MENTAL vs. MODEL THEORY

"Model theory is the theory of interpretations of formal languages" (Geoffrey Hunter).

"Archetypes are metaphors that drive us and serve as models" (Carlos Alberto Churba).

"You can never change things by fighting against the existing reality. To change something, create a new model that makes the current model obsolete" (Buckminster Fuller).

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Model Theory
Definition

Model theory is the theory of interpretations of formal systems and how these systems relate to each other. It is also concerned with how interpretations vary when we vary a formal system.

Model theory tries to connect the formal (the syntactic) with its possible interpretations (the semantic) through the notion of truth. Model theory is considered to be the modern version of semantics.

In model theory, the definition of meaning is: 1) the meaning of a term is the referent of every possible domain, situation or world; 2) the meaning of a sentence is a truth value in every possible domain, situation or world.

• A formal system S is a set of sentences defined by means of a formal language L. It consists of initial statements (axioms) and rules for deducing new statements (theorems).
  
  
• A formal language L is defined by: 1) a set of symbols (the language alphabet); 2) a set of rules for forming sequences of symbols that constitute well-formed formulas (fbf's). A formal language can be identified with the set of its fbf's.
  
  In model theory the first order predicate calculus (with a finite number of predicates) and with an additional predicate which is the equality or identity relation (x = y) is normally used as a formal language. This formal language is called a "first-order language" and a formal system constructed with this language is called a "first-order system".
  
  First-order languages are formal languages that deal with objects in a domain that have a certain structure (based on properties and logical relationships between objects). Each first-order language determines the class of structures that conform to that language.
  
  A formal language can be defined completely without reference to any interpretation.
  
  
• A language L can also be considered a formal system, where axioms are the rules of grammar and theorems are the possible sentences of L. A formal system can also be considered a language in itself.
  
  
• An interpretation of a formal system S is an assignment of meanings to its symbols and formulas. Examples:
  
  
  • The set of natural numbers is a model for Peano's axioms.
    
    
  • The set of natural numbers is a model for group theory under the operation of addition.
    
    
  • The set of real numbers (except zero) is a model of group theory under the product operation.

Using first-order predicate logic as its formal language, model theory is considered a branch of mathematical logic. But model theory is best situated as an interdisciplinary area between mathematics, philosophy and computer science.


Characteristics

• Model theory is metaphysically and ontologically neutral and open with respect to possible interpretations of formal systems. "Model theory is completely agnostic about what kinds of things exist" [Hodges, 1997].
  
  
• The more generic or simple a formal system is, the more interpretations it has. Group theory is perhaps the most representative or most important formal structure, since it has models in numerous fields of mathematics.
  
  
• One usually speaks of standard and "non-standard" interpretations, depending on whether the interpretation is the most usual (or known) or not, respectively. But these two concepts are not really part of model theory. For example, the standard model of Peano axiomatics is the natural numbers.
  
  
• Symbols in formal language that have a fixed interpretation are called "protosemantics". For example, the logical connectors of the first-order predicate language. In this case, not every component of a formal system is interpretable.
  
  
• An interpretation of a sentence of a formal system can be true or false, depending on the domain. For example:
  
  
  • The statement ∃x | x*x = 2 is false in the domain of rational numbers and true in the domain of real numbers.
    
    
  • The statement ∃x | x*x = −1 is false in the domain of real numbers and true in the domain of complex numbers.
  
  Therefore, in model theory, truth is relative. This is its most fundamental principle.
  
  
• The sentences of a formal system can be: universally valid (true in all domains), contradictory (false in all domains) and contingent (neither universally valid nor contradictory).
  
  
• A formal system in which all models are isomorphic is called "categorial", i.e., the formal system represents a set of isomorphic models.
  
  
• Model theory also studies mathematical structures (groups, rings, bodies, etc.) by providing meanings to the first-order sentences that formalize these structures. In this sense, model theory is closely related to universal algebra. Model theory generalizes universal algebra because it allows relations, whereas universal algebra only allows functions. According to Chang & Keisler [2012], model theory is an extension of universal algebra:
  
  
  model theory = universal algebra + mathematical logic
  
  Universal algebra provides semantics to structures. Mathematical logic provides the syntax.
  
  
• Model theory is subdivided into several branches such as: finite, infinite, classical, abstract, categorical, geometric, computational, etc. model theory. Finite model theory is closely related to universal algebra and computability theory (in particular, complexity theory).
  
  
• A theory is a formal system that is semantically complete in the sense that any model that satisfies the sentences of the theory also satisfies any other sentence that is a consequence of that formal system.
  
  
  • A first-order theory is one that uses a first-order predicate language.
    
    
  • A model of a theory is a structure where the formal sentences of the theory are interpretable and where the sentences can be regarded as statements about the model.
    
    
  • The objects of study of model theory are models of theories in a formal language.
    
    
  • A theory is consistent if it does not lead to contradictions, that is, if for every pair of formulas (Φ, ¬Φ) of the formal language only one of them belongs to the theory.
    
    
  • A theory is complete if for every pair of formulas (Φ, ¬Φ) of the formal language at least one of them belongs to the theory. That is, a complete theory is the one that contains every sentence or its negation.
    
    
  • If all the models of a theory T are isomorphic, then the theory T is complete.
    
    
  • If a complete theory T has a finite model, then all its models are isomorphic.
    
    
  • A theory is categorical if all its models are isomorphic.
    
    
  • A theorem of a theory is a provable formulation. The notion of truth is not equivalent to the notion of provability in first-order axiomatic theories by Gödel's incompleteness theorem.

Löwenheim-Skolem Theorem (LST)

It is a metalogical theorem of first-order predicate logic, like Gödel's incompleteness theorem. In its first version, due to Löwenheim in 1915, it states:

If there exists an infinite model for a formal system, then there exists a numerable model for this formal system.

The generalized version for transfinite numbers, due to Skolem, in 1920, states:

If L is a first-order language and has a transfinite cardinality model k, then it has at least one cardinality model ≤ k (top-down TLS) and one cardinality model ≥ k (bottom-up TLS).

Gödel's Completeness Theorem (1929)

This theorem is fundamental in mathematical logic and has a close connection with model theory, since it establishes a correspondence between truth (semantics) and provability (syntax) in first-order logic:

There exist first-order systems in which all logically valid formulas are provable. The first-order calculus is powerful enough to deduce all valid formulas.

Henkin's theorem (1944) is also a first-order completeness theorem. It is a simpler version of Gödel's completeness theorem:

A first-order theory is consistent if and only if it has a model.

To prove that a first-order theory is consistent, it suffices to prove that it has a model. In mathematics, all theories are consistent.

Consistency is a syntactic notion. The notion of model is semantic.


Skolem's paradox

The so-called "Skolem's paradox" [1922] is an apparent paradox of the TLS:

An axiomatization of set theory in first-order logic that is consistent has a numerable model.

The paradox consists in the fact that set theory does not refer to numerable sets and yet has a numerable model. Therefore, axiomatic set theory has a large number of possible interpretations, including non-isomorphic interpretations.

Cantor discovered in 1874 the existence of non-numerable sets, such as the power of the natural numbers and the set of real numbers. Zermelo proved it in 1908 with his axioms of set theory. But the axioms of set theory were supposed to have only one model: all sets, finite or infinite.

Skolem described his discovery as "a paradoxical state of affairs," although he did not really consider it an antinomy like Russell's paradox. Skolem himself [1922] explained that there was no such contradiction: in the context of a specific model of set theory, the term "set" refers to a concept of the model and not to the classical concept of set theory.

The TLS was the first major result of model theory. But it was Skolem who deeply analyzed its impact on the philosophy of mathematics and metamathematics. Among these philosophical consequences or conclusions are the following:

• Relativism.
  The concepts of set theory are relative. Relativity lies in the fact that a set may have a property in one model and not have it in another. Examples:
  
  
  • Numerability and non-numerability are relative notions. A set may be numerable or non-numerable, depending on the model. For example, the property that the set of all subsets of the natural numbers is not numerable in one model, but there is another model in which it is numerable. The TLS seems to lead to a paradox that all sets are numerable in some model.
    
    
  • If E is an axiomatization of the real numbers R, we know that from E one can infer that there are more real numbers than natural numbers, that is, that the cardinality of the real numbers is greater than the cardinality of the natural numbers.
    
    But, according to the TLS, E has a model of the same cardinality as the set of natural numbers N, i.e., an infinitely numerable model. That is, the sentences of E can be interpreted as describing a model that has the same number of elements as N. This model is non-standard and top-down.
  
  Skolem saw the TLS as the impossibility that absolute truth could be captured by a system that founds mathematics. The TLS generalizes the relativity of all mathematical notions.
  
  
  • "The true meaning of Löwenheim's theorem is precisely this criticism of the undemonstrable absolute" [Skolem, 1941].
    
    
  • "A consequence of this state of affairs [the TLS] is the impossibility of an absolute categoricity of mathematical notions" [Skolem, 1958].
    
    
  • "All concepts of set theory, and consequently of all mathematics, thus become relativized. The meaning of these concepts is not absolute, it is relativized to the basic axiomatic world" [Skolem, 1941].
    
    
  • "Relativism is inevitable because of the general nature of Löwenheim's theorem" [Skolem, 1941].
  
  
• Foundation of mathematics.
  The notions of set theory being relative, axiomatic set theory fails as a foundation of mathematics, for its truths are relative. "It is clear that axiomatization in terms of sets was not a satisfactory foundation of mathematics" [Skolem, 1922].
  
  
• First-order language.
  The weakness of the first-order language, on which all mathematical theories had relied for their formalization.
  
  
• The concept of model.
  The questioning of the model concept. If the model concept is accepted, there is the possibility of non-standard models.
  
  
• Cardinality.
  The relativity of the concept of cardinality, which makes it necessary to reinvestigate it. Consistent first-order theories cannot control the cardinality of their models. And they cannot have only isomorphic models. If a theory has a numerable model, it also has non-numerable models. For example, Peano's axioms of arithmetic have to have non-numerable models.

Despite all this, Skolem still believed in the axiomatic system, even regarding relativism as positive. "A relativistic conception of fundamental notions is clearer than the absolutist, Platonist conception that dominates classical mathematics" [Skolem, 1958].

Skolem underwent an evolution in his mathematical thinking. Initially he was an intuitionist "absolutist" or Platonist. At this stage Skolem thought that primary mathematical truths should be clear, natural, absolute, unquestionable, and connected to our faculty of intuition. In fact, Skolem was one of the founders of finitism in mathematics. Skolem was skeptical about the existence of non-numerable sets and considered the TLS as evidence that his skepticism was justified. In any case, Skolem was in favor of a formal language for expressing mathematical ideas.


Putnam's view

Hilary Putnam's article "Models and Reality" [1980] marked a renewal of interest in model theory and in Skolem's paradox in particular.

Putnam analyzed Skolem's paradox in depth, arguing:

• The concepts of set theory are relative. And this circumstance is a problem that affects natural languages and scientific theories. Everything must be "skolemized", that is, everything must be relativized, because semantics can never be fixed. No theory can formalize its own interpretation.
  
  
• The semantics of model theory fails as a theory of meaning. And the reason it fails as a theory of meaning is that it fails the principle of compositionality: the meaning of a sentence is a function of the meaning of its components.
  
  The problem arises when combining reference with truth. If we change a term in a sentence, we change the semantics of the sentence, but model theory says that the truth (the meaning of the sentence) is preserved. This goes against the principle of compositionality.

These considerations led Putnam to reject metaphysical (Platonist) realism and adopt the position of anti-realist semanticist. "The world does not choose models or interpret languages. We interpret our languages or nothing does" [Putnam, 1980]. Putnam's solution was to choose a semantics in which use and reference are closely linked: we have the interpretation insofar as we understand its use.


Theory of institutions

The theory of institutions was introduced by Joseph Goguen and Rod Burstall in their seminal paper [Goguen & Burstall, 1992]. It is a universal categorical model theory that attempts to formalize the intuitive notion of "logical system" without reference to any particular logic. The logical entities it uses are entirely abstract. The thesis of the theory of institutions is that every particular logic can be formalized as an institution.

The concept of institution arose in computer science, specifically in the area of algebraic specification, as a response to the proliferation of particular logics (first-order, higher-order, equational, Horn clausal, infinitary, dynamic, intuitionistic, temporal, etc.). These logics are used to solve problems such as concurrency, overloading, exceptions, etc. In addition, the logic in mathematics varies depending on the theorem to be proved, although first-order predicate logic is usually used.

The concept of institution is based on abstract categorical entities from category theory. Institution theory can be considered as a synthesis between model theory and category theory.

The formalization of a logical system includes syntax, semantics and an axiom (called "satisfaction") that relates them. This axiom tries to express that the truth (semantics) is invariant with respect to the change of notation.

Informally, an institution consists of:

1. A collection of signatures (each signature defines a notation) for constructing sentences in a logical system.
   
   
2. For each signature there is: a) a collection of sentences; b) a collection of patterns; c) a satisfaction relation of sentences by patterns.

When signatures change (via morphisms), the satisfaction of sentences by models changes consistently.

Institutions can be applied in the development of specification languages, programming languages, database theory, and artificial intelligence.


MENTAL, a Universal Model
Model theory is a sophisticated branch of mathematical logic. We compare it to MENTAL in the following respects:

• Universal model.
  MENTAL is a universal formal language and a universal model (or metamodel) that manifests itself in particular models in different domains: mathematics (foundational model), computer science (computational model), cognitive psychology (model of the mind), depth psychology (archetypal model), philosophy (model of deep reality or philosophical categories), etc.
  
  
• Protosemantics.
  MENTAL is a universal model with protosemantics, i.e., with fixed primary semantics. The universal model of MENTAL is the universal grammar and language that underlies everything. MENTAL is the culmination of model theory, as all models are connected by protosemantics.
  
  In MENTAL no meanings are assigned to symbols because they have fixed, absolute semantics because they are associated with primitives. Their meanings are primary or protosemantic. For example, {...} is a set, (...) is a sequence, the symbol ∈ indicates belonging to a set, etc. In model theory, on the other hand, a different meaning could be assigned, leading to relativism.
  
  However, the symbols of the MENTAL primitives can be modified if desired, even replacing them with names. For example,
  
  ⟨( set(x) =: {x} )⟩
  
  Only the names used, which have secondary meanings, are assigned meanings. The primitives are the container, the "skeleton" or structure. The names are the content. All surface manifestations are based on the same primary semantics. If we only have as reference the formal (superficial) systems, all disconnected from each other, then everything is relative. If there is a primary system, then there is an absolute frame of reference.
  
  Names admit of many interpretations. The interpretation is open, including the literal one. For example, {a b c} is an expression representing a set consisting of 3 elements (a, b and c). These elements can be the letters themselves (literal interpretation) or they can represent other expressions. There are infinite sets of 3 objects, but they all share the same structure, such as:
  
  {1 2 3} {Pepe Juan Luis} {⟨a b c⟩ 17 Juan}
  
  Assigning an interpretation to names can be done using a dictionary. If natural language names are used, then the interpretation becomes obvious. For example, Pepe/man (Pepe is a man).
  
  
• Truth and demonstration.
  In MENTAL interpretation is not associated with truth because truth is something outside language. Nor does it refer to demonstration because in MENTAL inferences are automatic.
  
  
• Particular models.
  By the very nature of primitives, every formal system created with MENTAL is categorial, that is, all its models are isomorphic. Particular models can be more easily defined for two reasons:
  
  
  1. Because MENTAL connects syntax with semantics. Every formal system incorporates a primary semantics. In this sense, an expression is a model of itself. A formal language in isolation, without semantics, is meaningless. Semantics is inexpressible because it belongs to the internal world; we can only connect it through syntax as an archetype that connects the internal world and the external world.
     
     
  2. Because MENTAL is a more general, flexible and expressive language than first-order predicate logic. This makes it easier to establish relationships between formal systems. MENTAL transcends first-order languages. In fact, it is a generalized predicate language not tied only to logic.
  
  
• Relativism.
  Model theory is the culmination of the abandonment of the search for a primary, original, deep, archetypal, absolute semantics and the surrender to theories of the superficial and relative. It is the culmination of a mistaken approach: the one that goes from the superficial (the formal system) to semantics (the deep).
  
  The way should be the opposite: going from semantics to syntax, from the generic to the specific, from the deep to the superficial. The true meaning of Gödel's theorem is that it is impossible to go from the superficial to the deep.
  
  If "everything" is subject to interpretation, this leads to relativism, for there is nothing solid to be grounded on. In this sense, mathematics would have no foundation at all. There must be a fixed, Platonist, absolute and universal protosemantics as provided by MENTAL. Model theory is the theory of relativity of formal systems.
  
  Model theory, and TLS in particular, has caused much confusion about the nature of semantics, about truth, and about the grounding of mathematics. The correct way is to ground everything on the archetypes of consciousness.
  
  
• Possible generic and particular expressions.
  The names that appear in a generic expression as parameters do not admit more than one interpretation, since the names are irrelevant when representing any expressions. For example, the following two expressions are equivalent:
  
  ⟨( f(x y) = (x+y x*y) )⟩
⟨( f(u v) = (u+v u*v) )⟩
  
  Any particular expression of MENTAL is a manifestation of a category of expressions. For example, the expression {a b c} is a manifestation of the category "set of 3 elements", a category expressed by the generic expression ⟨{x y z}⟩. And a+b is the manifestation of the category "sum of 2 elements", whose category is ⟨x+y⟩.
  
  
• Truth and semantics.
  In model theory, truth is a semantic concept and is relative to the interpretive domain. But semantics has little to do with truth. Semantics has to do with the internal (mental) world, the world of all possibilities and where the concept of truth is meaningless.
  
  But the problem of mathematical truth is a consequence of the language used to express mathematical truth, not of mathematics itself. All the problems, paradoxes, obscurities of mathematics come from the language used. With a universal language, with a language of consciousness, with a profound language, the problems disappear, are clarified or simplified. Problems only appear at a superficial level.
  
  Model theory is a complex, Ptolemaic, superficial view. With MENTAL we have a Copernican, Platonist, absolutist, deep view. What is inconsistent is to use model theory as a theory of meaning. Meaning cannot be formalized. Any attempt in this direction is futile. Model theory is just one more attempt.
  
  Every formal expression always has an associated semantics, even if it is primary. Semantics "sustains" syntax, for syntax is a manifestation of semantics. Syntax is not self-sufficient. In MENTAL, model theory is the assignment of additional secondary semantics to primary semantics.
  
  
• Cardinality.
  Skolem said that the concept of cardinality needed to be revised. It has been suggested that it is possible to get rid of or ignore the notion of non-numerability.
  
  The philosophy we hold here is as follows:
  
  
  • The cardinality of a set is its number of elements. This number can be finite or infinite. If it is finite, the cardinality is quantitative. If it is infinite, the cardinality is qualitative. An infinite set can be included in another infinite set. For example, the even numbers have the same qualitative cardinality as the natural numbers, and the even numbers are included in the natural numbers.
    
    
  • The continuous is associated with synthetic consciousness. The discrete is associated with analytic consciousness. According to the principle of descending causality, the discrete is a manifestation of the continuous. And the finite is a manifestation of the infinite. For example, the natural numbers are a manifestation of the real numbers and the real numbers are a manifestation of the continuum.
    
    
  • Cantor's transfinite numbers are meaningless. They all have the quality of being infinite and quantitative relations cannot be established. This justifies that in the TLS indirectly all transfinite cardinals are made equivalent.
  
  
• Institution theory.
  The theory of institutions rests on category theory, a controversial branch of mathematics because it has been called complex, too abstract, and lacking in conceptual substance. Institution theory is even more complex than category theory. It is another complex and futile attempt to formalize semantics. Semantics can only be formalized by primary archetypes.

First-order predicate logic vs. MENTAL

First-order predicate logic, also referred to simply as "predicate logic" or "predicate calculus", is that which contemplates objects with a finite number of predicates and statements with quantifiers that reach only object variables. There are no predicate variables.

First-order predicate logic has been used to formalize most mathematical theories: set theory, number theory, group theory, and so on.

In first-order predicate logic there are two types of formulas: open ones, which have free variables (not affected by any quantifier) and closed ones, which have bound variables (those affected by some quantifier).

First-order predicate logic, as we know it today, has its origins in Frege, in the 19th century, with his Conceptography, where he presented the first system of predicate logic, although with a notation different from the present one.

First-order predicate logic has two metalogical theorems:

1. Gödel's completeness theorem, already mentioned above.
   
   
2. Gödel's undecidability theorem. A system is decidable when there is an effective method for deciding whether any formula of the language is logically valid or not. Propositional logic is decidable, but first-order predicate logic is undecidable.

If we compare MENTAL with the language of first-order predicate logic, we see that the latter is a limited language. It only allows logical relations between objects in a domain. But there are more relations than strictly logical ones. Logic is only one of the dimensions of reality.

TLS highlights the weakness of the first-order language itself. If model theory is formalized using MENTAL, with full semantics, then all formal systems are categorial and Skolem's paradox disappears.

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Addenda
History of model theory

• Model theory goes back to Charles Sanders Peirce and Ernst Schröder. But Löwenheim's theorem of 1915 is considered to be the official beginning of model theory, since no one until then had raised the issue of the various interpretations or models associated with a formal system. Semantics then began to play a role in logic.
  
  
• Skolem completed and generalized Löwenheim's theorem in 1920, which was thereafter called the "Löwenheim-Skolem theorem" (the top-down version) referring to transfinite cardinalities, in which he used the axiom of choice to reduce the universe of discourse of the original model to a numerable set of elements.
  
  Since the proof of this theorem depended on the controversial axiom of choice, in 1922 Skolem presented a special case of the theorem, without reference to this axiom, and close in spirit to Löwenheim's original theorem of 1915.
  
  
• In 1930 appeared Gödel's completeness theorem, which is considered a great result of model theory.
  
  
• In 1935, Tarski proved the ascending form of the TLS. Tarski was the great promoter of model theory. The term "model theory" was first used by Tarski in 1954. To Tarski we owe basic concepts of model theory, such as the concept of truth and the definition of formal theory.
  
  Tarski's semantic methods −developed in the 1950s and 1960s, together with his disciples at Berkeley University− radically transformed metamathematics, consolidating it as a strict science.
  
  Tarski's main idea consisted in replacing the symbols of a certain theory by expressions of another theory, so that the axioms of the first one became theorems of the other, thus establishing a formal relation between both theories. Tarski's model theory studies the properties that are inherited from one theory to another in the course of these transformations, comparing the scopes of these theories. Today, model theory can be considered the culmination of this strategy, which has strengthened the ideas and methods of algebra.
  
  
• Abraham Robinson's (1974) nonstandard analysis is also considered an important contribution to model theory.
  
  
• Abstract model theory" is a generalization of model theory that emerged in the 1970s. It studies the formal properties of extensions of first-order logic and its models. It is based on the concept of abstract logic, which is an abstraction of the concept of truth: it is a relation between sentences of a certain kind and structures of another certain kind. The starting point of the study of abstract models is Lindström's theorem. In 1974, Jon Barwise axiomatized the theory of abstract models.

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