"Wisdom consists in the investigation of first causes or principles" (Descartes).
"Principles are not shown, but their truth is directly perceived" (Aristotle).
"Primitive components must be taken as simple as possible if order and clarity are to be produced" (Frege).
The Semantic Primitives Model
Lexical semantics and structural semantics
The semantic primitives model is a formal model of the semantics of a given domain based on:
The initial selection of a finite set of primitive concepts called "semantic primitives". This set constitutes what is also referred to as lexical semantics.
The establishment or definition of some combinatorial mechanisms of these primitive concepts to express any semantics related to the domain. These are the "derived concepts", "semantic derivatives" or, simply, "derivatives". These combinatorial mechanisms constitute structural semantics.
Assuming a certain formal representation, i.e. a formal language to express the primitives and their combinatorics, what is called "compositional semantics" must be verified: the meaning of a composite semantic expression is derived from the lexical semantics of its components and from the structural semantics (the semantics of the relations between the components).
Ontology
Ontology is the branch of philosophy that studies being, what exists, the essence of things. In the context of a given domain, ontology is concerned with concepts and their relations.
A structured set of semantic primitives of a given domain constitutes an ontology for that domain if it allows one to specify:
The classes or categories of things that exist (the concepts).
The properties of those concepts.
The structure of the relationships between the concepts (hierarchies, common properties, etc.).
Features of the Semantic Primitives Model
The following aspects and features can be distinguished in the semantic primitives model:
Level of abstraction
The two extremes that can be considered are:
Low-level primitives.
Very basic, almost atomic concepts are used. Their main advantage is that they allow a high level of detail to be specified, but they have many drawbacks:
In general, a large number of primitives are required.
A large combinatorics is required to cover all the semantics of the domain.
Derived expressions are more complex. The larger the combinatorics, the greater the complexity.
Expressivity is poor. To achieve a higher level of expressiveness one would have to resort to derived expressions to define concepts of a higher level of abstraction.
Their scope of application is restricted to the domain, i.e., there is usually no possibility of using them in other domains, due to their little or no generality.
High-level primitives.
Generic concepts are used, high level of abstraction, intelligible, easy to understand and use. Its advantages are:
In general, a small number of primitives are required.
Less combinatorics is required to cover all domain semantics.
Derived expressions are simpler.
There is greater expressiveness.
The possibility of use in other domains is opened up. The higher your level of abstraction, the greater the number of possible application domains.
They favor the normalization of derived expressions (canonical forms).
The main drawback is that there may be limitations that can lead to ambiguities. Depending on the nature of the high-level primitives used, two things can happen:
That it becomes impossible to specify beyond a certain level of detail.
That it may be possible to specify the desired detail by combinatorics of the high-level primitives. This is only possible when the primitives are generic.
Selection
In principle, in a given domain, different sets of primitives could be selected. In general, there is no unique set. The selection is basically going to depend on factors such as:
The simple or complex nature of the domain.
The structure and relationships of the concepts in the domain.
The level of expressiveness and abstraction required.
The combinatorial mechanisms that can be used to express the derived concepts.
The selection of the set of primitives should be made using the following criteria:
Their number should be the minimum possible.
The primitives must correspond to clear, simple and intuitive concepts.
Structure
Ideally, the primitives should be independent of each other and there should be no combinatorial constraints. In this case, the primitives can be considered as the conceptual dimensions of the domain.
This orthogonal dimension model produces a great simplification in the formalization of a domain. In this respect, there is an analogy with modern physics, which considers dimensions to simplify the description of physical laws.
In the case where primitives were not independent of each other, there must be a clear structure (e.g., subsystems of primitives and relationships between these subsystems).
Simplicity
In general, the primitives to be selected should be simple, as simple as possible. But there are two types of simplicity:
Definitional simplicity.
Primitives are chosen purely on the basis of reductionist criteria, even if they are not very intuitive.
Conceptual simplicity.
Primitives are chosen exclusively on the basis of conceptual criteria, i.e., they must be simple and at the same time understandable and intuitive, even if they are reducible to even simpler, but less intuitive, concepts.
Definitional simplicity is usually associated with a low level of abstraction. And conceptual simplicity is usually associated with a high level of abstraction. It is certain that Einstein was referring to this when he said "Everything should be made as simple as possible, but not simpler".
An example of definitional versus conceptual simplicity is that of logical operators [see Addendum].
Granularity
Indicates the number of semantic primitives used.
Low granularity: few semantic primitives.
High granularity: many semantic primitives.
In general, there is an inverse relationship between granularity and abstraction level:
Low abstraction level: requires or implies high granularity.
High abstraction level: requires or implies low granularity.
Combinatorial
Ideally, no combinatorial constraints to form derived concepts.
Between primitives.
Between primitives and derivatives.
Between derivatives.
Combinatorics produces hierarchical structures. Primitive concepts would be level 1. Concepts resulting from the combinatorics of primitives would be level 2, etc.
Combinatorial constraints basically occur when semantic primitives are not independent of each other, due to: the existence of subsystems of primitives, the existence of primitives subordinate to others, etc.
From a psychological point of view, combinatorial constraints affect negatively, especially in domains such as programming languages (where freedom and maneuverability are fundamental), causing the programmer, discomfort and some confusion.
Ideally, the combinatorics should be performed with the primitives themselves, i.e., lexical semantics and structural semantics should coincide.
It is desirable that the combinatorics produces new, unexpected concepts, i.e., generates creativity. Creativity is all the greater the higher the level of abstraction of the primitives.
Type
The selected primitives can be operational or descriptive, intensive or extensive, data or process, structural or functional, etc.
Syntax
Once the set of semantic primitives and their combinatorial mechanisms have been established, there is freedom with respect to the syntax to be used.
Ideally, there should be a direct relationship between semantics and syntax, such that:
A syntactic construction that clearly identifies each semantic primitive is associated with it.
Given a syntactic construction, its semantics can be immediately known.
In addition, it is desirable that the syntax can be modified, always respecting the two previous points.
Domains of Application of the Semantic Primitives Model
The semantic primitives model has been applied, more or less formally, to many domains. In some, low-level primitives have been used, in others high-level primitives, but we must take into account a general trend that is clearly observed in science: the unification of different domains, through the use of conceptual primitives of a generic type that facilitate the understanding of the world.
In certain cases it is stated that they use "semantic primitives", but only in the sense of base or fundamental concepts. For an authentic formal model to exist, the combinatorial mechanisms for generating (or deriving) the semantics of the domain from the primitives must be provided. In other cases there is only a partial formalization.
Prominent domains where the model of semantic primitives has been formally applied are:
Natural linguistics.
Schank's conceptual dependency model.
Anna Wierzbicka's model of semantic primitives.
Informatics.
Models of computation (e.g., the Turing machine).
Programming languages.
Structured programming.
Abstract syntax.
Entity-Relationship Model.
Molecular Computation.
Ideal Software Machine.
The AMOA (Action-Modifier-Object-Attribute) model of user functions.
Essential programming language.
Specification languages.
Mathematics.
Throughout the history of mathematics, only one semantic primitive has been used as the foundation of all mathematics: numbers, sets, structures, categories, and functions. But behind this primitive there are other subordinate but not explicit concepts. But there has not been, so far, a proposal of a set of semantic primitives as a theoretical-practical foundation of all mathematics.
Artificial Intelligence (AI).
The IPS (Information Processing System) model, by Herbert Simon and Allen Newell.
Soar. Symbolic cognitive architecture for the development of AI systems.
Discovery systems.
Cognitive models of the mind.
The hypothetical "language of thought".
Model of Semantic Primitives vs. Formal Axiomatic Systems
There is an analogy or correspondence between the model of semantic primitives and formal axiomatic systems:
Semantic primitives
Formal axiomatic systems
Primitives
Axioms.
Combinatorics
Rules of inference
Derivatives
Theorems
Axioms are simple, minimal and indispensable concepts from which it is possible to deduce the theorems of the formal system. They would correspond to the semantic primitives of a domain.
The rules of inference correspond to combinatorics.
Theorems are deduced from axioms (or other theorems). They would correspond to derived concepts, by combinatorics of primitives or other derivatives.
The idea of founding a domain on the basis of a minimum number of principles or axioms has its origin in Euclid's "Elements", where all geometry was founded on a minimum number of principles or axioms (the famous 5 postulates), an idea that was an intellectual milestone of enormous importance, since it showed that it is possible to found and build the knowledge of a domain on the basis of a small set of initial principles.
Two properties must be fulfilled in a formal axiomatic system:
Consistency. It must not be possible to derive two contradictory theorems, i.e., stating one thing and the opposite. Current formal axiomatic systems are not identified with truth, as in Euclid's time, but with consistency.
Completeness. Every expression of the domain must be provable its truth or falsity by means of axioms and rules of inference.
Similarly, in the case of the model of semantic primitives, these two properties must also be satisfied:
Consistency. Any derived expression must be well-formed. For example, the expressions a((b and a+++ are not consistent.
Completeness. All domain concepts are derived from primitives and combinatorial mechanisms.
The Model of Universal Semantic Primitives
The question now arises as to the possibility of searching for, identifying and abstracting a set of universal semantic primitives, i.e., valid for all domains of the formal sciences, especially mathematics and computer science. If successful, we would have a unifying paradigm, i.e., a way of seeing the world always through primitive concepts of a universal type.
The characteristics of these universal semantic primitives would be as follows:
They must be necessarily simple at the conceptual level, but when combined together they produce the complexity of the world.
Their level of abstraction must be supreme, so that they can cover the totality of the domains.
Their granularity (the number of primitives) should be as low as possible.
For maximum simplification, the structural semantics, i.e., the combinatorial mechanisms of the primitives, must be the same as the lexical semantics (the semantics of the primitives).
They must be independent of each other, i.e. orthogonal. Therefore, they can be considered as:
Dimensions of the mind (semantic dimensions).
Degrees of freedom of the mind (as they delimit our expressive possibilities).
"Keys" of the mind.
Set of mental instructions.
Mental DNA.
Primitives, by their universal nature, must incorporate generic mechanisms of operational and descriptive, intensive and extensive, data and process, structural and functional, and so on.
The syntax should be simple and should have a biunivocal relationship between syntax and semantics.
If all these characteristics are met, then you have an environment where creativity would be supreme, the maximum possible.
As a consequence, we would have a global or universal ontology and epistemology, that is, a vision applicable to all domains of science, which would constitute a more solid foundation for the specific domains of human knowledge, since it would provide:
A common semantic basis for all domains.
A means of integrating, relating and communicating the different domains through that common base.
An understanding and positioning of things within an overall conceptual framework.
A foundation for artificial intelligence, by equating human mental resources or mechanisms with those of the computer.
Addenda
Examples of primitive selection
Selecting a suitable set of primitives, each with its level of abstraction, in a given domain is not an easy task, even in the simplest and clearest domains. One can choose high-level, low-level, or a combination of both.
Partnership
For example, in knowledge representation, in the clear and structured domain of kinship, one can choose:
Low-level primitive relations.
The most obvious ones are: father, mother, son and daughter.
Derived relationships would be: brother, sister, grandfather, grandmother, cousin, cousin, etc.
If facts are available at the primitive level, it is easy to deduce derived relationships. On the other hand, if facts are available at the level of derivatives, there would be some ambiguity. For example, if the fact is "Ana is Pepe's cousin," then there would be four possible interpretations:
"Ana is the daughter of herman(o/a) of (p/m)adre de Pepe"
High-level primitive relationships.
One could use: parent (instead of father and mother) and direct descendant (instead of son and daughter).
Derived relationships would be sibling(s), grandparent(s), prim(s), descendant and ascendant.
In the case of the previous fact ("Ana is Pepe's cousin"), the expression is simplified by using more generic primitives, with two possible interpretations:
"Ana is a direct descendant of herman(o/a) of Pepe's progenitor"
Propositional logic
In this domain it is possible to use a single two-parameter (p and q) low-level semantic primitive: "neither p nor q" (p↓q), called "Peirce's arrow or "Quine's dagger", defined by means of the following truth table:
p
q
p↓q
0
0
1
0
1
0
1
0
0
1
1
0
This primitive is equivalent to: (p∨q)' = p'∧q' = NOR (Not Or).
By means of this single primitive it is possible to express other logical operations (derivatives):
This is a representative example of definitional simplicity. If a higher level of abstraction approach with conceptual simplicity had been adopted, then "negation" and "conjunction" would have been chosen as primitives:
The disjunction would be defined as derived from the previous two: p∨q = (p'∧q')'
The implication would be defined as: p→q = p'∨q
One could also have chosen "negation" and "disjunction" as primitives and defined conjunction and implication as derivatives:
p∧q = (p'∨q)'
p→q = p'∨q
One could have used the dual logical operation to NOR:
(p∧q)' = p'∨q' = NAND (Not And).
This is the so-called "Sheaffer's bar" (p|q), whose truth table is.
p
q
p|q
0
0
1
0
1
1
1
0
1
1
1
0
Bibliography
Dekker, Paul J.E. Dynamic Semantics. Springer, 2012.
Eco, Umberto. La búsqueda de la lengua perfecta. Crítica, 1993.
