(null) to replace any primiting.
DEC 17, 1990
Abstract. This paper illustrates major flaws in a paper by C. A. Hilgartner entitled A New Formalized Language Based On Entirely Non-Traditional Premises. A close analysis of the material presented by Hilgartner shows that the presented material contains many ambiguities and inconsistencies. The proposed system fails to qualify as a formalized language and the language as a system is incoherent. Claims based upon the system are therefore unfounded.
Background. In 1975, C. A. Hilgartner wrote A New Formalized Language Based On Entirely Non-Traditional Premises and submitted it to the Notre Dame Journal of Formal Logic. My understanding is that the paper was initially accepted in 1976, and then later rejected by that auspicious journal. Subsequently Hilgartner included a major portion of the material as an appendix to another paper You can't get there from here!, which was published in Eco-logos, 4th quarter 1978.
The appendix, Appendix V, was not completely published, but was made available as a supplement which could be ordered separately from Eco-logos. Pages 46 through 49 and the top of page 50 from Appendix V, sections 1. A. through 1. F. is an almost verbatim copy of A New Formalized Language... pages 6 through 8B sections II. A. through II. F. This material purports to be the formal specification for a new formalized language.
To make referring to the material easy, I reproduce the appropriate sections here exactly as they appeared in the original work.
[page 6]
II. RUDIMENTARY NOTATING
A. MARKINGS
As the allowed markings which make up this non-Aristotelian
languaging, our logiccing selects the ten arabic numeralings, the Latin
and Greek letterings (printed in various type-facings, e.g. italic,
roman, bold-faced, etc.); and, to punctuate, various parenthings,
e.g. ( , ) , [ , ] , < , > , { , } , etc.
B. INDEXING
Our logiccing modifies some markings by means of other markings,
which he calls indical markings or indexings. Given the marking
(terming) T, he may write these indexings as left or right
subscriptings and left or right superscriptings, e.g.
d c
T .
a b
When he so chooses, he may also write the indexings as a right
subscripted finite string, e.g.
T .
abcde
[page 7]
Our logiccing regards an indexing as touching the indexed marking.
Likewise, where he writes unindexed markings adjacent to each other, as
in a terming such as
terming ,
he regards marking as touching its neighboring marking to the left.
Indexed markings written adjacent to each other, e.g.
T T ,
a b
or
d c h g
T T ,
a b e f
do not touch.
Where markings touch, our logiccing intends that we read them
together ("as a whole"); where markings do not touch, he intends that
we read them in some serial ordering, viz. first one, then the next,
etc., with the particular ordering specified explicitly or by the setting.
C. SYMBOLING AND EXPRESSING
Our logiccing regards a marking c as a symboling provided
that c qualifies as some allowed marking other than a parenthing or an
indicial marking.
An expressing, then, comprises an indexing symboling or a
linearly-ordered, finite arraying of (touching or spaced) unindexed or
indexed symbolings.
[page 8]
D. THE PRIMITIVE TERMINGS (PRIMITINGS) OF THIS NON-ARISTOTELIAN
LANGUAGING
As his undefinded termings or (to neologize, creating the verbing
to primit from "primitive terming") the primitings of his non-Aristotelian
languaging, our logiccing chooses to use the terming suggested by
Korzybski [5], namely, structure, order, and relation. As expected,
in discursive settings he takes these as verb-forms, e.g. to structure,
to order, to relation.
To anticipate later developments, our logiccing believes that he can
perhaps define or specify any one of these by means of the other two, but
that he cannot further specify any of them by wording (writing, talking),
To notate, our logiccing can signify these primitings by the
symbolings S , O , and R respectively.
Please note that any solitary primiting, unindexed or indexed,
automatically qualifies as an expressing. No unindexed
defined terming, however, qualifies as an expressing (cf. below, defined terming).
Please note also that, initially at least, our logiccing distinguishes
between S , O , and R ONLY ARBITRARILY, so as to convenience himself.
E. GENERALIZING AN PARTICULARIZING
Under certain circumstances, our logiccing may replace markings
with other markings. For example,
a) He may generalize by using T (terming) to replace ANY primiting;
b) He may particularize by using any one of the primitings to replace
T ; or
c) He may particularize by using (null) to replace any primiting.
[page 8A]
F. WELL-FORMED
Our logiccing uses the terming well-formed to signify a judging,
namely that judging which in traditional western Indo-European
language (such as English or symbolic logic) we signify by expressings such as
"complete sentence" or "grammatically-correct expression," etc.
a) Any primiting which stands alone, unindexed or indexed, qualifies
as well-formed.
b) An expressing compounded of more than one primiting can qualify
as well-formed, provided that
i) It consists of at least four well-formed expressions;
ii) Each of the three primitings appears therein at least once;
iii) That primiting which occupies the initial (viz. left-hand)
positioning of the expressing appears therein at least once and
at most once;
iv) The primitings bear some indical marking (e.g. they appear
printed in different type-facings, or bear an indexing, etc.)
relevant to some hierarchical ordering, and arranged so as to
qualify as well-formed, the expressing must show the initial
primiting as assigned to a positioning in the hierarchical
ordering higher by at least one than that occupied by the
remaining primitings.
An expressing compounded of defined terming (c.f. below) too can
qualify as well-formed, provided that
[page 8B]
1) It consists of at least four expressings;
2) Our logiccing may regard each of these constituent expressings as
particularizing a primiting (or, conversely, our logiccing may generalize
each constituent expressing by replacing it with a primiting);
3) The generalizing expressing (composed of primitings) qualifies
as well-formed.
For years Hilgartner has been referring to these papers and has been claiming to have developed a new formalized language which includes general semantics principles; he particularly claims the alleged new language is entirely free of "identity". Moreover, he claims to have "applied" the "language" and to have achieved various benefits from the new system. Unfortunately, a close examination shows that none of Hilgartner's claims are warranted. Hilgartner himself has acknowledged in private correspondence that his remarks "leave the non-standard grammar in a rudimentary condition".
When I first read Hilgartner's paper back in 1978, I was quite excited by its apparent goal, although I could see that the paper was technically inadequate to the task. I could describe the paper as technically flawed, but seeming to hint at great possibilities, and I began a correspondence with Hilgartner in an attempt to resolve the technical inadequacies. Suffice it to say, that correspondence has left me most unsatisfied, while Hilgartner has continued to make public claims for his language. At this point, thirteen years later [1991], it appears that Hilgartner is neither willing nor capable of correcting the technical inadequacies in his presented system. Accordingly, I present here to you my analysis of his system in a manner that you can see for yourself just how it fails. Perhaps this public exposure will be the stimulus necessary for Hilgartner to attempt the technical corrections.
First, I shall present my understanding of the language or notation from the abstract point of view of what Hilgartner seems to be trying to accomplish. Second, I shall return to A New Formalized Language... and detail specific problems with the statements Hilgartner made.
1. My understanding of what Hilgartner is trying to accomplish.
In Hilgartner's project he seems to seek to instantiate some general semantics principles into the obligatory structures of the new language. Tense, for example, is obligatory in English. One cannot say an English sentence without first choosing the tense of the verb. It seems Hilgartner's main perspective or focus is "Organism in the environment as a whole" taken together with dating and indexing. Everything will be in respect to "one particular organism transacting with its environment at a date".
I think Hilgartner seeks to devise a new language whose obligatory structures contain parts which represent the organism, the environment, and the transacting by the organism. He seems to require a four part structure, each of which represents one of "organism", "environment", "foreground", and "background". Hilgartner's approach to transacting appears heavily influenced by Gestalt Psychology; he has represented transacting from the point of view of the organism. By having a part which selects what the organism is focusing on (the foreground), and a part which the organism is excluding, concealing, not focusing on, etc., (the background), he has divided transacting into two components. In addition, he uses indices to represent the individual organism and the date-time of the representation.
I feel that Hilgartner wanted the language to be particularly useful for devising maps of a given set of circumstance (one particular organism transacting with its environment at a date). His goal seems to me to have been to devise manipulation rules which would change the linguistic maps (statements in the new language) in ways that represented putative changes in the territory. For example, one desired rule would represent the organism's changing its focus from that which it was perceiving as a figure to that which it had previously relegated to the background ("reversing figure and background"). The desired manipulation "rule" would allow swapping figure and background terms. Hilgartner calls this "negating", but does not provide examples of well formed formulas illustrating applying the rule, so I cannot provide a specific example. Also, the organism could contract its focus to look at a particular aspect of what had been represented by the foreground term. The desired manipulation "rule" in this case would allow expanding the foreground term into a full formula by a process called "particularizing", (-- supplying the particulars); the organism would then shift its focus to "figure" one part of the newly expanded term. Again, Hilgartner provides no specific examples. Additional "rules" were desired for other manipulations with the terms, including generalizing, componenting, substituting, and specifying.
However, these manipulations would be strongly influenced by another aspect of general semantics. General semantics uses the terms 'structure', 'order', and 'relation' as undefined terms. Each is thought of as "definable" in terms of the others in a special way.
For example: a relation is an ordering of two structures. Also, an ordering is presented as a relation between two structures. Moreover, a structure is thought of as a complex of ordered relations. The pattern of these inter-connections is that one term is "defined" using one of the other two taken twice and the remaining one taken once. For example: R S S O, where R stands for Relation, S stand for Structure, and O stands for Order, and the order of the writing of the symbols means the first symbol is defined in terms of the other three.
These terms remain "undefined" when they are taken as the starting point for building complexes. But, their use is not totally un-restricted. Once one term is chosen, the way the others can be used is restricted. I think Hilgartner writes S O O R because a structure is determined by two orderings related somehow. Which of these undefined terms is chosen first is arbitrary.
Finally, I believe Hilgartner hopes to embody the general semantics formulation of "non-identity" into the new language where he defines 'identical' as "the same in all and every respect". Hilgartner defines tacit identity as the organism's "non-verbal failure to distinguish" between different happenings or doings, and attributes the cause of tacit identity to the noun-verb distinction. Hilgartner seems to conclude that eliminating the noun-verb distinction will eliminate tacit identity. Hilgartner, it seems to me, has chosen to eliminate the nouns because nouns in English appear static or "self-identical", whereas verbs seem to connote continual change. So, according to my reading, Hilgartner has chosen to emphasize verbs in his non-standard notation. Mind you, he is not discarding nouns and keeping verbs; he is eliminating a two term grammar by doing away with the verb/noun distinction.
So, we can understand in principle what the structure of the language is desired to do, and we can use this "understanding" to abstractly follow the sense of what Hilgartner says in his papers. But when it comes right down to whether or not what he actually said accomplished any of the goals, that is another question.
Here is the second part, in which I detail significant problems with A New Formalized Language....
We are all familiar with Bob Pula's very concise formulation showing the relation of general semantics to other disciplines. (See my paper Levels or perspectives on the use of language.) Bob says in effect:
However, the general semanticist does not exclude the first perspectives. Those are necessary aspects of general semantics. In any semantic reaction situation, we will be interested in the grammatical, the logical, and the semantic aspects of the situation as well as the semantic reactions per se.
Let us use these four levels to evaluate the degree of attainment, as I see it, regarding Hilgartner's goals and purpose.
Formal languages are concerned with the first two levels as described by Bob Pula and interpreted formal languages the first three levels.
The grammatical aspects must be specified to describe a formal language (as opposed to an informal or natural one). A formal language may be completely specified, whereas an informal one may be incompletely specifiable, or open ended. To completely specify the grammatical aspects of a formal language we must do the following.
1. We specify what tokens are used in writing down the language. Because the language is formal, we can and must specify all the tokens that may be used.
Hilgartner, it seems to me, has not adequately met this level of criteria in II. A., which he calls "MARKINGS", because he leaves unspecified various marking by using the term 'etc.', to indicate others not listed. Some tokens of the language are specified, but given the name 'markings'.
To be precisely specified the defining of MARKINGS should identify the letters and numbers as "atomic markings", and the remainder should be completely listed and divided into connectives and punctuation. Hilgartner gives the eight parenthesizing punctuation tokens '(', ')', '[', ']', '<', '>', '{', and '}', but gives no connective symbols and neglects to include the very significant tokens the comma and the space. I gave him the benefit of the doubt and included these in his "etc.", but on condition that II. B. be eventually fixed to specify these precise requirements. If a token is not listed, how is one to decide if it is a proper one or not? If it is listed the rule is simple. A listed token is proper, one not listed is not allowable. By including 'etc.', the rule becomes ambiguous and not specifiable - undecidable by reason of ambiguity.
2. We specify how those tokens are to be combined in groups to form some defined structure. In traditional languages the kinds of objects so defined have been called terms, factors, expressions, formula (well formed formula), sentences, etc. If the language is to be formal, the specification must be complete and un-ambiguous. It is necessary that the specification for any object composed of more than one token of the language be complete and un-ambiguous so that it is decidable whether an arbitrarily complex object is or is not a defined structural object in the language. This just means that the grammar of the language is stated in such a way that it is always possible to tell if or if not and why or why not an expression is grammatical or not. The simple two-valued distinction is enforced at this level. While the language may not be two valued itself, (when we get to semantic issues), whether an expression is grammatical or a-grammatical must be decidable without appeal to interpretation or semantic reactions. Decidability is inherently a two valued question. Is the sentence grammatical? Yes or no only.
At this level, it seems to me, Hilgartner has inadequately stated what are the composite structures, how to build them, how to decide when one is grammatical or not, at least not in a way that is decidable on the basis of grammatical considerations alone.
In II. B., Hilgartner first defines 'indexing', by referring to a "terming", which he has not yet introduced. How a "terming" is constructed of markings is not specified. We could know what indexing is provided we already knew what terming is, but we don't because terming has not been presented yet.
If we defer understanding how to write down a terming, and assume we know indexing now in terms of the undefined "terming", we could say that we conditionally know writing indexing. But, Hilgartner further explains that indexing may be written as a "right subscripted finite string". Fine, but what is a "string"? The term 'string' has not been introduced or explained. Does it mean what it conventionally does in formal languages? -- A concatenated list of tokens? If so, it should have been introduced. Ok, let's tentatively presume that that is what Hilgartner had in mind. So, Hilgartner is saying that indexing can be written in two ways. One as pre- and post- super- and sub- script notation, and one as a sequence of indices "strung" together in the post-sub-script position on a "terming".
But the example he shows has more markings in the post-sub-script string than are given on the terming with pre-and post- super- and sub- scripts. What is the relation between the two ways of writing indexing? Hilgartner has just introduced another ambiguity. Can "the same thing" be written in both formats? How does one transform from one to the other? Since, I presume, an indexing is not limited to a bare marking (single token), how are we to tell which of the markings in the post-sub-script string correspond to which of the four positions?. If nothing else were wrong with the language, this, in itself, is a fatal flaw that prevents the specification from being complete; it prevents deciding what is an indexing.
So, in the first two sections, MARKINGS is (almost) well defined, and INDEXING is ambiguously "defined" in terms of the not the yet introduced terms 'terming' and 'string'.
In section II. B., Hilgartner discusses touching and not-touching. Marking separated by one or more spaces are not-touching, while markings not separated by a space are regarded as touching. Also, indexing markings are regarded as touching the marking ("terming") to which they are applied. "Terming" is still undefined, and this defining of indexing suggests that indexing may apply to "terming" as well as marking. If, as I suspect, the defining of terming yields that marking is a special case of terming, then 'terming' should be used in the defining of indexing and not 'marking'. The present formulation raises more questions than it resolves.
A distinction between "a serial ordering" and "as a whole" is also introduced; touching markings are to be "read" as a whole while marking not touching are to be "read" with a serial ordering. Obviously this cannot apply to our speaking the terms, since it is not possible to verbalize touching markings "as a whole". The whole business of touching is unnecessary as a separate defining in the language, especially since it is not used later. It would be appropriate in discussing the meaning of 'concatenating', which should be used in defining 'stringing'.
If the marking were classified into atomic markings and punctuation, we could divide the atomic markings into lettering and digiting; we could then say that symboling is concatenated stringing of lettering; numbering is concatenated stringing of digiting, and finally that terming is unindexed symboling or symboling indexed by numbering. However, what Hilgartner does say does not allow us to conclude this.
In II. C., Hilgartner introduces the terms 'SYMBOLING' and 'EXPRESSING'. He defines a SYMBOLING as a marking that is not a parenthesis or an indexing. This is unnecessarily convoluted because the MARKINGS were not divided into atomic markings and parenthesis. In consideration of our familiarity with standard formal languages, we may reasonably conclude that Hilgartner intends that a SYMBOLING be simply an atomic marking.
Ok, we could go back to section I and define MARKING as divided into two categories -- SYMBOLING and PARENTHETICAL MARKINGS.
Hilgartner defines EXPRESSING as an indexed symbol, or a finite sequence of indexed or unindexed, touching, or spaced symbols. The problem with this definition is that a single un-indexed symbol is a finite sequence (namely one) of indexed or unindexed, touching or spaced symbols. Does Hilgartner intend to eliminate this case? If not, the definition contains a redundancy. If so, it is not written in such a way as to eliminate that case. This definition also introduces, for the first time, a structure containing concatenated symbols. Are we to be without a name for that structure? There are too many levels of structure being introduced at once at this point.
Let me take a higher level point of view for a moment. In this defining of structures that make up the formulations by which we "get a-hold" of the language, there are certain "natural" break-points in which "little pieces" are put together into "medium sized pieces", which are in turn put together into "bigger sized pieces", which are themselves put together into even bigger sized pieces, for as many levels as are efficient to capture the overall structure of the language. The "natural" breakpoints have to do with the ways we combine "markings" into larger gestalts. So far, Hilgartner has missed these "natural" breakpoints. He has too many "little pieces", and tries to go all the way to big pieces without enough intermediate sized pieces.
If we were to follow the customary style, in which these types of ambiguities have been eliminated, we would arrive a natural and efficient sequence.
But Hilgartner doesn't do this...
In section II. D., Hilgartner discusses the motivational influence of structure, order, and relation, and identifies the latin letters S, O, and R as designating, or referring to these respectively. He defines S, O, and R as primitings. Unfortunately, this step begins to introduce a semantic notion by having certain symbols that stand for things, whereas the grammatical aspects of the language have by no means been completely specified.
But, here Hilgartner does something completely invalid. In II. D., he says that no defined term can be an expression if it is not indexed. Unfortunately this introduces a fatal inconsistency. Any marking written to signify a defined term without indices can be written as an expressing (II. C.). But if it is written as an unindexed defined term, it is not an expressing (II. D.). So, such a marking is both and expressing and not an expressing. Did Hilgartner mean (to have said) no defined terming can be a single un-indexed primiting? That is to say, that re-defining primitings is not allowed?. If he is not being completely inconsistent, then he seems, at the very least, to be adding to the defining of 'expressing'. As a matter of form, and for clarity of communication, the defining of 'expressing' belongs all in one place.
Hilgartner has begun to mix grammatical rules with semantic rules. Having already defined expressing as a grammatical rule, he is now also defining expressing as a semantic rule, and the two rules apply to the same marking in exactly the opposite way; the two rules contradict each other. Our ability to decide whether a marking is an expression has just been taken away from us. By one rule we can say that it is. By the other rule, we can say that it is not. Whether a marking is an expressing or not, then, is no longer decidable.
In section II. E., Hilgartner defines generalizing and particularizing.
Generalize by replacing a primiting with a terming. Particularize by replacing a terming with a primiting.
This would be fine, except he has never defined "terming". Our knowledge of both generalizing and particularizing is, therefore, contingent upon finding out what terming is.
As a special case of particularize, Hilgartner allows substituting
(null) for a primiting. Now this introduces another big ambiguity. Our customary
sense for null would have us believe that this special case of primiting can be
stated as deleting a primiting from the expression. But, on the other hand, he
could mean the greek letter phi. Do all primitings particularize to phi? What
about an expressing consisting of a single primiting? does it "go
away" as in deleting it? Does that make a single blank an expression? Or,
does the primiting become the marking phi. But, if all primitings can become
phi, what is left is an expressing with four instances of phi and with no
primitings in it. Some examples would be very helpful at this point. Well,
Hilgartner is just not clear about what he intends for the character .
Is it a symbol in the language, in which case he should have written '',
or is he using the null set symbol to mean eliminate the primitive? That would
make a blank an expression. Is that right? [Not normally!]
The baggage of ambiguity inconsistency, and contingent unexplained terms long ago prevented Hilgartner from accomplishing his goals. But let's continue.
Section II. F. introduces the notion of WELL-FORMED.
II. F. b. ii) poses a problem because it requires that each of the three primitives appear in [an expressing] at least once, but II. F. b. i) allows that it be composed of four well formed expressions, which by recursion could each contain three primitings. II. f. b. iii) seems to eliminate much of II. F. b. i).
If these are to be jointly satisfied, then the composite well formed expression must contain four components, each of which is an indexed primiting. Suppose any one of the terms were composite. Then it must be composed of at least three primitives each of which must appear at least once. This makes a composite formula composed of at least 6 indexed primitives with the first primiting appearing only once, but the expansion of any one term required the three primitings to appear at least once. There is only one way to do this -- expand only the first primiting into a composite term having the same first primiting. But by II. F. b. ii), each of the terms could be a composite term. Consequently one of II. F. b. ii) or II. F. b. iii) fails to hold except in this special case.
The various parts of II. F. b. "fight" each other to limit the form of well formed expressions as any expression beginning with one primiting and having a sequence of the other two (all indexed). Nothing in this definition tells anything about how the primiting should be grouped with parenthetical marking. --That would go a long way to resolving the difficulty.
The definition of well formed started off poorly enough with the recursive, but inconsistent specification, however depends further upon the totally new and unspecified notions of hierarchical ordering and some assignment function related to the order of the primitives.
We cannot say whether or not an expression is well formed until we know what a hierarchical ordering is, and what assignment functions are. The assignment function shows how a logician assigns the primitives to their positions in the ordering. Ok, now we are introducing a connection between semantic reactions and well-formed. It is the semantic reactions of the logician which determines how he assigns the primitives to their respective positions in the ordering. Unfortunately, semantic reactions are not something that can be written down. They cannot comprise a formal language structure. We must embody terms to represent those semantic reactions in some kind of rule or formula. So, from a grammatical point of view well formed is unreachable because it includes pragmatic or semantic reaction considerations just as expression was unreachable from a grammatical point of view because it included semantic notions. From a logical point of view well formed is unreachable because it is inconsistent. From a semantic point of view it is unreachable because it depends upon unspecified (and unspecifiable) semantic reactions.
Not only that, but Hilgartner used the term 'ordering' in the phrase 'hierarchical ordering'. Is this the undefined term which 'O' arbitrarily stands for? Or is it our usual notion of the term as in the order of '1, 2, 3'? If so, how is this to be applied to symbolings? Do digitings (concatenated) "stand for" our usual numbers with their associated ordering system? The semantics of the language has not even begun to be stated.
It is pointless to go on with more analysis of what was said, or what it was intended to mean.
Natural languages are not completely specifiable, whereas formal languages are. The rules for writing down statements in a formal language, unlike natural language rules, are un-ambiguous and complete. Hilgartner has not presented any un-ambiguous rules.
Clearly, Hilgartner, in attempting to get to the heart of some individual semantic reactions has sorely neglected the grammatical, logical, and semantical aspects of the language. The result is grammatically ambiguous, logically inconsistent, and semantically inadequate, resulting in incoherence at the level of induced semantic reactions. This in itself would not be reason enough for this public expose, but for the fact that Hilgartner has frequently and publicly claimed to have developed a new formalized language, and to have applied it in many areas. Unless Hilgartner revises his formal language specification to correct the ambiguities and inconsistencies I have pointed out he is unjustified in claiming he has done the job. In 13 years no corrections have been made; no progress has been demonstrated.
Moreover, my recent work has shown that Hilgartner's goals themselves are not possible. I have shown that elimination of all forms of "Identity" would result in a language that cannot be used for communication. Such a language could not be used for time-binding at all. [See my paper The Impossibility of Non-Identity Languages, General Semantics Bulletin 55, (1990).] In addition, one of the technical structures chosen by Hilgartner for his language seems to violate a fundamental general semantics principle. [See my paper Hilgartner's Violations].
Progress in time-binding includes ferreting out and discarding ideas which may have seemed promising, but which, on closer analysis, prove to be misdirected. Remember Rene Blondot, who claimed to have discovered subsequently discredited "N-RAYS". In mathematics there is a controversy over whether mathematical structures are "discovered" or "invented". Well, Hilgartner has neither discovered nor invented a new formalized language.
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