it* .. as said ! .. ”.. already contains ..” (3.318 , 3.333 , 4.2 , 4.431). – As for to d i s agree with ””both-and” possibi-litiesit would mean (merely) either ”tautology” or ”contradiction” (4.46 , 4.464).

But how (on earth) ””both-and” possibility” is .. ! can be consistent?

””Both-and” possibility” of a proposition*, containing in it*, ”.. is thinkable ..” (2.203 , 3.02) if and only if ”falsity” is replaced, substituted by an other proposition, by a ..”t r u e” proposition. To deny the ”truth” of a (contingent) proposition, to assert the ”falsehood” of a proposition, presuppose that there exist an other ”true” proposition which substitutes the ”falsity” of the (other) first proposition. – An other ”true” proposition stands for, represents ”a positive fact” (2.06) that the sense of the first proposition ! d i s- agrees (2.222 , 4.031). If there is no other, and necessary ”true” proposition, there would be ! ”nothing” that ! exists in reality the first propositon’s sense to disagree. One can say that a universal Turing mac-hine would not stop, halt (when trying to find ”nothing ”existing””); and if would halt, then ! contradic-tion, i.e. the result would be → ”a fact that express a fact which is not a fact”, .. a Turing machine in case so to speak blows up (by refuting its result). – For better or worse a Turing machine cannot halt.

To deny, denying the ”truth” of a (contingent) proposition –

”.. determines a logical place other .. o u t side .. the denied proposition ..” (3.4 , 3.41 ,3.411 , 3.42* , 4.064 , 4.0641, below a remark of that* paragraph.).

If a locigian deals with the totality of well ordered propositions, all possible true and false, ”false” subs-tituted by ”true”, propositions ”.. must already be given .. in logical space ..” (1.13 + 3.42 , 4.023 , 4.06, so-me parts of paragraphs 1.13 and 3.42 are compound.) This is done in Tractatus by the fact that well or-dered propositions of correspondence theory contain their all ””both-and” possibilities”.

A well ordered proposition ”.. contains the prototype of its own argument ..” (3.333), that is the ””both-and” possibilities”; and in order to be able to be ”false” a valid proposition points out of itself to an ot-her propositionto ”.. a logical place other .. with .. different meanings ..” (3.333 + 4.0641 + 5.55, some parts of paragraphs are compound. Italics is mine.) – whose ”truth” is absolutely possible.

Actually propositions 3.332 , 3.333 and 4.0641 / 4.064 say the same (thing). That a well ordered propo-sition ”.. already contains the prototype of its own argument ..” (3.333), and consequently ”.. cannot be contained in itself ..” (3.332) anymore says the same than (above quote) paragraph 4.0641 / 4.064. This emerges when Wittgenstein says in paragraph 3.333 –

”.. If .. we suppose that the function F(x) could be its own argument, then there would be a proposition ”F(F(fx) )”, and in this the outer function F and the inner function F must have different meanings; for the inner has the form φ(fx) , the outer the form ψ(φ(fx)). Common to both functions is only the letter ”F” , which by itself signifies nothing. ..”. (3.333, italics is mine.)

That ”..the outer function F and the inner function F must have different meanings ..” (3.333) says that ”.. the outer function F and inner function F ..” are vis-a-vis, with respect to each other ”.. a logical place other ..” (4.0641). In other words if a well ordered proposition tries to ”.. be its own argument .. be con-tained in itself ..” (3.332 + 3.333, some parts are compound.) it inevitably points out of itself to ”.. a lo-gical place other .. outside ..” (4.0641) of itself.

This means that if anybody a logician tries to access the ”.. prototype of its own argument ..” (3.333) i. e. all the possibilities of both ”truth” and ”falsity” of ”.. truth-arguments of propositions”(5.01) of their own, that undertaking, intention will be out of reach. To wit, to reach that intention would mean that a well ordered proposition of correspondence theory agrees with an ! other valid ! true proposition, that ma-kes the first ”agreeing” proposition false

a logician would reach a vicious circle by that a ”truth” proves, deduces ”falsehood”.

As such, above said, presupposes that the law of contradction holds true; in Tractatus. I think that also other two fundamental laws of logic, the law of excluded middle and the principle of identity, hold true in Tractatus. Sure, there exists ”.. a logical impossibility” (6.375), by the fundamental laws of logic, cor-respondence theory to be contradictory. It is consistent by itself, by that those other (contingent) ”true” – must – exist – propositions ”.. cannot be said” (4.1212).

This, supposed that the law of contradiction is like an axiom, argues that Wittgenstein’s ”fundamental thought” (4.0312) indicates also the second incompleteness theorem. How? Paragraphs (4.0312 , 4.128 , 6.4321) say in my book that not a single well ordered proposition of correspondence theory can be de-duced a priori (in logic) since to deduce a single well ordered proposition would mean that also all the others ”.. must already be given ..” (3.42). Well ordered propositions of correspondence are all interre-lated; yet they form by their ”truth” values compatible, consistent groups (4.46 , 4.464) which exclude each others, making falsity of contingent propositions of other (excluded) groups possible. – Thus, that not a single (group of) well ordered contingent proposition(s) can be deduced a priori (in logic) means that the consistency of correspondence theory can not be proved either. – On the other hand cor-respondence theory is consistent by itself precisely due to the fact its valid proposisitions are ”.. inex-pressible ..” (6.522, italics is mine.), and so ! there is no need to prove its consistency.

(Perhaps groups of well ordered contingent propositions can / must be done in alternative logics, with not (all) the fundamental laws of logic; or (completely) outside of any special logic (only) with the fact: ”We cannot give a sign the wrong sense” (5.473 , 5.4731 , 5.4732 , 5.4733 , 5.5563). – A fact by the logic of given theory of ”truth” (2.024 , 4.061.) all propositions* of universal-language are already well or-dered by the fact that they* are independent of reality; it’s just to set up them* to compatible groups.)

Before last lines, and some conlusions, a synopsis of valid language of correspondence theory.

The valid language of correspondence theory made in universal-language ”.. from within .. (4.114) app-lying correspondence theory to that language, independent of ”reality” by the logic of correspondence theory itself, is a model (2.12 , 4.01), a simulation inside universal-language, how the relation between universal-language and reality functions completely correct. – In order to be that valid, consistent well ordered language* (of correspondence theory) which contains all its* truth-possibilities, tautological, contradictory and especially contingent (4.46 , 4.464) IT* MUST POINT OUT OF ITSELF* TO THE INITI-AL UNIVERSAL-LANGUAGE WHERE IT* WAS MADE .. WHERE IN THE INITIAL UNIVERSAL-LANGU-AGE ARE BY ”.. A LOCICAL NECESSITY ..” (6.375) ”TRUE” PROPOSITIONS .. Y E T – T O – B E – M A D E – A N D – P R O V E D.

Since these necessarily ”true” propositions ”.. cannot be said” (2.18 , 4.06 , 4.12 , 4.121 , 4.1212 , 4.128) al-though the valid language of correspondence theory contains the possibility to express them as well (3.13), the valid language of correspondence theory made inside universal-language is not possible to be set out, access or decipher – to avoid opening ”pandora box” of contadictions. – Even if ”.. A proposition .. does contain the possibility of expressing .. its sense” (3.13), i. e. that proof-theoretically correct ”sense” (4.2) just because of the fact that a well ordered proposition ”.. already contains the protype of its own argument ..” (3.333), contains its ””both-and” possibilities”, to fulfill #that ”.. possi-bility of expressing .. its sense” (3.13)# is not possible. To wit to fulfill #it# presupposes a proposition to contain in itself in order to compare itself with all its truth-possibilities, and therefore to be driven in the mess of contradictions.

If the valid language of correspondence theory could, or will be, possible to decipher then it will show as groups (4.46) of true propositions which exclude each other; in other words the propositions of the-se groups exclude each others truth. – How could ! all each other excluded groups to be said, expres-sed? – This might be in the midst of P vs. NP (question).

That the valid language of correspondence theory, made inside universal-language, can not to be set out is the the core of ”the mystical” in Tractatus. – That ”.. the world ..is” (6.44 , 6.45 , 6.522), by the ne-cessarily ”true” but ”.. inexpressible ..” (4.1212 , 6.522) propositions, is ”the mystical” of Tractatus. These have to be ”true” but ”inexpressible” propositions represent, stand for the reality of ontological realism. ”The mystical” of Tractatus is not as much that ”.. a limit of the world .. the metaphysical subject” (5.631 + 5.641, some parts are compound.), but ! the inexhaustibility of ”truths” contained in universal-langu-age. It is indeed, definitely amazing that all the possible well ordered propositions of correspondence theory which represent reality* ! since the dawn of time, as it* ! have been or ! could have been, are .. already contained in universal-language (1.13 , 3.42). And yet none of those (contingent) propositions can not be set out, can not decipher by the means of logic (5.551 , 6.4321); even if they ! All ! Exist.

“To see a World in a Grain of Sand
And a Heaven in a Wild Flower,
Hold Infinity in the palm of your hand
And Eternity in an hour.”
― William Blake, Auguries of Innocenc

What’s my conclusion.

First, ”the facts” by means of correspondence theory, as a theory of (ontological) realism, are alike jig-saw puzzle made and to be made piece by piece applying scientific method as it evolves.

Second, there is no ”totality” of all valid propositions of correspondence theory to be set out, decipher inside universal-language, since that ”totality”* is ! broken, has ! no limit throughout pointing necessari-ly out of ”itself”* back to universal-language, where the task to express all valid propositions of corres-pondence theory was intended to fulfill. There is no limit of the valid, well ordered propositions of cor-respondence theory to be drawn and expressed; no limit to be drawn ”.. to the expression of thoughts .. in language .. from within ..” (Preface / (4.112 , 4.114)). Wittgenstein’s aim fails since ”.. we cannot .. say what we can n o t think” (5.61, italics is mine, except ”say”.). Therefore the answer to ask the question of all the valid propositions of correspondence theory demolish .. ! the question (6.5). A ”.. riddle does not exist ..” (6.5) if ”it” is proved to be such that not ”.. possible to answer ..” (6.5). The question, ”.. The riddle ..” (6.5) of the existence of the world, reality by (all) well ordered propositions of corresponden-ce theory is not ”.. possible to answer ..” (6.5) .. ! Yes or No. It ”.. can be shown ..” (4.1212), the fundamen-tal laws of logic stand, that there is necessarily true (contingent) propositions of correspondence theo-ry which represent ”.. that .. the world is ..” (6.44), but ”.. t h a t .. the world is .. cannot be said” (4.1212 + 6.44, some parts are compound.). Otherwise either correspondence theory falls or the law of contra-diction falls. – In the very end there is nothing to be set out in(side) universal-language, but what re-main is the universal-language / CA / appendix ! without any limit to ”.. make ourselves .. pictures of the reality” (2.1 + 4.06, some parts are compound.) – Thus what David Hilbert said in radio 8. 9. 1930, „Wir müssen wissen, wir werden wissen!“ / (https://www.ardaudiothek.de › swr2), is alive notwith-standing Kurt Gödel’s results in 1931, which inhere (in my book) to that ”wissen”.

Thirdly. What about ”.. a limit of the world .. the metaphysical subject” (5.632 + 5.641, certain parts are compound.)? Does ”it” exist or not, or does ”it” exist and be a necessary but ”.. inexpressible ..” (6.522) truth. Or consistently, as in there is no limit of all the valid propositions of correspondence theory the-re is .. not ”.. the meta-physical subject” (5.641) either. Or when correspondence theory, as a theory of (ontological) realism, says that reality (be it defined ”so or so”) exist independent of ”.. thinking, presen-ting subject ..” (5.631), it is not possible (as a corollary of ”no limit of the world of valid propositions of correspondence theory”) to say where the ”.. thinking, presenting subject ..” (5.631) ends and reality be-gins (5.633).

Fourthly, the answer to a view that Tractatus would end up to a circular argument is ! No. No, because of the proved fact that there already exist* in universal-language all possible true propositions ”tobemadeandproved. It*’s the trick of the of logic of the correspondence theory to be consistent. It* intertwines, weaves with that –

”.. pictures of the reality .. cannot be said .. in advance .. all at once ..” (4.06 + 4.1212 + 5.47, some parts are compound.)

Tractatus would end up to a circular argument, as an analysis of correspondence theory, if there could not be any valid propositions of that truth-theory.

APPENDIX

”Tosi” is ”true” in English, ”epätosi” is ”false”; ja = and. (Accidentally in Finnish.)

The Circle(s) above demonstrate how the valid, well ordered (universal-)language BA* of the corres-pondence theory (2.06 , 2.1 → (2.21 , 2.2221 , 2.222 , 2.223 , 4.031 , 4.0311 , 4.06) → 4.25 , 5 , 5.01) is do-ne, and how IT* is, in universal-language CA; applying correspondence theory in universal-language CA.

The valid, well ordered (universal-)language BA of the correspondence theory, made in universal-lan-guage CA – vis-à-vis, self-referential, in the same universal-language defined ”reality” – consist of the well ordered propositions of correspondence theory that contain the possibility of (provability of) their both ”truth” and ”falsity” (2.06 , 2.1 , 2.11* , 2.18 , 2.201 , 2.202 , 2.203* , 3.02* , 3.13 , 3.318 , 4.06 , 4.1 , 4.2 , 4.21 , 4.25 , 4.3 , 4.4 , 4.41 , 4.431 , 5 , 5.01).

Anyway, in the respect of Wittgenstein’s aim (Preface, (4.112 , 4.114)), the well formed propositions of BA cannot constitute the totality of ”thoughts” / ”pictures of the reality”, i. e. the well ordered propositions that can be proved true or false (1.13 , 2.06 , 2.063 , 2.1 , 2.12 , 2.141 , 3 , 3.12 + 3.14 , 4 , 4.001 , 4.01 , 4.06 , 4.1 , 4.25 , 5 , 5.01).

To ! a g r e e with the possibility of its* (provability of) both ”truth” and ”falsefood” a ! contingent pro-position* (4.2 , 4.21 , 4.25 , 4.3 , 4.4 , 4.41 , 4.431 , 5.01) points outward of itself* → CB , TO AN ! OTHER PROPOSITION THAT IS NECESSARILY ”TRUE” ((3.332 , 3.333) 4.0641 / 4.064), in order to make the ”f a l s i t y” of the first, initial, proposition* p o s s i b l e, ”.. its sense”* (2.221 , 4.031 , 4.0311) to ! d i s – a g r e e (2.222) with some(thing) existing, t r u e (positive) fact of the world, reality.

Therefore, necessarily ”true” propositions substitute ”false” (propositions), well ordered propositions of correspondence theory can be seen to include solely all the true propositions. So, the logic of corres-pondence theory is truth-preserving. To preserve their contingency as well true propositions constitu-te various sets, groups which groups e x clude each others (4.46 , 4.464); d e n y, contradict the truth of propositions that belong to other groups.

Therefore, consequently the circle BA disappears and transforms (BA (CB)) to different (sized), by their ”truths” compatible, groups of true propositions. Each group is ”.. a model of the reality ..” (2.12 , 4.01). They exclude each others. These sets, groups of true propositions, are due to their logical natu-re ”anumerical” (4.0312 , 4.128 , 6.4321), uncountable. They cannot be enumerated with any (general) method of logic.

Anyway these different sized – by their ”truths” compatible – groups of true propositions exists ”.. in logical space ..” (1.13 , 3.42) of correspondence theory. If not (their existence), there would be no logic of correspondence theory either (5.552 , 5.5521). These each other excluding groups of true propositi-ons are not equal. They contest with each other.

The winning group is persistently expanding group of true propositions of science.

Given that .. we will.

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