Abstract
This paper presents IEML, Information Economy MetaLanguage, a constructed language with the same expressive power as a natural language and with computable semantics. Distinguished from pragmatic and referential semantics, linguistic semantics have not yet been completely formalized. Only its syntagmatic dimension has been mathematized in the form of regular languages. Its paradigmatic dimension remained to be formalized. In order to complete the mathematizing of language, including its paradigmatic dimension, I have coded linguistic semantics with IEML. This article introduces its 3000-word dictionary, its formal grammar, and its integrated tools for building semantic graphs. For the future, IEML could become a vector for a fluid calculation and communication of meaning—semantic interoperability—capable of de-compartmentalizing the digital memory, and of advancing the progress of collective intelligence, artificial intelligence, and digital humanities. I conclude by indicating some research directions.
Introduction
How can we enhance collective intelligence (Levy, 1997, 2010, 2011; Mulgan, 2017; Baltzersen, 2022) at a time when it emerges largely from a process of stigmergic communication (Bonabeau, 1999; Dorigo et al., 2000; Heylighen, 2016a and 2016b) in a shared digital memory? To address this issue, I approached it from a librarian’s perspective, considering how to organize memory in a manner that would maximize three key factors: the possibilities for original data description, effective translation and communication, and the automation of information processing. To make progress in this direction, I required a next-generation documentary language designed to function as a semantic metadata system for the global digital memory. Categorizing data using this language would not only guarantee semantic interoperability but also foster automatic reasoning and machine learning. For the language to effectively serve this role, it must have computable semantics, which entails the following properties: a. It is capable of translating natural languages. b. It can encode knowledge graphs, ontologies, and data models. c. It is unambiguous. d. The syntactically valid character sequences in this language have a one-to-one relationship with concept networks (concepts being defined by sentences in the language). e. This language has an algebraic structure: its expressions are functions whose roles can be filled by variables and constants. The language features a coherent set of functional mechanisms for generating, transforming, and recognizing concept networks.
However, no existing language meets these criteria. Natural languages do not satisfy requirements (c), (d), and (e). The constructed language that comes closest to our criteria is Lojban (Woldemar, 2016), which is based on formal logic and has both a regular syntax and unambiguous meaning. Yet Lojban lacks properties (d) and (e). Specifically, different sequences of words can carry the same meaning, and its expressions are not algebraic functions that can include both constants and variables. Neither programming languages (such as C++ or Python) nor standard file formats (like XML or RDF) possess property (a). Finally, existing document languages (Dewey, Rameau, Library of Congress headings, Colon classification, etc.) date back to the printing era and, at best, they only meet criterion (c). That’s why I had to create IEML from scratch.
From a scientific standpoint, the creation of a language with computable semantics sits at the intersection of philosophy, linguistics, symbolic artificial intelligence, and collective intelligence. Such a transdisciplinary approach can give rise to numerous misunderstandings. To dispel some of them right from the start, I would like to clarify what I mean by the terms “semantics” and “concept.”
Semantics cannot be confined to the linguistic field and obviously depends on the practical context and the modes of veridiction (or judgment) at stake. When an utterance is pronounced, it takes on meaning on at least four levels. That of the mental representation governed by its grammatical structure and the meaning of its words (a network of concepts), the logical plane of its reference to a state of things (it is true or false), and the practical plane of social interaction (it is a move in a language game). These three meanings—linguistic, referential, and pragmatic—echo the Trivium of Roman antiquity and of the Western Middle Ages: grammar, dialectics, and rhetoric. 1 Finally, the fourth plane of semantics, which we might call phenomenological—that of motor patterns, sensitive images and emotions—is rooted in memory and personal experience rather than in the structures of language or other social conventions. These four co-dependent levels determine the overall semantics of an utterance. When I speak of “calculating semantics,” I’m referring only to the linguistic level, since the logical and pragmatic levels are already largely formalized, and the phenomenological level doesn't seem to be amenable to formalization.
It may come as a surprise that I place concepts at the linguistic level. Indeed, conceptualization is well attested in animal cognition (Laurence and Margolis, 2015), so it exists independently of symbolization. But language enables concepts—that is, general categories—to reflect on themselves, while multiplying their number and complexity. What is the signified of a linguistic symbol, if not the reflection of a concept by a signifier? When I speak of concepts, then, I'm not referring to the fixed natural signs of an innate language of thought à la William of Occam or Jerry Fodor, but to the cognitive counterparts of the signified of a language: stars of relations at the intersection of several systems of differences, which are transformed according to local contexts or ontologies. The words “concept” and "signified" function here as two sides—cognitive and linguistic—of the same coin.
Even before exposing, in all their complexity, the problems that I had to resolve to formalize linguistic semantics and the details of the solution, I would like to underline the correspondence between the functioning of languages and the mathematical structure of IEML. How do languages work? Let’s for the moment set aside the obstacles of ambiguity and misunderstanding in order to outline the main process. On the receiving end, we hear a sequence of sounds that we translate into a network of concepts, thus giving meaning to linguistic expression. On the emitting side, from a network of concepts we have in mind—a meaning to be transmitted—we generate a sequence of sounds. Language functions as an interface between sequences of sounds and networks of concepts. The sound chains can be replaced by sequences of ideograms, letters, or gestures as in the case of sign languages. The quasi-automatic interfacing between a sequence of sensorial images (auditory, visual, and tactile) and a graph of abstract concepts (general categories) remains constant among all languages and systems of writing.
This reciprocal translation between a sequence of images (the signifier) and networks of concepts (the signified) is the very principle of the IEML mathematical model, namely, a mathematical category 2 that organizes a correspondence between an algebra and a graph structure. The algebra regulates the reading and writing operations on the sequential texts, while the graph structure organizes the operations on the nodes and the oriented links of the concept networks. Each text corresponds to a concept network, and the operations on the texts dynamically reflect the operations on the concept graphs. To mathematically encode chains of signifiers, I used a regular language (Chomsky, 1957; Chomsky et al., 1963). Such a formalism makes it possible to automatically transform sequences of symbols into syntagmatic trees—reflecting the structural dependence between words in a sentence—and vice versa. Even though the syntagmatic tree is indispensable for understanding the meaning of a sentence, it alone is not sufficient. Indeed, each linguistic expression lies at the intersection of a syntagmatic axis and a paradigmatic 3 axis. The syntagmatic tree represents the internal semantic network of a sentence, while the paradigmatic axis represents its external semantic network and, in particular, its relations with sentences having the same structure but also having some different words. To understand the phrase “I choose the vegetarian menu,” one must of course recognize that the verb is “to choose,” the subject “I” and the object “the vegetarian menu” and know moreover that “vegetarian” qualifies “menu.” But one must also know the meaning of the words and know, for example, that vegetarian differs from carnivore, flexitarian, and vegan, which implies going beyond the sentence to situate its components in taxonomic systems and semantic oppositions, those of language and those of various practical fields. The establishment of semantic relations between concepts thus implies that one recognizes syntagmatic trees internal to sentences, but also paradigmatic matrices external to sentences. This is why IEML algebraically encodes not only the syntagmatic trees, but also the paradigmatic matrices where words and concepts take their meaning.
In short, each sentence in IEML is located at the intersection of a syntagmatic tree and paradigmatic matrices. In addition to a regular grammar, IEML relies on a dictionary of about 3000 words—without synonyms or homonyms—organized in little more than a hundred paradigms. These words have been carefully chosen to allow the recursive construction of any concept, or paradigm of concepts, by means of sentences.
It should be noted that the spirit of IEML's grammar is closely aligned with Montague semantics (Montague, 1970). Indeed, the meaning of an IEML sentence is strictly determined by the way its constituents—words and phrases—are combined with each other (principle of compositionality), and the ontologies encoded in IEML are compatible with the theory of logical types.
Given the role of language in human intelligence, artificial intelligence could benefit from a qualitative leap through improved data categorization and a new semantic computability. A language that is understandable by humans while being executable by machines would bring to smart contracts, and to the various blockchain systems, the transparency they lack today. As a contribution to the digital humanities, IEML is intended to be a semantic coordinate system introducing human sciences to a new regime of scientificity. Finally, the metaverse would no longer be limited to an imitation of three-dimensional space and could make the world of ideas perceivable in an immersive and social way. We can think about all of these changes from the perspective of a long-term cultural evolution.
On the technical level, it is an agile and decentralized project. An open-source editor is already available which includes the IEML dictionary with English and French translations, a parser supporting some computable functions on language expressions, and a system displaying paradigms visually in the form of tables. A platform for collaborative editing and sharing of concepts and ontologies remains to be built. Intellectually and socially, however, this is an ambitious project, because the development, maintenance and use of a possible semantic protocol based on IEML would require a coordinated long-term research and training effort.
This article is divided into three parts. I will present successively (a) the main issues of the problem of mathematizing language; (b) the solution I found to this problem, IEML; (c) the possible applications of IEML, notably for collective intelligence, artificial intelligence, and digital humanities.
The problem: Mathematizing linguistic semantics
How do we create a language with computable semantics? First we need to know exactly what we are talking about. Therefore, we will start by listing the major functions of languages to make sure that our mathematical language is able to fulfill them. Once this is done, we will look at the structure of languages and identify the three axes—syntagmatic, paradigmatic, and semiotic—that organize linguistic semantics. Then, I will focus on distinguishing linguistic semantics from referential and pragmatic semantics. Referential semantics has already been formalized by mathematical logic, and pragmatic semantics is the domain of game theory and systems theory, which are well advanced in their mathematizing. Only the domain of linguistic semantics—namely, that of conceptualization—is not yet formally modeled. We can then recognize that, in this well-defined domain, the syntagmatic axis has already been mathematized in the form of regular languages. What remains to be mathematized is the paradigmatic axis by means of a code (IEML) capable of matching the relations between sounds (the sequences of phonemes) to the relations between meanings (the networks of concepts).
Functions of languages
Let us begin by distinguishing philological languages from other types of languages. The concept of “philological language” was introduced by Louis (Hjelmslev, 2000) to designate languages capable of “saying everything,” whether they are constructed or natural. For example, Esperanto or IEML are constructed languages. Nevertheless, because they have the same expressive properties as a natural language, they are philological languages. On the other hand, the programming language Python, which is also a constructed language, is not a philological language because it is not able to say everything. It is a specialized language.
The narrative function
The first character of a philological language is its narrative capacity. In order to play its practical and referential roles, the language must be able to evoke complex scenes in which several actants interact. The following quotation, taken from Lucien Tesnières’ posthumous work Eléments de Syntaxe structurale (Tesnière, 1959, chapter 48, p. 102) explains this property of philological languages well: «The verbal node (…) expresses a whole little drama. Like a drama, in effect, it comprises (…) a process and, most often, actors and circumstances. The verb expresses the process. (…) The actants are beings or things (…) participating in the process. (…) Circumstances express the circumstances of time, place, manner, etc.». In a given place, during a certain time, several human or non-human participants, each with their own intentions and in their own way, play distinct roles in unfolding a process in the mind of a listener or reader. The sentence structure is a grammatical reification of this scripting: the verb determines the state or process, the subject initiates the action, the object interacts with the subject, etc. With narrativity comes the ability to describe real scenes or interactions but also the possibility of fiction.
Because sentences are able to fit together like Russian dolls, they can evoke complex situations. Actants, circumstances or modes of action can themselves be described by sentences and this in a recursive way. Marcel Proust pushed this property of languages to its limit in In Search of Lost Time. Similarly, quotations and narratives can be embedded indefinitely, as in One Thousand and One Nights or in Jan Potocki's The Manuscript Found in Saragossa.
The dialogic function
Language must not only allow the evocation of social interactions, it must also provide the interlocutors with the means to situate each other in these interactions. Therefore, all languages have grammatical persons, that is, ways of saying I and we, you, he, and they. A living language is learned and practiced in dialogue. With dialogue comes questioning, answering, persuading, reasoning, demonstrating, and the various roles that people can play in a multitude of conventional social games (Bakhtine, 1978, 1981).
Dialogue can cross languages since philological languages can translate other philological languages as well as a multitude of other more specialized codes and languages. Certainly, a language is closely adapted to the life of its interlocutors, it reflects their universes of discourse and their social games. Still, it is always capable of interpreting other languages, with an inevitable margin of betrayal, and of evolving through borrowing, abandoning, and creating.
The referential function
Linguistic expressions must be able to point to references, and in particular to extralinguistic realities, at all levels of complexity: word, phrase, complex sentence, text. Such is the role of proper names. This is also the role of demonstratives and deictics (Benveniste, 1966 and 1974, see in particular chapters 20 and 21 of volume 1): this, that, I, you, now, tomorrow, here, there..
More generally, language is a means of categorizing and indexing the universe of its interlocutors. All languages function as a system of semantic addressing of the memory—individual and collective—and thus as an instrument for the coordination of action.
Language is not only a tool for referencing the extralinguistic world, it is also capable of self-reference. This is notably the role of grammatical anaphora, for example, when a pronoun refers to a noun quoted earlier. This self-referential capacity of the language also comes into play in commentary and dialogue. It extends to self-explanation: in a dictionary, words are explained in a circular way, so much so that a philological language is always its own metalanguage.
Structure of languages
Components: Phonemes and words
We know that languages evolve, differentiate, and hybridize, but for the purposes of analysis, we will consider the structure of a language at a given moment. A language is built from some sound units (between 30 and a 100): phonemes, which have no meaning and are generally divided into consonants and vowels. 4 Units of meaning—called morphemes—are made up of sequences of phonemes. What most people call words are actually clusters of several meaning units. For example, the word «tartlets» contains: the concept of «tart», the diminutive «let» and the plural mark «s». It is considered that speakers of a language master a vocabulary of between three thousand (minimum proficiency) and 30 thousand (maximum proficiency) lemmas, the latter being canonical words, independent of conjugations, inflections, cases, etc. Since my goal here is only to give a broad outline of the structure of languages, and although the reality is more complex, I will call “words” the elementary units of meaning and “sentences” their grammatical composition in compound units of meaning.
The syntagmatic axis
According to Roman Jakobson (Jakobson, 1963) and the structuralist school of linguistics, a language crosses two axes: syntagmatic and paradigmatic. Let’s start by defining a syntagm: it is a sequence of words—a sentence or a phrase—belonging to a language whose grammatical rules allow us to reconstruct the dependency tree and the role of each leaf in this tree. Let's consider the sentence: “I’m repainting the little room in blue.” Figure 1 The syntagmatic axis: a string of sounds is organized in a dependency tree.
The use of conjunctions, punctuation marks—written or spoken—as well as some grammatical rules code the hierarchies of dependencies between sentences and parts of sentences, between main and subordinate clauses. Their place in the phonetic sequence or the use of cases allow us to recognize the grammatical role of words: verb or process, subject or first actant, object or second actant, addressee or third actant, locative complement, etc.
I can only understand a sentence if, from a sequence of phonemes, on the one hand, and from the knowledge of the dictionary and of the grammatical code specific to my language, on the other hand, I reconstruct a syntactic-semantic nesting scheme whose topological structure is that of a tree. From this tree structure, in which the grammatical role of each word is determined, I infer the scene or model that my interlocutor had in mind. Conversely, I pronounce a meaningful sentence when, from a mental model, I first build a syntactic tree of concepts represented by words (a “deep structure” according to Noam Chomsky) which I then transform into a sequence of sounds following the grammatical rules of my language. Of course—except in the case of deliberate literary creation—these different logical steps are taken quasi-simultaneously and unconsciously.
The paradigmatic axis
The paradigmatic axis is “perpendicular” to the syntagmatic axis. In general, paradigms are groups of substitutions on the nodes of a syntagmatic tree. Figure 2 The paradigmatic axis: each word in a sentence is selected from a matrix of alternative possibilities relevant to the context.
Let's consider again the sentence “I am repainting the little room in blue.” By keeping the present tense and the grammatical object (second actant), I can vary the person and the number of the subject (or first actant) and substitute for “I,” the series: “you, he or she, we, you, they” while harmonizing the verb. I can also vary the tenses by keeping the mode: “I repainted” (preterite), "was repainting” (imperfect), “will repaint” (future). I can also, still for the same sentence, keep the mode, the tense, the grammatical person and the object but replace “to repaint” with “to wallpaper,” “to furnish,” “to decorate,” etc. In the latter case, I have varied the lemma or, if one prefers, the concept of the action. However, I can’t do it just any old way. How could I “debunk” or “tickle” a room, for example (except in a highly metaphorical sense)?
This means that there is a paradigm of actions—and therefore verbs—that are suitable with the word “room” in the grammatical object role (or second actant). Similarly, I can paint a wall, a fence, or a pole, but a cloud, a political constitution or a nightmare would be much more difficult to paint. Therefore, there is a paradigm of items in grammatical object role that are appropriate with the verb “to paint.”
A set of words that can be substituted for each other in the same syntagmatic role within a given sentence constitutes a paradigm. The words of the same paradigm have in common their semantic suitability with the constant parts of the sentence, but they differ according to a system of symmetries or oppositions. A paradigm is thus a system of variations on a background of semantic constants.
Many words in a language are associated with other words that designate complementary syntactic/semantic roles in a standard way. For example, the grammatical subject of the verb to steal (the crime) is a thief and the object of the theft is a larceny. When we speak of a vehicle, the person who operates it is a driver or chauffeur. These systems of suitability between words in a language have been studied particularly under the names of lexical functions (Melchuk, 1982, 1998; Polguère et al., 2003; Leo, 1982), collocations (Tremblay, 2014), and frames (Fillmore, 1968, 1982). Dependency relations between word uses—relations that reflect semantic and practical connections—are rightfully part of the conventions of languages and should be systematically included in dictionaries worthy of the name.
We can also classify under the broad umbrella of paradigms sets of words that keep their fundamental concept—the lemma is identical—while changing their grammatical role. For example, “metaphor” (a noun that can be used in the role of an actant), “metaphorize” (a verb, that is, used in the role of an action), “metaphorical” (an adjective, that is, used to qualify a noun), and “metaphorically” (an adverb, that is, used to qualify a verb).
In sum, paradigms correspond to these three cases: - They are substitution groups for inflections or grammatical roles for a constant lemma; - they are substitution groups for words in a sentence and for a constant grammatical role; - finally, they are more or less complex systems of co-occurring words (or phrases) as in lexical functions, collocations, or frames.
Let us remember that each node of a syntagmatic tree belongs to a paradigm, that is, to a matrix of alternative possibilities relevant in a certain context. A paradigmatic matrix describes a domain of semantic variation structured by a system of symmetries or oppositions. Additionally, paradigmatic variables are generally co-dependent: the choice of a word in a syntagmatic role conditions the choice of words in the other roles. In all cases, a paradigm appears as a field of semantic variations that are appropriate to a given context.
The semiotic axis
Some linguists add to the two syntagmatic and paradigmatic axes a third axis, semiotic, which concerns the relationship between the signifier and the signified or, if one prefers, the relationship between the sound (the signifier) and the meaning (the signified) of linguistic expressions Figure 3 The semiotic axis: a signifying chain corresponds to a network of concepts, that is, a syntagmatic tree crossed by paradigmatic matrices.
Meaning involves at least three actants who play distinct roles. A sign (a) indicates something: the referent (b), for someone: the interpretant (c). The linguistic symbol is a particular type of sign, divided into two parts: the signifier and the signified. We say that a linguistic symbol is conventional because, in most cases, there is no analogy or causal link between the sound and the concept: for example, between the sound “tree” and a plant genus. The meaning (the signified) of the sound “tree” is governed by the English language and is not left to the choice of the interpreter. However, it is in the context of a particular enunciation that the interlocutor understands the referent of the word: is it a syntactic tree, a palm tree, a Christmas tree…? Let us remember this important distinction: the signified is determined by the language but the referent identified by the interpreter depends on the context. We sometimes make the mistake of attributing the connection between the signifier and the signified to the interpreter because natural languages are ambiguous, and the interpreter often uses his or her understanding of the context to determine the signified of an equivocal signifier… But the fundamental intermediary between the signifier and the signified is none other than the linguistic system itself.
It has been noticed for centuries that, except for a few rare cases of words whose sound evokes the meaning (the name “cuckoo” reminds us of the song of the bird), the phonetics of a word has no direct relationship with its semantics. The relationship between the sound and the meaning of words is purely conventional and depends on the language. However, the independence of sound and meaning in speech should not be exaggerated. The more poetic the language, the more prosodic aspects such as rhythms and assonance—especially rhymes—contribute to the creation of meaning. Moreover, deciphering the syntagmatic dimension—that is, reconstructing a dependency tree including identifying the grammatical role of leaves—relies heavily on word order, the use of ear-sensitive cases, punctuation, and tonal accent. It is not a question here of a direct term-to-term correspondence between signifying units and signified units, but rather of an analogy or resonance between sound structure and grammatical structure. Finally, phonetics plays a role in the recognition of the paradigmatic dimension, for example, when it allows us to identify a constant part and a variable part in a word. Let’s note, for example, the alternation «muchacho/muchacha» where the semantic constant and the gender variable is easily recognizable. Similarly for the alternation «tart/tartlet»: the semantic constant as well as the size variation are audible. But to designate a mini-mountain, we do not say a «mountainlet» but a hill. Mountain and hill are nevertheless part of the same paradigm of topological accidents and geographical descriptions; they have roughly the same shape (foot, slope, summit) while distinct from each other on a small/large scale of variation. None of the above, however, is sensed at the phonetic level. In this vein of paradigms that are impossible to recognize by sound, we can evoke colors: green, red, yellow, and blue are opposed to one another yet in the same color paradigm, and their belonging to the same paradigm is not sensed phonetically. If phonetics often helps to encode and decode the syntagmatic axis in natural languages, it only rarely plays this role for the paradigmatic axis. Let us keep in mind this last point, which will prove to be crucial for the solution of the semantic computation problem because, in IEML—and contrary to the situation that prevails in natural languages—all the words of the same paradigm have a phonetic invariant in common (see the color paradigm in IEML in Table 4).
Speech semantics
Let's now delve deeper into the question of semantics in general, going beyond what was covered in the introduction.
Pragmatic semantics or interaction
Language functions primarily as a tool for exercising, regulating, and representing social interactions. In fact, even personal thought—which does not go beyond the limits of our innermost being—takes the form of a conversation. According to Lev Vygotsky (Vygotsky, 1934), thought is the result of an internalization of dialogue. When we ask ourselves about the meaning of a sentence, we must therefore first note its pragmatic and dialogical dimensions: the type of language game (Wittgenstein, 1973) played by the interlocutors, the circumstances of its enunciation, its potential effects. In sum, the enunciation of a sentence is an act (Austin, 1962; Searle, 1969). And the pragmatic meaning of the sentence is the effect of this act as it is registered in the collective memory of the interlocutors. Pragmatic semantics is rooted in the relationship between a speech act—more generally a symbolic act—and a social situation. It is mainly related to game theory, law, sociology, and even systems theory.
Referential semantics or veridicality
At the level of pragmatic semantics, a sentence is more or less relevant or effective, felicitous or infelicitous, depending upon its effects in context. But for an utterance to be capable of such a practical meaning, it also must most often be capable of describing a reality, whether it is exact or inaccurate, serious, or fictitious. Like states of consciousness, propositions are intentional (Husserl, 1950; Searle, 1983), that is, they point to a reference. Pragmatic semantics records the way in which an utterance modifies a social situation, whereas referential semantics focuses on the relation between an utterance and a state of things. A proposition is true or false only at the level of referential semantics. It is also at this level that its truth value is transmitted—or not—from one proposition to another in logical reasoning. Let us note that if language were not inclined to carry referential semantics—that is, of representing reality and telling the truth—it could not play its pragmatic role. It would be impossible, for example, to state the rules of a game or to evoke social interactions. Referential semantics, which are more akin to the exact sciences and logic, stands out against the backdrop of objective reality. Here, linguistic expressions describe and index the world of the interlocutors and allow logical reasoning.
Linguistic semantics or conceptualization
Just as pragmatic semantics has referential semantics as its condition of possibility, referential semantics in turn can only arise on the basis of linguistic semantics. Indeed, a sentence must first have meaning in a certain language and evoke a certain mental representation—a concrete or metaphorical scene—in order to compare this conceptualization with reality and declare it true, false, or half true. The linguistic meaning of an expression comes from the words it is composed of and the meaning assigned to them in the dictionary. It also comes from the grammatical roles that these words play in the sentence. In sum, linguistic meaning emerges from the relations of inter-definition, suitability, similarity, and difference between words in the dictionary and the grammatical relations between words in the sentence. The addition or subtraction of a circumstantial complement, the replacement of one verb by another, of singular instead of plural, changes the meaning of the sentence and produces a different narrative. Linguistic meaning is based on systems of difference. At this level of semantics, the meaning of a sentence is determined by its relationship to the language, and even by its relationship to its translation into other languages. Anterior to the question of truth, it has more affinities with literature than with logic.
Phenomenological semantics or sensorial-emotional grounding
When someone speaks to us, provided we are proficient in the language, it resonates—interacting with the three preceding levels—with a complex array of memories, subliminal motor impulses, sensory images, and emotions, which obviously vary from person to person. This level of semantics corresponds to the notion of bodily and subjective grounding of meaning and does not appear to be formalizable.
Calculating linguistic semantics
We have just discussed the layered structure of meaning. In a single move, an utterance manifests the four semantics: pragmatic, referential, linguistic, and actual personal grounding. It is only for the analysis that they must be distinguished. But referential semantics and pragmatic semantics are already formalized and phenomenological semantics does not appear to be formalizable. During this analysis, we carefully distinguished linguistic semantics, whose formalization remains incomplete. As we will see, the problem of semantic computability has been solved by the invention of the IEML linguistic code, which makes it possible. The semantic calculus must in principle take an algebraic form. What does this mean?
Algebraic calculation
Algebra can be defined as the art of mechanizing symbolic operations. It implies the adoption of a coding system for variables and operations as well as the use of regulated sequences of operations—algorithms—to perform calculations. The application of an algorithm to a set of input variables leads to result variables as output. Since symbols are made up of a signifier part and a signified part, it must be emphasized that algebraic operations are applied to signifiers in a mechanical way, that is, without taking into account the signified. The algorithms are blind to the semantic contents of the symbols they manipulate. Even when we perform multiplication by hand, we always follow the same routine, whatever the numbers multiplied. Signifiers manipulated by algebraic operations can be assimilated to material pieces like tokens, marbles, or pebbles. The word calculus itself comes from the Latin calculus, meaning pebble because the ancient Romans used small pebbles to perform arithmetic operations on abacuses. The progress of algebraic coding and the improvement of automatic calculation procedures generally mark a leap in the scientificity of the domain to which they apply.
Algebraic language modeling
What about the algebraic modeling of language? As far as its pragmatic dimension is concerned, the mathematical theory of games and the systems theory, which took off after the Second World War, are already well advanced. The algebraic coding of logic is almost complete, and obviously supports the progress of computer science.
The main issue of logic is the truth of propositions—that is, their correspondence with states of affairs—as well as the validity of reasoning from true propositions. In contrast, the theory of regular languages, as developed by pioneers Stephen Cole Kleene (Kleene, 1956), Noam Chomsky and Marcel-Paul Schützenberger (Chomskyet al., 1963), concerns only the internal structure of languages, that is, their syntax. A regular language consists of a finite set of symbols (its alphabet) and word formation rules, which are sequences of symbols. It should be noted that regular languages can be composed in hierarchies of internal complexity and that their “words” often correspond to what one would want to call sentences or texts. With a certain idealization, the syntagmatic structures of natural languages can be described by means of regular languages. The essential point of the algebraic handling of syntax is that well-formed expressions of a regular language can be produced or recognized by machines (finite automata). The theory of regular languages developed at the same time as computer science and accompanied the design of programming languages and compilers.
However, game theory, logical calculus and formal syntax are not sufficient for an algebraic modeling of semantics. The first concerns pragmatic semantics, the second referential semantics and the third formalizes only the syntagmatic axis of linguistic semantics. It remains to encode the paradigmatic axis.
Philosophers have long noticed that natural languages—because they are irregular and ambiguous—are ill-suited to mechanical calculations about concepts. They tried (from Ramon Lulle to G.W. Leibniz, 1989 5 ) pairings between concepts and numbers in order to subject the concepts to arithmetic-type calculations. But if all numbers are concepts, all concepts are not numbers. The codes and operations designed to facilitate arithmetic calculations have proven inadequate for conceptual calculations. Purely combinatorial algebra has also proved inadequate. It is impossible to reconstruct all conceivable concepts and their relations from a small number of elementary concepts using only combinatorial methods.
To carry out the program of semantic computation, we must undoubtedly use an algebra, in the most abstract sense of the word, but an algebra specially designed to encode the linguistic matter of concepts. Since languages have a syntagmatic dimension, we must necessarily use a regular language. But it must be a regular language able to encode paradigms: the substitution groups on the nodes of the syntagmatic trees, the internal symmetries of these substitution groups, the play of constants and variables in the construction of concepts and their relations. IEML was invented to meet this challenge.
The coding of the signified by the signifier in IEML
Remember that computers are robots that execute algorithms. They only recognize sequences of 0’s and 1’s in memory boxes, perform arithmetic and logical operations on these sequences, and arrange the results in other memory boxes. From beginning to end, they manipulate only signifiers and this is why they are well adapted to the processing of regular languages. The regular languages actually used today are programming languages. But the “semantics” of programming languages is none other than the execution of the operations they command. None of them approaches the expressive capacity of a philological language, which allows us to talk about anything and everything and to perform many other illocutionary acts beyond giving instructions to a machine. To overcome this limitation, I decided to build a language which is simultaneously (1) philological, (2) regular, and (3) whose paradigmatic structures would be automatically decodable; in short, a language with computable semantics.
Both the signifiers and the signified of a language are organized by systems of differences, symmetries, and oppositions. Since languages are conventional, there is nothing to prevent the construction of a philological language whose syntagmatic and paradigmatic structures are coded by a regular arrangement of signifiers. The difference system of signifiers of a language with computable semantics must reflect the structure of its difference system of signified. By following this method, one does not obtain a phonetics of words (or sentences) that resembles their meaning—as in Plato’s Cratylus or other linguistic utopias—but rather relations between sounds that bear an analogy to relations between meanings.
A solution: IEML
IEML allows for semantic computation because the semantic symmetries between its signified are encoded by syntactic symmetries between its signifiers. The varying roles of morphological (to generate words) and syntagmatic (to generate sentences) functions create regular paradigmatic matrices. The letters of the IEML alphabet correspond to positions in symmetry systems, just like the cells of the paradigmatic matrices of its words and sentences. The language as a whole corresponds to a regular algebraic structure (a non-commutative ring) which allows the automation of a large number of functions.
My description of IEML will be divided into four parts. First, I will present the language’s alphabet and the uniform morphological function that uses this alphabet to build the dictionary’s words. In the second part, I will explain the syntagmatic function that takes the words of the dictionary as input to build sentences. In the third part, I will explain the system of syntactic relations—represented as a graph in the machine—which follows from the morphological and syntagmatic functions. In addition to these last two, link-building functions are used to weave networks of semantic relations between concepts (i.e., between words or between sentences), which corresponds to the text level in linguistics or to ontologies and knowledge graphs in computer science. Finally, the fourth part will address the handling of reference (and language self-reference) in IEML.
Words
The dictionary
In order to make the language optimally computable and easy to understand, the IEML dictionary does not exceed 3000 words. These words have been carefully chosen to facilitate, as much as possible, the recursive creation of new concepts by means of sentences. I have taken into account that the exact sciences have already standardized their special codes and languages and that these codes (e.g., chemical elements) can be noted in reference. Therefore, while the basic vocabulary of IEML is suitable for describing any meaning, it was specifically designed to be used by the fields of literature, humanities and social sciences. A wide variety of metaphysical, epistemological, anthropological, and existential perspectives have been considered. In particular, IEML words can express both “Oriental” and “Occidental” philosophies.
In order to keep the language as unambiguous as possible, no two words in the dictionary have the same meaning (no synonyms) or form (no homonyms). IEML words are currently translated into French and English, but they can be translated into any language. Note that the words stand for concepts that can be presented as verbs, nouns, adjectives, or adverbs, depending on their inflection and their role in the sentences. The IEML dictionary has two types of words: content words or lemmas, which express concepts or general categories, and function words.
The content words, grouped into paradigms, cover a large number of domains such as climates and landscapes, continents and regions of the globe, seas and oceans, sky and meteorology, animals and plants, technical functions, anthropological functions, types of relationships, units of time, values and ideas, signs and semiotic functions, data curation and critical thinking, feelings, body parts, etc. I particularly recommend the exploration of paradigms related to the human sciences and their objects. 6
The function words, also grouped in paradigms, are used: - To inflect lemmas, such as genders and numbers for nouns or modes and tenses for verbs; there are 80 distinct inflections in IEML; - to specify the grammatical roles of sentences by means of prepositions or cases, such as “above,” “below,” “on the left,” and “on the right” for locative complements; there are one hundred and 31 prepositions in IEML, each one being attached to a particular grammatical role; - to establish junctions between concepts, sentence members or phrases such as “and,” “or,” “but,” “because,” “therefore,” etc.; there are 29 junctions in IEML.
Its wide range of function words allows IEML to express fine nuances and to reduce the number of lemmas needed to define complex concepts.
The alphabet
IEML, being a regular language, is based on a set of six primitive symbols, or alphabet: (E U A S B T). All IEML expressions are composed of punctuated sequences of these six characters. Each of the letters is a word in itself and represents a particular meaning when used alone. But when a letter is used in the composition of another word, it serves to mark a paradigmatic invariant or a position in a symmetry group.
The Six Primitive Symbols (Capital Letters).
The morphological function
The words are generated recursively by a three-role, non-commutative multiplicative operation that concatenates the words of the lower layer. Only words of the same layer can be multiplied.
At layer one we have words of the form: "U:S:E:. U:B:E:. U:T:E:. A:S:E:. A:B:E:. S:T:E:." etc. Words are generated by operations of the form, “U:⊗S:⊗E: = U:S:E:.”
At layer two we have words of the form: “U:S:E:.U:B:E:.U:T:E:.- A:S:E:.A:B:E:.S:T:E:.-,” etc. Words are generated by operations of the form: “U:S:E:. ⊗ U:B:E:. ⊗ U:T:E:. = U:S:E:.U:B:E:.U:T:E:.- ”
We stop at the sixth layer, which is more than enough to generate a dictionary of 3000 words. Punctuation marks (: . - ' , _ ;) are used to identify the successive layers, from layer 0 (:) to layer 6 (;). In homage to Spinoza, the three roles of the morphological operation are called substance (the first place), attribute (the second place), and mode (the third place).
There is an additive operation, which is a simple union of sequences of the same layer. The additive operation is commutative, but the order of the letters is normalized to ensure that no two formally different expressions have the same meaning.
Multiplication is distributive over addition. For example, “U: ⊗ (S: ⊕ B: ⊕T:) ⊗ E: = U:S:E:. ⊕ U:B:E:. ⊕ U:T:E:.”
On a mathematical level, the algebra resulting from the 6-symbol alphabet, the non-commutative ternary multiplication and the addition has a non-commutative ring structure. 8
Lower case letters
The 25 Lowercase Letters.
The tetrad wo. wa. wu. we. is generated by the function “(U: ⊕ A:) ⊗ (U: ⊕ A:).” To understand its meaning, we must relate to the fundamental sensory-motor cycle. Let us begin with the virtual act of orientation, which takes place in the mind (wo.). From this virtual place we act (wa.). As a result, our current environment is transformed and a new reality manifests itself (we.). Finally, from the actuality of the external event, we perceive (wu.) by returning to interiority. Note the two intertwined oppositions:
Wa. Act (inside → outside)/wu. Perceive (outside → inside).
Wo. Orient (inside → inside)/we. Appear (outside → outside)
The hexad y. o. e. u. a. i. is generated by the function: “(U: ⊕ A:) ⊗ (S: ⊕ B: ⊕T:).” The corresponding words represent six kinds of actions. The row generated by “U: ⊗ (S: ⊕ B: ⊕T:)” displays three virtual (or latent) actions: to know (y. Related to sign), to want (o. Related to being), and to be able (e. Related to thing). The row generated by “A: ⊗ (S: ⊕ B: ⊕T:)” expresses three actual (or manifest) actions: communicating or saying (u.), using signs to express oneself, committing (a.) to someone and/or something and doing (i.) something in the material world.
The hexad j. g. h. c. p. x. is generated by the function « (S: ⊕ B: ⊕T:) ⊗ (U: ⊕ A:). » The corresponding words represent six kinds of actors. The row generated by “S ⊗ (U: ⊕ A:)”shows two kinds of actors related to the sign: ideas (j.) in the virtual world and messages (g.) in the actual world. The row generated by “B: ⊗ (U: ⊕ A:)” displays two types of being-related actors: subjects (h.), or abstract forms of subjectivity in the virtual world, and as individuals (c.) who exist in the actual world. The row generated by T: ⊗ (U: ⊕ A:) displays two types of thing-related actors: objects (p.), in the virtual world, and physical bodies (x.) in the actual world.
The Ennead s. b. t. k. m. n. d. f. l. Is generated by the function « (S: ⊕ B: ⊕T:) ⊗ (S: ⊕ B: ⊕T:) ». The corresponding words represent nine dimensions of human experience.
Reflection “s. = S:⊗S:” Signs and symbols fuel the mind’s power to think, understand, form judgments, and generate other symbols. Deduction, induction, interpretation, and imagination are achieved by combining signs. The idealist traditions, the phenomenology that studies the structures of consciousness, as well as much of the meditative philosophies of India, are based mainly on the act of reflection.
Language « b. = S:⊗B: ». Signs lead to communication. A message is sent to a subject that gives it meaning. The philosophy of language and communication focuses on speech, its powers, its limits and the media in which it is embedded.
Memory « t. = S:⊗T: ». By the gesture of inscription, a sign is reified into a thing. Without any means of recording, there is no memory and therefore no notion of time, history, or evolution. In the same way, the great religious traditions based on sacred texts are partially related to this conceptual archetype.
Society « k. = S:⊗B: ». A community self-organizes through symbolic systems. Members of a clan may have a totemic animal. We fight under the same flag. We exchange economic goods by agreeing on their value. We obey the law. As good citizens, we commit to keeping our promises and contracts. We are bound by conventions. Sociology has highlighted the symbolic functions that build society.
Emotion « m. = B:⊗B: ». Beings orient themselves towards other beings. Desire, love, hate, fear, indifference, or compassion are emotions that flow between us. Pious people, poets, artists and comedians know this central role of emotions in human life. Psychoanalysis, much of clinical psychology and common sense emphasize the importance of emotional connections.
World « n. = B:⊗T: ». Humans express themselves in their physical environment. They inhabit it, they work in it with tools, they name things and give them value, creating an ordered world, a cosmos. Philosophy, anthropology, law, and other disciplines of the social sciences and humanities resonate with this concept, which simultaneously implies the reality of the non-human universe (the thing) and the cultural shaping of its order (the being).
Truth « d. = S:⊗T: ». A descriptive message contains a representation of reality. A proposition is true if it is a correct description of a state of things. The traditions of logical and analytic philosophy are primarily interested in the notion of truth and the problems of reference. Similarly, epistemology and cognitive science place the construction of true belief, or knowledge, at the heart of their approach.
Life « f. = B:⊗T: ». Matter comes to life. Living creatures are embodied. Eating and drinking are among the most obvious ways in which a thing is transformed into being. Many philosophical schools emphasize the lived experience of the subject. Empirical, biological, and medical traditions find their concepts in the living body and its functions.
Space « l. = T:⊗T: ». A point connects to other points. Geometric and physical spaces are built on relations and proximities. Geometers, topologists, atomists, materialists and physicists base their concepts on space.
As in the case of the other letters, s. b. t. k. m. n. d. f. l. can be used in composed words to represent positions in an ennead, that is to say, any dialectic of nine concepts.
Let us remember that, like primitive symbols, lowercase letters have a particular meaning when used alone, but denote a paradigmatic invariant or a place in a symmetry group when used to compose a word. By combining multiplication and addition, it is possible to create as many morphological functions (and symmetry systems) as desired.
We could have chosen other primitives and arrived at other concepts as results of the first multiplications. But my goal here is less a matter of demonstrating indisputable truths than of arranging a system of primitive notions pointing to all directions in the semantic space, and capable of generating a large number of notions derived from the first layers. Not only are languages conventional, but they cannot not be conventional.
I would like to reiterate that the primitives of IEML (both uppercase and lowercase letters) are not intended to construct all possible meanings through a combination of their specific significations (which is impossible), but rather to provide the basic semantic symmetries from which grammar rules can represent any complex semantic symmetry in the paradigms one would like to construct. These primitives also do not claim to be ‘natural.’ If there are truly universal semantic primitives, they were discovered by Anna Wierzbicka 9 (Wierzbicka, 1996) following a study of numerous natural languages. It’s worth noting that all one hundred elementary concepts from Wierzbicka’s Natural Semantic Metalanguage are included in IEML’s basic dictionary.
Word paradigms
Each word in the IEML dictionary is part of one and only one paradigm. A paradigm is generated by a morphological function in which some functional roles are occupied by a constant while other roles are occupied by variables, with an addition acting as a substitution group. There are no more than three variable roles in a morphological function, which is thus presented as a one-, two- or three-dimensional matrix. A paradigm is visualized by one or more 2D tables. Words in the same table have a constant part (they have the same letters in the same functional roles) and are distinguished by their variable parts. Words belonging to the same row or column also have a constant part and a variable part. These sets of constant and variable signifiers in functional roles encode the symmetry relationships between the signified of the words. Below are two examples of word paradigms.
Some of the Components of Human Development.
The six columns correspond to invariants in attribute: types of knowledge (invariant y.), psychic orientations (invariant o.), skills (invariant e.), signs (invariant u.), social roles (invariant a.) and technical functions (invariant i.). The nine rows correspond to invariants in substance and decline the meaning of the letters s. b. t. k. m. n. d. f. l. according to their attributes.
Colors.
The invariant letters e. in the second place and i. in the fourth place are used to identify the color paradigm. The variables U: and A: represent the white and the black while the variables S: B: and T: represent the three primary colors blue, yellow, and red. The principal nuances are in substance and the secondary nuances in attribute.
Sentences
Algebraic structure
Note that the ring of words, defined by an alphabet and two operations, includes in principle an immense number of variables but only three thousand of them have a lexical meaning. The number of words is kept “small” to facilitate both automatic semantic computation and cognitive management of the language for humans. Sentences also have a non-commutative ring structure, but more complex than that of words. Their alphabet is none other than the dictionary of 3000 words. Sentences are generated recursively by a nine-role, non-commutative multiplicative operation and additive operations that correspond to commutative (and, or) or non-commutative (but, because…) junctions.
The syntagmatic function
A sentence is enclosed in parentheses and the closing parenthesis is followed by a period ( ).
The syntagmatic function—which generates sentences in IEML—has nine roles, numbered from 0 to 8. 0. The root contains the verb (or the noun in case of a noun phrase) to which the other roles are subordinated. 1. The initiator corresponds to the subject in traditional grammar or the first actant in modern grammar. 2. The interactant corresponds to the direct or indirect object complement in traditional grammar or the second actant in modern grammar. 3. The recipient corresponds to the “dative” or third actant of modern grammar. 4. The cause—in a broad sense—is the fourth actant, which corresponds to a role of instrument, causality, or ontological structure (type of, part of, etc.). 5. The time specifies the temporality of the sentence (duration, moment, etc.). 6. The place specifies the spatial dimension of the sentence (locative complement). 7. The intention specifies the purpose, motivation, or social context of the action. 8. The manner specifies the qualitative, quantitative, possessive, and other dimensions of the root.
Some roles, except the root, can be empty and each role of a sentence, except the last, ends with a comma. Roles 4, 5, 6, 7, and 8 can be specified by means of prepositions belonging to the function word paradigms respectively assigned to these roles.
Sentences have four types of components: concepts, inflections, prepositions, and junctions. - Concepts—represented by full words in the dictionary or by subordinate phrases—are preceded by a hash #. - Inflections, which precede the full words they qualify and are preceded by a tilde ∼ specify the grammatical class of the concept (verb, noun, adjective, and adverb). They also specify gender, number, and other determinants for nouns as well as mode, tense, aspect, and other determinants for verbs. - Prepositions are cases that specify grammatical roles. They are always preceded by a star *. The preposition is analogous to a parameter of the grammatical role. - Junctions can be commutative (“and,” “or inclusive,” “or exclusive”) or non-commutative (like “but,” “therefore,” etc.). They are preceded by an ampersand & and the phrases, roles, or concepts that are joined are enclosed in square brackets [ ].
In the following examples, IEML words and phrases are replaced by their English identifiers to facilitate reading. However, the reader should keep in mind that IEML sentences are built recursively from the words of the dictionary and that they are displayed in the machine as punctuated sequences of the six symbols of the IEML alphabet.
«In the house, the mother tells a story from a book, to her child, with love, among joyful laughter, before she falls asleep ». The sentence in quotation marks above is a literary translation of the IEML sentence that follows, not the other way around ( 0 root ∼indicative ∼present #storytelling, 1 initiator ∼defined ∼feminine #mother, 2 interactant ∼undefined #storytelling, 3 recipient ∼defined ∼feminine #child, 4 cause *instrument ∼undefined #book, 5 time *before #sleep, 6 place *in ∼defined #house, 7 intention *colored by feeling #love, 8 manner *with &and [∼plural #smile happy ∼plural #caress] )
The semantic emphasis
By default, the semantic emphasis (Langacker, 1987–1991) of a sentence is on its root. However, it is possible to put the semantic emphasis on another role of the sentence by means of an exclamation mark. For example, the sentence below, with the exclamation mark in role 2, reads: «The story that the mother tells her child from a book, with love amongst joyful laughter, before she falls asleep». If the exclamation point had been in role 3, the sentence would have read, «The child whose mother tells a story in a book, etc.» ( 0 root ∼indicative ∼present #story, 1 initiator ∼defined ∼female #mother, 2 interactant 3 recipient ∼defined ∼feminine #child, 4 causative *instrument ∼undefined #book, 5 time *before #sleep, 6 place *in ∼defined #house, 7 intention *colored by feeling #love, 8 manner *with &and [∼plural #joyful smile ∼plural #caress] )
When the semantic emphasis is on a syntagmatic role with an empty concept, it designates the prototypical concept corresponding to this role. For example, the prototypical concept of the initiator (or subject) of the verb “to commit a crime” is “criminal.” The object, victim, or place of a crime can be referred to in the same way. This notation minimizes the number of words needed to express concepts. It can be used to make explicit the “frame” (in Fillmore’s sense) of a concept and certain lexical functions. The notation is presented as in the example below. @node fr: commettre un crime en: to commit a crime ( 0 root ∼verb #to commit a crime ). @node fr: un criminel en: a criminal ( 0 root ∼verb # to commit a crime, 1 initiator ).
We proceed in the same way to define the object, the victim, or the scene of a crime by marking respectively with an exclamation mark the empty roles of the interactant, the recipient and the locative complement.
Sentence paradigms
A sentence paradigm is a sentence composed of two parts: a constant part and a variable part. The variable part can concern one, two, or three (at most) roles. Variables are presented as #concepts between braces and separated by a semicolon.
In the example below, role 1 has a substitution group of five variables. “1d:/#/1” indicates that the paradigm has only one dimension (one variable role) and that this role is the initiator (role number 1).
@paranode
fr:paradigme des fonctions psychiques
en:psychic functions paradigm
1d:/#/1
( 0 #mental health symptoms, 1 {#psychic autonomy; #subjective dimension; #emotional dimension; #cognitive dimension; #personal memory } ).
The Most General Psychological Functions in an Ontology of Mental Health.
I used the fundamental symmetries of IEML to order the universe of discourse of mental health symptoms. The top row corresponds to the emergent qualities of the psyche in its most virtual dimension (autonomy) while the bottom row corresponds to the most actual or concrete dimension, which supports the other psychic functions (memory). Between these two dimensions, the ascending and descending information is coded and decoded on a subjective (sign), emotional (being), and cognitive (thing) mode, as we see in the three intermediate rows. The three columns of the table roughly correspond to sign/being/thing, and the binary subdivision within each of the 15 rectangles corresponds to a division between the most virtual aspects (for the top cell) and the most actual (for the bottom cell). 11
To build such a table in IEML, it is sufficient to organize the paradigms by means of a tree structure in which the variables of a higher level (more general) paradigm become the constants of a lower level (more detailed) paradigm. For example, to create the three columns of the fifth row concerning memory, we write the following paradigm.
@paranode
fr:paradigme des symptômes liés à la mémoire personnelle
en:paradigm of symptoms related to personal memory
1d:/#/2
(
0 #mental health symptoms,
1 #personal memory,
2 {#consciousness; #intermediate memory; #long-term memory}
).
To specify the type of symptoms concerning long-term memory, we write the following paradigm.
@paranode
fr:paradigme des symptômes liés à la mémoire à long terme
en:paradigm of symptoms related to long-term memory
1d:/#/4
(
0 #mental health symptoms, 1 #personal memory, 2 #long-term memory, 4 {#declarative; #non declarative} ).
Paradigm for perception problems in an ontology of mental health.
This table is generated by the sentence paradigm below.
@paranode
fr:paradigme des symptômes liés à la perception
en:paradigm of symptoms related to perception
1d:/#/1
2d:/#/2
(
0 #perception problems -, 1 {#hypersensitivity; #disability; #pain; #illusions; #hallucinations}, 2 {#sense of hearing; #sense of sight; #sense of smell; #sense of taste; #sense of touch; #cenesthesia} ).
Relations
Syntagmatic relations
An IEML sentence is a tree with a root role and leaf roles. Therefore, a sentence automatically creates a node of relations between the concepts that fill its different roles. The relationships between root and leaves are defined by the syntax: #B is the subject of #A, #C is the object of #A, #D is the locative complement of #A, etc. The links of the syntagmatic graph are labeled by the root-leaf relationship. The concepts that occupy the roles of a sentence can be words or sentences. If they are sentences, they deconstruct themselves into syntagmatic roles, and so on recursively, until we arrive at just one word. These nested compositional relationships are recorded by language syntax and allow hypertextual navigation in the conceptual tree.
Paradigmatic relations
The paradigmatic relationships connect the concepts that are generated by the same function, defined by its operations, constants, and variables. The graph that joins the concepts according to their paradigmatic relationships is not hierarchical, as in the previous case, but of a matrix type. The concepts generated by the same function have in common an invariant part and a domain of variation. Therefore, they belong to the same symmetry system. In IEML, the symmetry system of signifiers is analogous to the corresponding symmetry system of the signified. While syntagmatic relations connect concepts in trees, paradigmatic relations connect them in “horizontal” matrices, perpendicular to the hierarchical order. In sum, the fundamental syntactic grid of IEML crosses syntagmatic hierarchies and paradigmatic matrices in a regular fractal graph, a complex semantic network that can be called rhizome-like (Deleuze and Guattari, 1980).
Conditional links
Based on the fundamental grid just discussed, it is possible to generate additional relationships by means of conditional functions. To generate relationships between concepts in IEML, one proceeds in two steps: (1) writing a linking sentence, which represents a virtual relationship, and (2) formulating a function that will create actual relationships between specified conceptual nodes.
The link, which can be applied to a large number of nodes, simply lists "arguments" ($A, $B, $C…) and situates them in the sentence in reference position. 12 The link can be unary, binary, ternary, or even more since the sentence has nine distinct roles. For example, the link below, which is called “is the opposite of,” is constructed to join two nodes. It is possible to construct relationships with one to nine nodes (examples of unary and ternary links can be found in the dictionary).
The function that actualizes the link mentions the domain of variation of the arguments (usually a paradigm) and the conditions for establishing the relationships in terms of the contents of the syntactic roles. The elementary conditions can be connected by means of NOT, AND, and OR (the Boolean connectors). Here is an example concerning the relationship: “$A is the opposite of $B,” with the link first and the function second.
@link
args:($A, $B)
fr:est le contraire de
en:is the contrary of
template-fr:$A est le contraire de $B
template-en:$A is the opposite of $B
(
0 #to signify,
1 # word <$A>,
8 *contrary #word <$B>
).
Gradient Paradigm, Generated by the Function « F:.o.E:+S:+B:+T:.- »Reminder: « F: = U: ⊕ A: ⊕ S: ⊕ B: ⊕ T: ».
@function
type:word
link: is the opposite of
domain:($A in “F:.o.E:+S:+B:+T:.-,” $B in “F:.o.E:+S:+B:+T:.-”)
condition:
(
$A.substance == $B.substance
and
$A.mode.substance == "S:"
and
$B.mode.substance == "T:"
). “mode.substance” reads: “substance of the mode.” The same link “is the opposite of” can be applied to other domains (to other paradigms) but with different functions, e.g. those concerning spatial relationships (“up” is the opposite of “down”), temporal relations (“before” is the opposite of “after”), and so on. Functions can also connect links belonging to different paradigms.
The conditional function above automatically generates the following five sentences: “Evil is the opposite of good”; “Smallness is the opposite of greatness”; “Impossibility is the opposite of necessity”; “Rare is the opposite of frequent”; and “Absent is the opposite of present.”
In Table 6, it can be noticed that an IEML word can be translated by several expressions in natural languages, which describe the extension of its semantic field. It should also be remembered that the concepts have a nominal translation (here in the table), but also verbal, adjectival, and adverbial.
Our example concerned the creation of relationships between words. The creation of relationships between sentences follows the same principles, namely, first, writing a linking sentence with arguments in reference position and, second, writing functions that specify (a) the domains of variation of the arguments—usually nested sentence paradigms—and (b) the conditions for establishing the relationships in terms of syntagmatic role contents.
Finally, it is possible to specify the mathematical properties of each binary relationship created by a link: is it reflexive, symmetric, transitive?
The crucial point here is that semantic relationships are created automatically from syntactic conditions. This is obviously only possible because words and sentences belong to patterns of constants and variables in algebraic functions, and these patterns reflect semantic symmetries.
IEML is designed to facilitate the automatic generation of concepts and semantic relationships between concepts. This makes the language particularly well suited to the creation and exploration of semantic networks: data models, knowledge graphs, ontologies, and other hypertexts. Since concept networks in IEML are composed from the same compact dictionary of 3000 words and the same regular grammar, they are semantically interoperable.
Reference and self-reference
In IEML, the reference is enclosed in angle brackets < >. The reference can contain an IEML concept (as we just saw), it can also contain a “literal” such as a proper name, an address, a date, a GPS position, a number, a unit of measure, a currency, a geographical object, a URL, a URL pointing to an image, etc. The literal is a signifier that has no signified in IEML. Its meaning is therefore purely referential. Finally, the content of the brackets can be of mixed type if it is an IEML sentence that has literals as references.
The first 12 natural, whole numbers are nevertheless considered as IEML concepts. They have semantic relations with ordinal numbers, symmetries, and regular geometrical figures. Large geographical areas also exist in IEML. They can be assimilated to postal codes that can be used in semantic calculations. These codes in particular allow the determination of respective positions (North, East…) of the coded zones, as well as the location and grouping of countries, cities, and other geographical objects. For example, to say “Italy” in IEML, we write: (0 ∼ defined #Europe center-south <Italy>). To say “the number 292,” we write: (0 ∼ defined #number <292>). To designate a proper noun: (0 ∼ masculine #sculptor <Rodin>). To designate a hyperlink: (0 ∼ definite #hyperlink <https://ieml.intlekt.io>).
The reader will contrast the IEML approach with that of the Semantic Web, in which URIs are rigid designators—literals—and cannot be semantically computed from their material form. Of course, the IEML approach and the Semantic Web approach are not incompatible since valid IEML expressions or USLs (Uniform Semantic Locators) have a unique form and can be represented as URIs.
We have given above some examples of self-referencing in explaining the creation of conditional relationships. Each of the three thousand words in the IEML dictionary is defined by sentences using other words, and each complex expression in IEML (word, sentence, and semantic graph) refers to the circular inter-definition core of the dictionary. Moreover, this circular inter-definition is, as we have already noted, a feature of all philological languages. Finally, independently of the dictionary, IEML can refer to its own expressions in order to define relationships, to propose definitions, to develop reasoning or to make comments.
IEML is a philological language with computable semantics
It is now time to verify that IEML fulfills its promise of encoding semantics in a computable mode while manifesting the properties of a philological language.
A mathematical language
The syntax of IEML is mathematical. It is defined by two nested functions. For creating words, a morphological function with three multiplicative roles operates on an alphabet of six primitive variables. To create sentences, a syntagmatic function with nine multiplicative roles operates on an alphabet of 3000 words generated by the morphological function. The paradigms and their symmetries are nothing but the matrices generated by these functions when one, two, or three of their roles are variable. Each word in the IEML dictionary belongs to one and only one paradigm. A sentence, however, can belong to several paradigms and the sentence paradigms are freely created by the speakers. On the regular grid that crosses syntagmatic trees and paradigmatic matrices, the functions of link creations allow the weaving of semantic relations from syntactic conditions and the creation of as many hypertexts or knowledge graphs as desired. In short, the semantics of IEML is computable.
A philological language
The structure of an IEML sentence allows for the description of any complex interaction, and the use of the exclamation mark ! allows for emphasis of a particular role in the sentence. The verbs can be affirmative, negative, interrogative, or others, and they can go through the whole range of logical modalities and grammatical moods. The sentence is obviously recursive since the concepts that compose it can be actualized by words or sentences, sentences that can in turn contain sentences, and so on. IEML has the same narrative power as natural languages. IEML’s dictionary contains all kinds of deictics and its grammar allows for the explanation of reference (extralinguistic) and self-reference (references of linguistic expressions to other expressions) operations. As far as the explicitness of dialogical roles is concerned, IEML contains the three grammatical persons. Finally, our metalanguage allows, in principle, for the translation of any concept, story, or reasoning expressed in natural language. The IEML editor’s parser already allows us to write texts in IEML by using French or English words.
The current single-user open-source editor—a simple prototype—will have to give way to a platform where ontology developers and data curators can collaborate. In this new kind of meta-ontological encyclopedia, each concept could be read in all languages, plus be located in the paradigms and knowledge graphs in which it participates. One could see the concepts that compose it, the higher-layer concepts in which it is used, and according to which grammatical roles. This conceptual network—living, evolving, expanding, and differentiated according to an open multitude of practical purposes—would serve to address the immense flow of data that rises from human activities and the memorial background where it accumulates.
Some applications
I am working from the perspective of an artificial intelligence dedicated to the augmentation of collective intelligence. I designed IEML to serve as a semantic protocol, allowing the communication of meanings and knowledge in the digital memory, while optimizing machine learning and automatic reasoning. Imagine a multitude of communities with heterogeneous purposes and practices; each of them would self-organize from a database indexed in IEML according to an original and evolving knowledge graph. The accounting and value allocation modes of these communities would be defined by rules—or intelligent contracts—in IEML, thus understandable by humans and executable by machines. Despite the diversity of their conventions and expertise, these communities could share their public data, 13 their ontological modules and their formal rules in an interoperable network of knowledge bases, feeding machine learning. Between this ecosystem of interoperable knowledge bases and living human users lies a “no-code” deep learning layer which provides immersive multimedia access to metadata and data control, helping users to feed, explore, and analyze this new kind of open encyclopedia.
In this last section I will describe some of the applications made possible by the existence of IEML, but without any claim to prediction. Like any language, IEML has a communicative function. It can serve as a pivot language between natural languages, and thus between humans. It can also play the role of a communication language between humans and machines, remedying the ambiguities and errors that plague chatbots based on statistical learning from texts in natural languages. Thanks to its formal properties, IEML would allow computers to “understand” linguistic expressions much better than they do today. Finally, our mathematical language can serve as a code for machine-to-machine communication. Of course, in all its applications, the raw IEML code would only be manipulated by machines and the content would always be presented to—or provided by—humans in the form of natural language text, diagrams, or icons. This section on IEML applications is organized in three successive parts with closely intertwined themes. First, I discuss the organization of digital memory and what an IEML-based semantic metadata system could contribute to collective intelligence. I then discuss how IEML could augment artificial intelligence in both its symbolic and neural-statistical branches, while promoting neuro-symbolic integration. I conclude by discussing the prospect of a semantic coordinate system that would unify the human sciences by opening the way to a computable, yet inexhaustibly complex universe of meaning.
The organization of digital memory
Metadata and semantic interoperability
Collective human intelligence is largely based on a stigmergic communication whose medium is increasingly identical to the digital memory. But we still have to overcome the compartmentalization due to the multiplicity of languages, classification systems, disciplinary viewpoints, and practical contexts. If the organization of memory conditions the dynamics of our collective intelligence and if its fragmentation diminishes our cognitive power, we must take a closer look at metadata systems.
Information systems and software applications organize information through metadata. Metadata is information that organizes data. In the labyrinth of memory, it acts as an Ariadne's thread. Its function is to identify the data in order to better organize and retrieve it. It serves less to describe things exhaustively—it is not a question of making maps on the same scale as the territory—than to provide reference points for users to find what they are looking for, with the help of algorithms. The notion of metadata, which is very general, embraces the whole range of labeling systems that allow data to classified.
A distinction can be made between (a) material metadata, such as the format of a file, its date of creation, its author, or its license of use, and (b) semantic metadata, which relates to the contents of a document or a set of data: what the data are about; as well as their practical dimension: what the data are used for, by whom and in what circumstances.
Here we are mainly interested in semantic metadata. A semantic metadata system can be as simple as a vocabulary. At a higher level of complexity, it can be a hierarchical classification or taxonomy. At the most complex level, it is an “ontology,” that is, the modeling of a domain of knowledge or practice, which may contain several taxonomies with intersecting relationships, including causal relationships and automatic reasoning capabilities. The notion of ontology is now giving way to that of the knowledge graph, in which digital resources are accessed by a data model organized in a network of relations and with a controlled vocabulary. Data are no longer contained in the fixed tabular schemas of relational databases, but in the new graph databases, which are more flexible, easier to evolve, better able to represent complex models, and allow several different views. Like an ontology, a knowledge graph may be used for automatic reasoning (pertaining to classical symbolic AI), but also for machine learning if it is well designed and if there is enough data.
(Public Domain, https://commons.wikimedia.org/w/index.php?curid=1353062)
Semantic metadata is therefore an essential element of artificial intelligence devices Figure 4 An example of semantic network.
Because they structure contemporary knowledge, whose medium is digital, metadata systems represent a major issue at the scientific, cultural and political levels. Standards have been developed to put some order and transparency in the digital memory. To understand the contemporary landscape of metadata standards, it is important to distinguish between formats and models.
Formats
The system of standard formats and languages proposed by the World Wide Web Consortium 14 to reach the "Semantic Web" has existed since the end of the 20th century. The four main standards are XML, RDF, OWL, and SPARQL. XML describes the hierarchical structure of a document; RDF organizes resources in the form of graphs of relations; OWL programs ontologies for automatic reasoning; SPARQL allows querying of RDF graphs.
It is important to understand that the W3C standards system concerns file and program formats dealing with semantic metadata, but not the semantics themselves, that is, the categories, concepts, properties, events, and relations that are expressed in natural languages, with all the ambiguities, multiplicities, and incompatibilities that this implies.
Models
On top of this system of standard formats, there are standard models to deal with the actual semantic content of concepts and their relations. Indeed, the same conceptual model can be used to structure different data, thereby facilitating communication between different memories. Models like schema.org 15 for websites, supported by Google, or CIDOC-CRM, 16 developed by cultural heritage conservation institutions, are good examples. Standard models have been published for many fields, from finance to medicine to logistics. Unfortunately, there are often several competing models for a domain, 17 and the models themselves are hyper complex, to the point that even the specialists of a model master only a small part of it. Moreover, these models are expressed in natural languages, and most often in English.
The most sophisticated metadata model editing and management tools on the market (such as Pool Party, 18 Ab Initio, 19 or Synaptica 20 ) have no way of clearly distinguishing between words or terms in a particular natural language, and concepts or categories, which are more abstract and trans-linguistic notions. This is problematic because the same concept can be expressed by different words in the same language, let alone in different languages. Moreover, the same word can correspond to several concepts, even in the same language. Is the cardinal an ecclesiastical dignitary, a bird, or a gladiolus? Is a mole a burrowing animal, a blemish of the skin, or an infiltrating enemy agent? The words of natural languages are ambiguous and multiple, but recognizable by humans immersed in a particular context. But machines have a hard time making contextual inferences. Hence the need to uniquely identify the underlying formal concepts so that they can be interpreted by machines.
In practice, taxonomies, ontologies, knowledge graphs, and other metadata systems, implemented in real applications to organize datasets, only partially use standard formats and models. Users submit to these layers of standards as best as they can, in the hope that their data and applications will become the happy subjects of a semantic interoperability kingdom. But their hopes are often dashed. The ideal of the intelligent and decentralized Web of the late 1990 s has given way to Search Engine Optimization (SEO) more or less aligned with Google’s (secret!) knowledge graph.
Today, a large part of the digital memory is still in so-called relational databases, without a clear distinction between data and metadata, organized according to mutually incompatible schemas, poorly optimized for the knowledge, coordination, or decision support needs of their users. Various semantic indexing and querying models have been proposed for legacy RDBMS (see, for example, Tekli, 2018) where IEML could be used to compute relations and proximities between concepts.
Because of the multitude of disparate formats, models, and applications, not to mention language differences, it remains very difficult to allow different cultures, business domains or disciplines to communicate. Unfortunately, there is no semantic interoperability in areas as important to human development as health, education, or law. The same is true for skills and the labor market, with each sector and country cultivating different nomenclatures. The situation in supply chains, transportation, international trade, and the Internet of Things is no less chaotic. The prospect of smart cities recedes when investigating the messiness of urban metadata. All sectors should be mentioned, but I would particularly like to highlight the lack of semantic interoperability in the world of encyclopedias, museums, libraries, and humanities research. Thus, most fields of knowledge and practice today have a fragmented digital memory. As for semantic interoperability between sectors of activity and administrations, which is so important in emergency situations, disasters, or population mobilization, it lags far behind.
Towards a semantic protocol
Images, sounds, smells, places, and emotions shape human memory, like that of animals. But it is language that unifies, orders, and reinterprets our symbolic memory at will. This is true not only on an individual scale but also on a collective scale, by the transmission of stories and writing. Through language, humanity has gained access to reflective intelligence (Bickerton, 1995; Deacon, 1997). By analogy, I think that we will reach a reflexive collective intelligence only by taking note of the evolution towards a stigmergic communication via digital memory, and by adopting a language-writing adequate to the organization of this memory. What has already been achieved for space, time (Galison, 2003), and various units of measurement remains to be accomplished in the semantic sphere: to establish a universal and regular system of coordinates that would favor formal modeling and the harmonization of operations. The recording and computing power of gigantic data centers now makes this ideal physically attainable. The history of automatic computing is certainly marked by the evolution of hardware and applications, but also by an impressive layering of operating systems, codes, protocols, and standards. There is no reason why this multilayering of software standards should stop evolving.
Why not adopt IEML as a semantic protocol, since its formal properties support automatic reasoning and data mining by semantic proximity? IEML is format independent and can translate any model. The digital memory could then be organized in all possible ways according to a multitude of practical needs and a diversity of points of view, while allowing fluid exchanges between its different parts and facets.
Neuro-symbolic artificial intelligence
Neuronal and symbolic AI
The two branches of AI—neural and symbolic—have existed since the middle of the 20th century and they correspond to two cognitive styles that are also present in humans. On the neural side, we have pattern recognition, which corresponds to sensorimotor reflex modules, whether these are genetic or learned. On the symbolic side, we have explicit and reflective conceptual knowledge, often organized in causal models and which can be subject to reasoning. Since human intelligence is itself neuro-symbolic, there is no theoretical reason not to try to make the two branches of AI cooperate in hybrid artificial intelligence systems that combine learning and reasoning (Garcez and Lamb, 2020; D’Avila D’avila Garcez and Lamb, 2009; Lévy, 2021). The benefits are obvious and, in particular, each of the two subsystems can remedy the problems encountered by the other. In a hybrid AI, the symbolic part overcomes the difficulties of conceptualization, generalization, causal modeling and transparency of the neural part. Symmetrically, the neural part brings capabilities of pattern recognition and learning from examples that symbolic AI is missing.
The role of language in human cognition
But today, the two types of AI have in common a certain logical compartmentalization, difficulties of accumulation, integration, and recombination of knowledge. However, bringing together the two branches of AI will not magically help us to make the human knowledge accumulated for centuries accessible to machines, nor to facilitate the transfer of knowledge from one system to another. Yet, thanks to language, human societies are able to transmit traditions from one generation to another. Within a single generation, natural language allows silent perceptions and experiential skills to be transformed into shareable knowledge. Through dialogue, an expert in one field will eventually make himself understood by an expert in another field and may even teach him something. How can this type of cognitive transfer be reproduced in mixed collectives of machines and humans? What could play the integrative role of natural language in societies of artificial intelligences? This is where IEML comes in.
Many people think that, since the brain is the organic medium of intelligence, neural models are the unique key to its simulation. But which intelligence are we talking about? Let's not forget that all animals have a brain, yet it is not the intelligence of the gnat or the whale that AI wants to simulate or increase, but that of the human. And if we are “more intelligent” than other animals (at least from our point of view) it is not because of the size of our brain. Elephants have bigger brains than humans in absolute terms, and the ratio of brain size to body size is greater in mice than in humans. It is mainly our linguistic capacity, supported by a large prefrontal cortex as well as Broca’s, Wernicke’s and some other areas—unique to the human species—that distinguishes our intelligence from that of other higher vertebrates. However, these language processing abilities are not functionally separated from the rest of the brain; on the contrary, they inform all our cognitive processes and our memory is largely organized by a system of semantic coordinates provided by language.
Why doesn’t AI use natural languages to represent knowledge? The answer is well known: because natural languages are ambiguous. A word can have several meanings, a meaning can be expressed by several words, sentences have several possible interpretations, and grammar is elastic and changing. Computers are not able to correctly disambiguate natural language statements. For its human speakers, a natural language extends a net of predefined general categories that are mutually explanatory, and this common semantic network makes it possible to describe and communicate both multiple concrete situations and different domains of knowledge. But, due to the limitations of machines, AI cannot make a natural language play this role. This is why it remains today fragmented into micro-domains of practices and knowledge, each with its own particular semantics.
Semantics of common nouns and semantics of proper nouns
Yet, isn't symbolic AI expected to simulate human expressive and reasoning abilities, including the linguistic faculty? But despite its name of “symbolic,” before the invention of IEML, AI did not have an unambiguous and computable model of language.
As we have seen above, since Chomsky’s work, we have been able to calculate the syntactic dimension of languages, but their semantic dimension remained beyond the reach of computer science. This point is clear in the way the WWW Consortium’s “Semantic Web” handles the meaning of words. From the point of view of the scientific study of language, the semantics of words breaks down into two parts: the semantics of common nouns and the semantics of proper nouns. However, we will see that the WWW Consortium’s classical symbolic AI only has access to the semantics of proper names.
The semantics of common nouns or word-word semantics
Remember that a linguistic symbol (word or sentence) has two sides: the signifier, which is a visual or acoustic image, and the signified, which is a concept or a general category. For example, the signifier “tree” has as its signified: “a woody plant of variable size, whose trunk grows branches from a certain height.” The relationship between signifier and signified being fixed by the language, the signified of a word or a sentence is defined as a node of relationships with other signified. Verbs and common nouns (e.g., tree, animal, organ, and eat) represent categories that are themselves connected by a dense network of semantic relations such as “is a part of,” “is a kind of,” “belongs to the same context as,” “is the cause of,” “is prior to,” etc. We can only think and communicate in a human way because our collective and personal memories are organized by general categories connected by semantic relations.
The semantics of proper nouns or word-thing semantics
In contrast to linguistic semantics, referential semantics bridges the gap between a linguistic symbol (signifier and signified) and a referent (an actual individual). When I say that “oaks are trees,” I am specifying the conventional meaning of the word “oak” by placing it in a species-to-genus relationship with the word “tree”; therefore, I am strictly bringing common nouns semantics into play. But, if I say that “That tree in the yard is an oak,” then I am pointing to a real situation, and my proposition is either true or false. This second statement obviously brings common noun semantics into play since I must first know the meaning of the words and English grammar to understand it. But, in addition to the intralinguistic dimension, an extralinguistic or referential semantics are also involved since the statement refers to a particular object in a concrete situation. However, languages use certain words, such as proper nouns, that have no signified: their signifier does not point to a general category but refers directly to a referent. For example, the signifier “Alexander the Great” refers to a historical figure and the signifier “Tokyo” refers to a city (Frege, 1971; Kripke, 1980; Récanati, 1983).
In a classical dictionary, each word is situated in relation to other nearby words (the thesaurus) and is explained by sentences (the definition) using words themselves explained by other sentences, and so on in a circular way. A classical dictionary is mainly concerned with the semantics of common nouns. In contrast to an ordinary dictionary, which defines concepts or categories, an encyclopedic dictionary contains descriptions of real or fictional individuals with proper nouns such as deities, novel heroes, historical events, geographical objects, and so on. Its main function is to list and describe objects external to the linguistic system. It therefore records a semantics of proper nouns.
The simple distinction we have just made is sufficient to assess the extent to which pre-IEML symbolic AI mastered semantics. If we consider the ontologies of the Semantic Web (the standard in symbolic AI), we discover that the meanings of words and sentences do not depend on the self-explanatory circularity of a language (as in a classical dictionary), but on a reference to URIs (Uniform Resource Identifiers) that works according to the semantics of proper nouns (like an encyclopedic dictionary). Instead of relying on concepts that are already given in a language and that exist from the outset as nodes of relationships with other concepts, the scaffolds of the Semantic Web rely on concepts that are defined separately from each other using unique identifiers (Berners-Lee and Halpin, 2013). The flow of meaning through a network of signified is eschewed in favor of a direct relationship between signifier and referent, as if all concepts were represented by proper names. In the absence of a semantics of common nouns based on the grammar and the dictionary of a language, ontologies thus remain compartmentalized. In summary, before IEML, symbolic AI did not have access to the full cognitive and communicative power of language, but only to a rigid referential semantics combined with logic.
What IEML brings to artificial intelligence
A mathematical, standardized, and unambiguous language such as IEML could open new horizons for artificial intelligence in learning, reasoning, and communication. On the side of neural AI, which is currently in full expansion, the new research subfield of semantic machine learning shows that labeling or annotation using good ontologies favors generalization in machine learning (see, for example, Kulmanov et al., 2021). A fortiori, the use of ontologies in IEML could support more efficient learning because the units or “tokens” taken into account by the machines would no longer be sound units—characters, syllables, or words—in natural languages but concepts generated by a semantic algebra. This would lead to a more transparent and explainable neural AI. On the symbolic AI side, the use of IEML would bring a new semantic computation (the automatic creation of nodes and conceptual links) and a true linguistic semantics in addition to the logical functions and referential semantics (URIs) already in use. This language would also facilitate neuro-symbolic integration and allow models to connect and knowledge to accumulate.
Let us assume that, for a machine, “understanding” means projecting data onto conceptual graphs that model a domain of knowledge and practice. Therefore, by coding information flows in IEML, it is now possible to design generative AIs capable of understanding (1) their training data, (2) user prompts or commands, and (3) texts, codes, images, and other data they produce in response. This would move us from a probabilistic approximation of expected outcomes to a true controlled generation.
Given the role of speech in human intelligence, it is surprising that we hoped to reach a general artificial intelligence without having a computable model of language and its semantics. 21 The good news is that we finally have one. Artificial intelligence will only be able to sustainably increase human cognition if it becomes meaningful, interoperable, cumulative, integrable, exchangeable, and distributed. This means that we will not make significant breakthroughs in artificial intelligence without at the same time aiming at a collective intelligence capable of reflecting and coordinating itself in the global memory.
Digital humanities
The human sciences are today balkanized into disciplines and divided into theories that often do not even agree on their disagreements. This is a far cry from the division of disciplines along objective lines of levels of complexity (physics, chemistry, biology…) that prevails in the natural sciences and from the general agreement of the exact sciences on their methods and their organizing categories. Despite some exceptions, the degree of formalization of models and testability of theories in the human sciences remains low. This is reminiscent of the situation that prevailed in alchemy before the emergence of modern chemistry at the end of the 18th century. But the conditions are ripe for a scientific revolution (Kuhn, 1962).
Archives are progressively digitized and almost all human activities produce a growing stream of data. To process this data, we have efficient analysis and visualization algorithms as well as gigantic computing power. Finally, the modes of communication of the 21st century open up new avenues for teams of researchers to engage in dialogue, to share corpora and tools, to collaborate internationally, and to publish. Researchers in the humanities and social sciences are building databases for analysis, mining, and sharing. Research results are increasingly available online on researchers' blogs, some social networks, journal websites, but also in specialized search engines like ISIDORE 22 .
But the new conditions of research in the humanities and social sciences pose the crucial problem of a shareable and interoperable categorization of data and documents. As we have seen above, there are today many ontologies and semantic metadata systems, which vary according to language, discipline, tradition, and theory. These systems are often inherited from the printing era. In this context, IEML provides a programmable modeling and indexing tool, capable of ensuring semantic interoperability without standardizing the points of view. It becomes possible to collaboratively exchange models and sub-models between researchers speaking different languages and coming from different disciplines.
Just as material bodies are immersed in space and time, the human mind lives in the intangible element of meaning and value. The unified understanding of the physical universe through mathematical modeling and powerful tools of observation and measurement is only a few centuries old (Koyre, 1968; Lévy, 2021). It followed shortly after the massive use of movable type printing. The adoption of mathematical coordinate systems and common units of measurement for space and time has played a crucial role in the scientific and technical advances of the last three centuries, as well as in the improvement of the possibilities of concerted action of our societies. The emergence of the algorithmic medium could have an effect on the human sciences similar to that of the printing press on the natural sciences, provided that we recognize their fundamental unity and adopt a system of semantic coordinates supporting their mathematization. Let us specify that mathematization does not necessarily imply a reduction to the quantitative. Many fields of algebra and geometry, for example, formalize structures and symmetries rather than quantities. Just as the idea of a mathematical physics was unthinkable before the end of the 15th century, the idea of a mathematical semantics may seem implausible today. Yet IEML opens the way to a new scientific cosmos, welcoming to meaning, coherent, and computable, though inexhaustibly complex. A more fluid feedback loop between ecosystems of ideas and the communities that sustain them would bring us closer to the ideal of a reflexive collective intelligence at the service of human development. It is not a question of maintaining some illusion about the possibility of total transparency, and even less of claiming to reach a single true theory. The goal would rather be the critical exploration of an infinite universe of meaning.
Conclusion
There is now at least one language that lends itself to the calculation of its semantics. The scientific foundations have been laid for the establishment of a protocol capable of solving the problem of semantic interoperability, with all the consequences mentioned above for the improvement of collective intelligence, artificial intelligence, and human sciences. The function of such a semantic metadata system is to increase automatic processing in the service of a plural but interdependent collective intelligence. The road that remains to be traveled in order to reach an effective transformation of the digital memory is immense. Let's hope that a multidisciplinary network of researchers, linguists, and developers will get to work and that a consortium of universities, governments and companies will support their efforts.
Independently from the development of a new generation of neuro-symbolic AI models that “understand” what they are told and what they do, or the long-term prospect of its adoption as a protocol, IEML is already paving the way for a new kind of semantic engineering that supports several research directions. One of them concerns the design of information systems in which the rigorous conceptualization of data models by means of this mathematical language would allow the computation of a large number of analytic (arising from the construction of the concepts themselves), and synthetic a priori (arising from the relations between concepts) truths, even before the data are loaded (Kant, 1787). This cognitive layer prior to data recording could considerably increase symbolic reasoning and statistical learning on empirical data. A complementary research direction is the development of a method for producing ontologies or knowledge graphs in IEML, in order to best support user communities in reaching their goals and enhancing their practices. How to map a universe of discourse while avoiding blind spots? How to re-conceptualize data (and therefore make ontologies evolve) in a flexible way during the normal use cycle of an information system? By making observable the cognitive processes supported by a common semantic information system, we would progress towards a more reflexive collective intelligence.
Footnotes
Acknowledgments
Darcia Labrosse has made the creation of IEML possible through her constant support. Louis van Beurden Eng. Msc. Contributed to the research on IEML for several years and more particularly to the programming of the IEML parser and software library.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Canada Research Chairs Program; the Social Science and Humanities Research Council of Canada, (Grant number: 225427); and by the company Intlekt Metadata Inc.
Open Practices
The raw data and analysis code used to conduct analyses for this manuscript are available upon request. The questionnaires/research instruments used to collect these data are also available upon request. This study was not pre-registered in any data base.
