Merge:

External Merge = конъюнкция не-Х и не-не-Y, паракомплектная логика, "квантовая суперпозиция", нечеткость, "несуществующая" граница между незамкнутыми множествами.

Internal Merge = конъюнкция X = Y и X не = Y, параконсистентная логика, граница между двумя замкнутыми множествами (точнее, между одним замкнутым -- внутренним -- и вторым, внутри которого находится первое), которая принадлежит эксклюзивно, но обоим сразу.

рекурсия -- следствие возможности мыслить нечеткостями и противоречиями. в этом и состоит сдвиг, отличающий язык человека от языка машины и (известных) языков животных (если все-таки найдутся, как меня уверяют, животные, способные к языку с рекурсией, то я не против). UPD уточняется, что не всякая рекурсия, а только с endocentric labeling.

это про синтаксис, но в лексике то же самое (наша мысль про инконсистентность непрямых значений вкупе с мыслью Хинтикки-Санду о том, что непрямые значения не отличаются от прямых, и некоторых лингвистов и наших футуристов о первичности непрямых значений и "некомпозициональности".

кстати, теперь я, конечно, вижу, что между полюсами метафоры и метонимии (Якобсон 1956) именно логическая граница: метафора параконсистентна, а метонимия паракомплектна (как Internal Merge: pars pro toto, totum pro parte...). метонимия построена на исчезновении границы между разными элементами, а метафора -- на ее подчеркивании при отождествлении...

продолжаем 28/29 дек.: таки да:

Internal Merge leads to copying as it is to be expected under paraconsistent separation, whereas Ext Merge leads to smth else (the

*tertium datur*of the paracomplete separation), therefore, β as the name of the head is not correct: it is in fact a ("quantum") superposition of α and β.

29-30.12: вечер и ночь пропереписывался с Х.

31.12, "в соавторстве" и с учетом Ch., POP 2013 and Ch. forthcoming:

SO = syntactic object

1. There is a need to discern between two stages: formation of a new

SO (F-SO) and its participation in formation of the next SO. The first

stage is inconsistent, whereas the second is consistent.

Inconsistency is limited to the formation stage, where a conjunction

of two elements is established. Therefore, to become able to enter

into the next SO (of a higher level), the present SO must appear as

its consistent "projection". (I introduce here this term, projection,

in the sense of a consistent description of an inconsistent object;

formally speaking, these "projections" are like the results of

inconsistent topological separation described with the means of

consistent geometry). The inconsistent operation F-SO applied to two

elements requires that each of these elements can be treated as

consistent.

2. The present conception of Merge involves two stages: both formation

of a new SO (F-SO) and formation of its consistent projection to be

used at the next iteration of Merge. Let us call the latter

"consistentialisation" (sorry for these hasty and monstrous term!)

(C-SO).

3. The "glue" forming SO at the stage of F-SO is inconsistency (some

kind of contradiction producing an inconsistent pair). It does not

deform any element entering into the pair.

4. The two different kinds of Merge imply two kinds of inconsistency

(and require two different kinds of C-SO, see point 5 below). Namely,

4.1. IM implies paraconsistency (allowing a kind of Russell's paradox:

a set that is a member of itself): when classical logic would admit X

OR Y but not X AND Y, IM admits X AND Y.

[here and below: the logical connectives AND is conjunction, OR is

exclusive disjunction]

4.2. EM implies paracompleteness (allowing a kind of vagueness and

sorites paradox, similar to the quantum superposition as it is treated

in some quantum logics): when classical logic would admit X AND Y but

not non-X AND non-Y, EM admits non-X AND non-Y.

NOTES to 4.1 and 4.2:

-- In IM, if X is a subset of Y, it must be considered, in classical

logic, as either X or Y but not both.

-- In EM, the pair of X and Y must be considered, in classical logic,

as X AND Y, but there would be no room for non-X AND non-Y. However,

the whole produced by F-SO via EM is something tertium that is datur,

exactly non-X AND non-Y. To continue the comparison with the quantum

superposition, we can refer to Schrödinger's own example: the quantum

objects behave as dollars in bank account: they are still numerable,

whereas indistinguishable. In the same manner, the elements of SO

cease to be distinguishable at the next (higher level) Merge stage.

5. Two kinds of Merge require two kinds of C-SO. Namely,

5.1. for IM, C-SO produces "copies" (for the formal details, see

Mortensen on paraconsistent topological separation and Heyting

algebra),

5.2. for EM, C-SO produces (sorry for one more monstrous and hasty

term!) "one-sided images" (for the formal details, see Mortensen on

paracomplect topological separation and Brouwer algebra).

NOTES to 5.2:

-- We can provide a simple illustration, which, moreover, has

something to do with linguistics (verbal tense system): our perception

of time and our ways to speak about. Logically speaking, the Present

is the paracomplect border between two open sets, the Past and the

Future. It is the tertium between them that is absent from any

consistent picture but, nevertheless, real. When we consistently speak

about the present, we have to consider it as either within the Past or

within the Future. By the way, I have read in Cinque 1999 that (the

following wording is mine) the verbal tense system in the majority of

languages considers Present as the limit case of Past, whereas in some

rare langues, it is the limit case of Future (both situations are

logically equivalent of course).

-- Therefore, a group of {X, Y} produced by EM must look (for the next

stage of Merge), after C-SO, as either X or Y.

---------------

02.01.19:

(why not ZF)

1. Why ZF is not optimal from a minimalist point of view:

Let us consider formation of the set {X, Y} which is a set under the ZF axioms. If it is given that this set is an unordered pair, any ordering (required by EM) would appear only as a result of a procedure different from pairing itself. The same is true about copying. However, if {X, Y} is an inconsistent set of the kinds I have described earlier, the very procedure of unification of X and Y into a set would result into either asymmetrisation or copying.

2. There are, at least, striking problems with ZF axiomatics (if they are resolvable, one has to write a special study).

First of all, to remain under ZF, the formula Merge(X,Y) = {X,Y} is to be rewritten as Merge({X}, {Y}) = {X, Y} : both X and Y have to be sets as well; otherwise we are outside ZF. However, it is never claimed explicitly. There is a need, if we are within ZF, to state that {X} and {Y} are singletons. If so, however, we obtain a series of problems related to extensionality (when we have to explain what is the difference between X and {X}).

If X and Y are not sets at all, we are dealing with urelements and find ourselves on the shaky ground of set theories with urelements (where the axiom of extensionality does not hold) -- anyway, far enough from ZF (whereas closer to the original Z).

Probably, the most intuitively clear solution would be to postulate extensionality in the way X = {X}. However, in this case, X becomes the so-called Quine atom, and there is a rupture with ZF anyway; moreover, the Russell's paradox could become allowed.

Beside these problems with the foundation, there could be another series of problems with the axiom of unions. If we are under ZF, any higher-level set has to contain as its elements the elements of the lower sets. Thus, the SOs of each particular stage of Merge must be elements of the set corresponding to the whole sentence. This is not evident, at least.

Let us consider formation of the set {X, Y} which is a set under the ZF axioms. If it is given that this set is an unordered pair, any ordering (required by EM) would appear only as a result of a procedure different from pairing itself. The same is true about copying. However, if {X, Y} is an inconsistent set of the kinds I have described earlier, the very procedure of unification of X and Y into a set would result into either asymmetrisation or copying.

2. There are, at least, striking problems with ZF axiomatics (if they are resolvable, one has to write a special study).

First of all, to remain under ZF, the formula Merge(X,Y) = {X,Y} is to be rewritten as Merge({X}, {Y}) = {X, Y} : both X and Y have to be sets as well; otherwise we are outside ZF. However, it is never claimed explicitly. There is a need, if we are within ZF, to state that {X} and {Y} are singletons. If so, however, we obtain a series of problems related to extensionality (when we have to explain what is the difference between X and {X}).

If X and Y are not sets at all, we are dealing with urelements and find ourselves on the shaky ground of set theories with urelements (where the axiom of extensionality does not hold) -- anyway, far enough from ZF (whereas closer to the original Z).

Probably, the most intuitively clear solution would be to postulate extensionality in the way X = {X}. However, in this case, X becomes the so-called Quine atom, and there is a rupture with ZF anyway; moreover, the Russell's paradox could become allowed.

Beside these problems with the foundation, there could be another series of problems with the axiom of unions. If we are under ZF, any higher-level set has to contain as its elements the elements of the lower sets. Thus, the SOs of each particular stage of Merge must be elements of the set corresponding to the whole sentence. This is not evident, at least.

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