Language informs the internal structure of knowledge. This structure is both a product of the knowledge it contains, and is the resulting artifact of that knowledge. These human languages are themselves comprised of fragments that each have their semantic and terminological meanings. These meanings can be condensed into an intermediary form, which contains those same represented meanings and therefore has the same inherent knowledge structures. Given basic operational rules, these condensed intermediary language representations can be expressed as general purpose programming languages. Because this condensed notational form of language contains representations of the structure and content of the original human language, all fully faithful expressions of that intermediary form must also contain representations of the initial underlying knowledge structures. This paper puts forth that programming languages contain representations of the human languages that they are derived from, and the knowledge structures of those initial human languages can be found in the knowledge structures implied by the programming language itself as shown by both the Catala programming language and a hypothetical programming language derived from the human language Lojban.
Language informs the internal structure of knowledge
Knowledge organization and language are intimately intertwined. Thellefsen argues that knowledge organization must be informed by linguistic theory and semantics in order to represent the entirety of the knowledge within a given domain (2003, p. 211). The language used within a domain is called sub language or special language (Thellefsen, 2003). This definition of special language means more than just the words that are unique to that domain; it also encompasses the change in meanings of existing words when used within the given domain. For instance, the word "inheritance" has similar yet distinct meanings in the domains of law and computer science.
This difference in meaning between these two domains is because the word "inheritance" has both a semantic meaning and a terminological meaning. The semantic meaning of the word is common to both domains; the word in both domains evokes generational ownership and motion. However, the terminological meaning of the word in both domains are different. In the legal domain, "inheritance" is something very specific and deals with beneficiaries and estates. In computer science, "inheritance" means object oriented programs and generic classes of objects that have distinct instantiations as objects.
When considering the functions of a given word within the languages of specific domains, a framework for this analysis is useful. The framework used in this paper is KRL, a framework provided by Bobrow and Winograd, which provides a structured way of associating descriptions with conceptual entities as an organizational strategy for declarative knowledge (1977). In KRL, a description is a group of descriptors, which are each a statement of some fact that is either an observation about an object, or a fact about the object that is only useful when used as a comparator (Bobrow & Winograd, 1977). Because a description may be comprised of multiple descriptors, it allows describing a complex event through multiple points-of-view simultaneously (Bobrow & Winograd, 1977, p. 6). This ability of KRL to simultaneously describe a single event through multiple view points lends itself well to describing special language since studies of special language are concerned with both the terminological and semantic meanings of the given piece of language.
KRL is put forth by Bobrow and Winograd as a general purpose program language. They accomplish this by specifying the rules and operations under which groups of descriptions, called units, can be interacted with (1977, pp. 6-9). They propose a syntax and a grammar for KRL and note that "We believe that it is more useful and perspicuous to preserve in the notation many of the conceptual differences which are reflected in natural language, even though they could be reduced to a smaller basis set" (p. 7). Thus, KRL represents a way to reduce natural language to a rigorous formalized grammar and syntax via a notation set.
There are other general purpose programming languages that have begun life as a notation set as KRL has. APL, created by Kenneth Iverson is a salient example. Iverson's main thesis in creating APL is that "Mathematical notation provides perhaps the best-known and best-developed example of language used consciously as a tool of thought" (1980, p. 444). In fact, Iverson originally created the notation used by APL as a teaching aid, and the notation set was only implemented as a programming language after some years of use (1980, p. 445). In his work, Iverson notes that in general, the advantages of programming languages as tools of thought are that programming languages are universal (that is, they are general purpose), they are executable, and they are unambiguous (1980, p. 445). Iverson's treatment of APL as a notational tool of thought provides the mechanisms for combining a notation set with a programming language. Iverson suggests that a good notation must have at least a few common characteristics.
The first characteristic of a good notation is that is "must allow convenient expression not only of notions arising directly from a problem, but also of those arising in subsequent analysis, generalization, and specialization. (1980, p. 446). This is true of any general purpose programming language, as it is the definition of "general purpose." Iverson also specifies another facet of general purpose programming languages that lends itself to being a characteristic of a good notation; he puts forth that a good notation is suggestive. By this, he means that a good notation can "represent identities in brief, general, and easily remembered forms" (1980, p. 447). This means that a solution for a problem, given in a particular notation, can be recognized as the notational form of a solution for that particular problem and any other similar problem.
Much like how Bobrow and Winograd noted that their notational form of KRL was larger than strictly required (1977, p. 7), Iverson writes that APL, and indeed any good notation, should subordinate detail and be economic in its vocabulary (1980, pp. 448-449). By this Iverson means that a notation should be able to express a large number of ideas in terms of a relatively small vocabulary (1980, p. 449). The mechanism by which this expression happens is by introducing a set of grammatical rules to be used in coordination with the notation set (1980, p. 449). A set of grammatical rules that govern the use of a language is central to the idea of a language in general.
Finally, Iverson suggests that a good notation is amenable to formal proofs (1980, p. 450). In the narrowest interpretation of that statement, one need only look at the origins of Iverson's APL language as a mathematical teaching aid to see the practical effects of this characteristic. However, when considering a programming language itself as a notation set, then this characteristic instead means that within the grammatical rules of the language itself, precise and unambiguous statements can be made.
The characteristics of a good notation are also the characteristics of special language. This means that a notation (and its grammatical rules) comprises of both the vocabulary of the notation itself and the meanings of those notational elements. If a notation wasn't a special language contained within the general human language, then there would be no need for the notation to have a vocabulary or rules that are different than those of the general human language that it is derived from. The characteristics of a good notation are also the characteristics of general purpose programming languages. This means that a general purpose programming language is a notation for a human language, and is also a special language of that human language.
While the ability to be unambiguous is vitally important in general purpose programming languages and notations, it is often less important for human languages to be unambiguous at all times. The human language Lojban however, was constructed for logic based unambiguous human-to-human and human-to-machine communication (Hintz, 2014, p. 18). Because Lojban has an unambiguous grammar, it is trivial to parse and easy to learn. The regular morphology, minimal regular syntax, and the explicitly minimized semantic ambiguity in Lojban all contribute to it being well suited for accurate and efficient communication (Hintz, 2014, pp. 18-21).
Given the features of Lojban, it is clear that Lojban as a language also fits many of the characteristics of a special language as well as a notation and programming language. Even when treating Lojban as a notational language however, the semantics of the language are still important. This importance has the implication that the semantics of the original language are still present and important in the notational form of that language. This is true for all programming languages, and not only for condensed human languages that can function as programming languages.
In the programming language Catala for example, legal texts can be translated to executable forms. This translation is achieved by a notation that is referred to as "the Catala programming language" (Merigoux, Chataing, and Protzenko, 2021, p. 1). The stated aim of the creators of the Catala programming language is to "bring together lawyers and programmers through a shared medium, which they can understand, edit, and evolve, bridging a gap that too often results in dramatically incorrect implementations of the law" (Merigoux et al., 2021, p. 1).
It is seemingly obvious that legal language is a sub language or special language derived from a base general purpose human language. Catala has been proven to be correct in its core compilation steps by the F* proof assistant (Merigoux et al., 2021, p. 1). This implies that Catala is amenable to formal proof and that it can express ideas unambiguously. Catala as a notation is also suggestive because it has clear semantics and "compiles to a generic lambda-calculus that can then be translated to any existing language" (Merigoux et. at., 2021, p. 3).
In legal texts, there are several structures that are not typically present in other texts. The first of these atypical structures is "out-of-order definitions" (Merigoux et al., 2021, p. 3). In this type of structure, the general case is given first, followed by an enumeration of limitations or exceptions. The creators of the Catala language describe this structure as "relevant information is scattered throughout, and [one section] alone is nowhere near enough information ..." (Merigoux et. alo, 2021, p. 3).
The second atypical structure found in legal texts is "back-patching" (Merigoux et al., 2021, p. 3). In this type of structure, a section of text is modified in place by other text that comes after the section to be modified. A modified version of this structure can be made by combining "back-patching" and "out-of-order definitions" to create "out-of-order back-patching" which may change the entirety of a section of legal text based on one out-of-order piece of information (Merigoux et al., 2021, p. 5).
The final atypical structures found in legal texts are "re-interpretation" and back-patching re-interpretation (Merigoux et al., 2021, pp. 4-5). In these structures, a section of text can recursive or re-entrant and can back-patch preceding texts.
Finally, in the context of the special language of legal texts, the underlying logic model is one of default logic. This non-monotonic logic has been refined in the context of legal purposes as "prioritized default logic" (Merigoux et al., 2021, p. 5).
All of these textual structures are present in the designed elements of the Catala language. In fact, the main design goal of the Catala language is exactly to provide a programming language the uses prioritized default logic and is tailored for use in law by both its syntax and semantics (Merigoux et al., 2021, p. 5).
While Catala is a programming language specifically created for use in the legal domain, it may be useful in other domains. One could argue however that the application of Catala to a different set of problems is simply the application of existing law to the domain that the problems exist in. Catala is interesting in this regard as it has been used to verify the implementation of legal structures in both English and French law. This is novel because of the assumption that the English and French languages have different structures, semantics, and syntax. Juxtaposed against this one-to-many relation of Catala to both English and French, is Lojban existing as a human language and a programming language simultaneously. When compared to English or French speakers learning Catala or APL to express their ideas, it would be interesting to study native Lojban speakers (should any exist) learning how to express their ideas in Lojban as a notational form that a computer can execute and verify.
By looking at the concrete examples of Lojban as a constructed human language that can be simultaneously a programming language and Catala as a constructed programming language made to match the structures of legal texts, this paper has attempted to show that a programming language is simultaneously a notation and a special language derived from a general purpose human language and that the knowledge structures present in the original human language can be found in the resulting programming language.
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