INTENSIONAL LOGIC

"The meaning of the world lies outside of it" (Wittgenstein, Tractatus 6.41).

"A word is not a sign, a substitute for a thing, but the name of an idea" (Walter Benjamin).

"The sense determines the reference" (Frege).

"Of sense we say that it determines denotation or is a concept of denotation" (Alonzo Church).

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Previous Concepts and Background
Fregue: meaning and reference

Gottlob Frege −the considered father of modern logic−, in his 1892 article "Über Sinn and Bedeutung" attempted to formalize the semantics of linguistic expressions by means of the concepts of "sense" (sinn) and "reference" (bedeutung), trying to separate psychology (the subjective) from logic (the objective).

Sinn = sense, way of expressing meaning.
Bedeutung = reference, denotation.

Examples:

• "The planets of the solar system" is a way of referring to Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune.
  
  
• "The number of planets in the solar system" is a way of referring to the number 8.

According to Frege the following properties are satisfied:

• Different senses can refer to the same thing. For example:
  
  
  • "The morning star" and "The evening star" are two ways of referring to the planet Venus.
    
    
  • "Rational animal" and "human" are two ways of referring to human beings.
    
    
  • "Aristotle" and "Plato's most prominent disciple" are two ways of referring to Aristotle.
    
    
  • "Cervantes" and "The author of Don Quixote" refer to Cervantes.
    
    
  • The expressions 1+4 and 2+3 have different meanings and the same reference (5).
  
  
• The sense determines the reference. Two expressions that have the same sense must have the same reference.
  
  
• Expressions have meaning only by virtue of having reference. Without reference there is no meaning.

According to Frege's principle of compositionality, the meaning of a sentence is a function of the meaning of its components and their modes of combination (by the grammatical rules of language). In other words, the meaning of a sentence is completely determined by the structure and meaning of its components.

This principle operates in arithmetic expressions. For example, to calculate (a+b)×(c+d) we must first calculate < i>a+b, then c+d, and finally the product of both.

The principle of compositionality explains why we understand expressions that we have never heard: because we know the meaning of their components and because we know their combinatorial forms thanks to the fact that there is a universal grammar that manifests itself in particular languages. The principle of compositionality provides an important foundation in any theory of meaning.

Frege distinguished three types of expressions: nouns, predicates, and sentences:

• Nouns.
  A name designates an object. Its meaning is the name and its reference is the object designated by the name. For example, "Aristotle" designates or refers to Aristotle.
  
  
• Predicates.
  Frege called predicates "concepts," but without considering the traditional view of concepts as mental and subjective. He considered them functions that assign truth values to objects. The function "Predicate(object)" is V (true) if the object has that predicate, and F (false) if it does not. This simple idea opened an important door to the formalization of semantics.
  
  For example, in the expression "Venus is a planet", the object is Venus and "being a planet" is the predicate. It is expressed as "Planet(Venus)" and is V. Another example is "Frege is German" is expressed as "German(Frege)" and is V, and "Frege is Spanish" is expressed as "Spanish(Frege)" and is F.
  
  
• Sentences.
  A sentence has a meaning (it expresses an idea or thought) and its reference is its truth value. There is a function that assigns to each statement its truth value). Two sentences can have the same truth value and not have the same meaning. For example, 3+4 = 7 and 2+5 = 7 are both true (they have the same reference, V), but they mean different things.

Frege's model problems are:

• There are expressions that make sense but have no reference. For example, "The current King of France", "The largest even number", "Don Quixote".
  
  
• The sense of a sentence, its meaning, has nothing to do with its truth value. If the meaning of a sentence is its truth value, then all logical sentences would be synonymous, which is not true.
  
  Not every expression has a truth value. There are expressions that are not logical propositions and have no truth value. For example, the algebraic expression a+b or the definition of a function as f(x) = x+5.
  
  
• The same meaning can have several references, depending on the context. For example, the word "dough" can have a physical or a culinary meaning. They are polysemous words.
  
  
• Since semantics is also present in the reference of an expression, it is possible to ask, in turn, about its meaning and its reference. And so on, in a recursive process.
  
  
• The reference may be fixed or variable over time. For example, "The King of Spain" is variable because it needs a temporal context. In contrast, "The positive solution of the equation x²−4 = 0" is fixed, it does not depend on any context.

Frege was a semantic dualist, for semantics is in everything: in meaning and in reference. Frege attempted to formalize the semantics of linguistic expressions by means of functions of the type "sense(expression) = reference".


Intensional and extensional contexts

Frege came to the conclusion that it was necessary to distinguish between sense and reference of a linguistic expression because of the problem posed by the substitution of equivalent expressions (those having the same meaning). The expression A = B, where A and B are equivalent expressions, must be understood as meaning that the two expressions refer to the same object, but with different senses. They are two senses with the same reference, two ways of referring to the same object. According to Kant's conception, the expression A = B is synthetic (it provides information), and the expression A = A is analytic (it does not provide information).

Frege's substitution principle derives from the principle of compositionality and states the following: If A is part of C, B is not part of C, and B means the same as A the substitution of A for B in C produces an expression C' which has the same meaning as C.

If B is part of C, the meaning may be altered. For example, if in "Peter knows that Aristotle was Plato's most prominent student" we replace "Aristotle" with "Plato's most prominent disciple", the sentence becomes "Peter knows that Plato's most prominent student was Plato's most prominent student", which has a different meaning from the original.

Frege, by distinguishing between meaning and reference, made it clear that there are contexts in which an expression cannot be replaced by its equivalent because its meaning is altered. Today they are called intensional and extensional contexts. An extensional context is one in which the extension (the reference) is the only thing that matters. An expression can be replaced by another expression that designates the same thing. For example, in the mathematical expression 1+4 = 2+3, the two terms have different meanings, but are equivalent and interchangeable in another expression within an extensional context.

An intensional context is one in which one expression cannot be substituted for another expression designating the same thing because its meaning may be altered.

Intensional contexts are: modality, conditional, time, obligation, informational and cognitive. In general, expressions such as "It is known that ...", "It is necessary that ...", "It is possible that ...", "It is reported that ...", "It is said that ...", "It is believed that ...", etc. are intensional contexts and the substitution of one expression for another cannot be applied because it may alter the meaning.


Carnap: intension and extension

Carnap was one of the most prominent members of the Vienna Circle, a philosophical movement that sought to ground knowledge through the logical analysis of language and a criterion for identifying true and meaningful statements: experiential verification of the content expressed in the statements. This idea was inspired by Wittgenstein's Tractatus, according to which statements must reflect facts.

Carnap, in "Meaning and Necessity" [1956] observed that, under Frege's conception, there could be higher-order senses (sense of sense, etc.), with their corresponding references. And that Frege did not clearly explain the concept of "sense". He proposed to replace Frege's concepts of "sense" and "reference" by the concepts of "intension" and "extension", respectively.

• In a language it is necessary to distinguish between syntactic grammar (which specifies the possible combinations) and logical grammar (which specifies the admissible combinations, those that have meaning).
  
  
• The intention of an utterance is its meaning. To grasp this meaning it is necessary to carry out a logical-semantic analysis. Semantic analysis is based on the determination of the categories involved in the utterance.
  
  
• The intension of a predicate is the property represented by the predicate. The extension of a predicate is the set of objects that have that predicate. If predicates have the same extension, they are equivalent.
  
  
• The extension of a statement is its truth value. The determination of truth must be done empirically. Equivalent statements are those that have the same truth value.
  
  
• We must distinguish between logical truths (analytical and necessary) and factual truths (synthetic and contingent). Logical truths refer only to language itself.

Carnap introduced the concept of "state description" to a set of atomic sentences, each with a truth value. The intent of a sentence is a "range": the set of all state descriptions that have the value V (true). The extensions are the states (the truth values).


Church's intensional logic

Church's intensional logic [1951] is an attempt to axiomatize Frege's two types of meaning (sense and reference) by means of a general-purpose intensional logic, considering only the formal behavior of intension, not its concept. Church was the originator of intensional logic. He called Frege's "reference" "denotation".

Church used the notation Δ(s, d) to express the relation between sense (s) and denotation (d). This relation is usually called "presentation". With this notation one can express, for example, that the sense determines the reference (two expressions with the same sense have the same reference):

Δ(s, d₁) & Δ(s, d₂) → d₁ = d₂.

But this notation is not the most appropriate because there are expressions with sense that have no reference. Actually, since the sense determines the reference, one could use the notation Δ(s) = d, where d could be the empty expression.

Church also adopted Frege's principle of compositionality: the sense and denotation of a compound expression is a function of the sense and denotation of its components and their combinatorics.

Church's intensional logic is very complex because it includes a theory of types developed by him in 1940 [Church, 1940]. Church's type theory is based on two basic categories: 1) the category (ι) of individual terms; 2) the category (or) of truth values of sentences. Additionally, there is a third category (αβ) which are functions between two basic categories of the language: from expressions of type α to expressions of type β. For example, (oo), (oι), (o(oι)), etc.

• Formulas are understood as terms that have truth values.
• Predicates are formalized as functions from individuals to truth values, as Frege did.
• An individual, truth-value, or function has meaning ("concept", in Church's terminology).

Montague's semantics

Richard Montague, in the 1970s, extended and formalized Carnap's ideas. Montague's semantics is a theory of the semantics of natural language (English) and its relation to syntax. Montague treated intensions as functions, like Carnap.

According to Montague, the study of natural language belongs to mathematics and not to psychology. His system he presented as a "universal grammar". Montague's grammar is a formal system for describing natural language through a combination of several theories, including intensional logic:

• Natural languages are formalizable because they have the same logical-mathematical foundation as artificial languages.
  
  
• The purpose of semantics is to characterize the notion of true sentence (under a given interpretation), rather than the notion of "meaning".
  
  
• The intension of a sentence is a function of the context and returns the extension of the expression in that context (its truth value). The extension of a name is the object referenced by the name.
  
  
• There is a close relationship between syntax and semantics. In this relationship, the principle of compositionality is used: the meaning of a sentence is a function of the meanings of its components and their combinatorial forms. Each syntactic rule of composition is accompanied by a semantic rule.

The entity types Montague used were only two: objects and truth values. These types could be combined to create higher-order types.

In Montague's intensional logic, for every linguistic expression α there is another expression ^α denoting the intension of α. The prefix operator (^) is recursive, i.e., higher-order intensions can be expressed: ^^α, ^^^α, etc. Montague introduced the opposite operator (^v), such that ^v^α = α. This second operator can be applied to any linguistic expression denoting an intent to obtain the corresponding expression.

Following the Chomskyan revolution in the mathematical formalization of syntax, Montague's approach was also revolutionary in applying mathematical methods to semantics and its relation to syntax. But Montague's original theory was very complex and has been subsequently modified and extended by linguists, logicians, and philosophers.


Semantics of possible worlds

Semantics of possible worlds (SMP) is a family of methods that are used to model a wide variety of intensional contexts. SMP was a generic or universal idea that drove the development of numerous fields: linguistics (semantics of natural languages), philosophy, logic (intensional logic), computer science, and artificial intelligence. Sense and reference in SMP:

• The meaning of a linguistic expression is the same in all possible worlds. For example, "the richest man in the world" has the same meaning in all possible worlds, but different reference in each possible world.
  
  
• The sense of an expression (noun, predicate or sentence) does not determine its reference, but depends also on the possible world.

According to Carnap, a possible world is a set of states and combinations of states, where a complete description of states describes a possible world. "Complete description" means that it is known that for every predicate and for every object whether or not the predicate-object relation is fulfilled. A possible world is a consistent world, i.e., one that can be described without contradiction.

According to Wittgenstein, a possible world is a set of possible facts. The world is not a set of objects, but of facts. In this way he related language and world. Through language we conceive the world as it is, as it is not, and as it could be. The limits of the possible are determined by language.

Although the concept of the "possible world" has been part of philosophical terminology since Leibniz, modal logic has been the great driver of SMP.

In the 1960s, Saul Kripke presented the first formal SMP-based model for modal logic:

• A statement is necessary if it is true in all possible worlds.
• A statement is possible if it is true in some possible world.
• A rigid designator is a name that designates the same entity in every possible world where that entity exists.

For example, the expression 3+4 = 7 is true in all possible worlds (it does not depend on the context). In contrast, the statement "The number of planets is 8" is not true in all possible worlds.


Intensional Logic

We must differentiate between extensional logic and intensional logic:

• In extensional logic, only the truth value of logical expressions is considered. An expression has a truth value that is determined by the truth values of its components. The context is extensional and governs the principle of substitution of extensions: if two expressions have the same truth value, then one expression can be substituted for the other in a third expression without altering its truth value.
  
  
• In an intensional logic, the meaning of expressions is considered. The context is intensional. The meaning of an expression is determined by the meaning of its components. And the principle of substitution of one expression for another in a third expression does not apply because the meaning of the latter expression may be altered.
  
  The meaning of an intensional expression depends on the context under consideration. The value of the intensional expression in a particular context is the extent of the intensional expression in that particular context.

In classical logic, a statement is either true or false. In intensional logic a sentence is ambiguous when the context is not specified, so it is neither true nor false.

The sentences of classical logic are superficial, analytic and rational; they are extensional and have a truth value. The sentences of intensional logic are deep, synthetic and intuitive; they are intensional and cannot be assigned a truth value. They correspond to the two modes of consciousness.

Intensional logic is an extension of mathematical logic that makes it possible to formally describe entities whose meaning depends on the context. In general we speak of "indexical expressions", which refer to a certain state of affairs in a certain context.

Classical logic (propositional and first-order logic) is extensional and intension plays no role. Intensional logic is the most important extension of classical logic. Classical logic can be considered to be a particular case of intensional logic.

In first-order predicate logic, quantifiers affect elements. In intensional logic, quantifiers additionally affect the values of the properties of those elements.

The applications of intensional logic are mainly the semantic analysis of natural languages and the analysis of philosophical problems.

Examples of intensional logics are:

• Modal logic, the logic of necessity and possibility.
  
  
• Temporal logic, where the intensional context is time.
  
  
• The logic of relevance, the logic that captures certain aspects of implication that are ignored by the condition operator of classical logic.

Intensional logic violates principles of classical logic, principally the law of substitution of identities:

If a = b and P(a), then P(b) may not be satisfied.

For example, if we have the expressions "2+6 = 4+4" and "Peter calculates 2+6", then the latter expression is not equivalent to "Peter calculates 4+4". In classical logic every expression has an extensional interpretation, with a fixed truth value. The logic of "possible worlds" is an intensional logic where the truth values of the expressions vary in each possible world.

The intensionality of an expression A in a possible world M is the extension of A in that world M: Intensionality(A, M) = Extension. Two expressions A₁ and A₂ have the same intension if they have the same extensions in all possible worlds. The intension of a name N in a possible world M is the object it denotes in that world: Intension(N, M) = object.

The origins of intensional logic come from natural language research, where a sentence has a different meaning depending on the context. Many natural language sentences are ambiguous, that is, they can be interpreted differently depending on the context in which they are used. This has led many scientists to believe that natural languages cannot be formalized from a mathematical point of view.

For example, the sentence "Madrid is the capital of Spain" is true today, but in the past (before 1561) it was false. That is, its truth value is a function of the time context. We can also consider sentences that depend on several contexts (such as time, place, audience, etc.). For example, the sentence "The value of the temperature" depends on three contexts: place, day and time. The contexts are called "dimensions" or "possible worlds".

The context dependence of natural languages is due to practical reasons. When we speak there is an implicit context, which avoids having to express it at every moment.

Another feature of natural languages is that there are operators that change the context, such as "yesterday", "tomorrow", "north", "next", etc. We say, for example, "the temperature yesterday", "the north of Spain", etc.


Intensional and multidimensional programming

Intensional programming is programming based on intensional logic. Its main characteristics are:

• The evaluation (or computation) of an expression depends on the context (dimension or possible world) under consideration. The context varies over time and is represented at each point in time by a value. The standard model for intensional programming is called "eduction" (eduction) [Wadge et al. 1985], an abstract model based on the evaluation of expressions with respect to the context under consideration.
  
  
• The context can be modified by means of intensional operators. Higher order operators can be defined from these operators.
  
  
• Entities and contexts have infinite values. They are called "streams" and are series of data that are successively generated as time flows. The streams can be manipulated as if they were ordinary values. They can be added, multiplied, applied if-then-else, etc.
  
  
• Programming is basically declarative, but can also include imperative features. At the declarative level it avoids having to use an index to reference an element of a sequence.

Intensional programming makes it possible to formalize the dynamic aspects of certain systems that are dependent on contexts that change over time, such as physical phenomena that depend on one or more parameters (time, space, temperature, etc.). For example, we have an iron bar to which we apply a heat source at one of its ends. The temperature at each point of the bar is a function of position (distance from one of its ends) and time.

Applications of intensional programming are: interactive programming, real-time systems, multidimensional spreadsheets, attribute grammars, multidimensional database systems, parallel programming, adaptive programming, signal processing, image processing, etc.

Lucid [Wadge et al. 1985] was the first intensive programming language. It was created in 1974. Its main features are:

• It is oriented to declarative programming with streams (data flows). The entities "flow" (change their values) as time flows.
  
  
• An intensional variable is a stream that is defined by its first elements and the rules to create the following elements from the previous ones. The basic operators are: "first" (sets time t to zero), "next" (next time, t+1) and "fby" (followed by, expression of the next element). For example:
  
  
  • The statement n = 0 fby n + 1; defines the entity n (the natural numbers) as the data stream (0, 1, 2, ...).
    
    
  • The statement f = 0 fby 1 fby f + next f; defines the Fibonacci sequence (0, 1, 1, 1, 2, 3, 3, 5, 8, ...), where each element is the sum of the previous two.
  
  In these examples the defined functions have no arguments and denote infinite sequences of values over the time dimension. There are also no time indices, they are implicit.
  
  
• Intensional functions can be defined that operate on streams and return streams as results. Composite (higher-order) functions can also be defined.
  
  
• It differs from imperative programming languages in that it does not use the assignment operator to change the values of variables.

The new version of Lucid is GLU (Granular Lucid) [Ashcroft and others, 1995]:

• It is a multidimensional programming language. It supports multidimensional entities. Each multidimensional entity has an identifier (context or dimension name) and an associated stream.
  
  In the original Lucid language, the only existing dimension was time. GLU allows the user to specify their own dimensions, appropriate to the problem or system to be modeled.
  
  
• Operators and intension functions can be parameterized so that the dimension can be specified.
  
  
• Allows declarative and imperative programming. The declarative core is from Lucid. So GLU is a procedural extension of Lucid.
  
  
• It is a functional-intensional language. It allows defining declarative functions (without indexes) and recursive functions (with indexes). User-defined functions act on multidimensional entities and can return multidimensional entities as result.

For example, if we have a variable x that varies in dimension t according to the indices or points (0, 1, 2, ...), for example, (3, 5, 1, 6, ...) and we want the cumulative sums (3, 8, 9, 15, ...), it is expressed by means of

s = x fby.t next.t x + s

which indicates: s is the initial value of x followed (in dimension t) by the next value of x (in dimension t) plus the current value of s. This is an intensional expression used to describe all the cumulative sums of x along the dimension t. The contexts are the points or indices, in dimension t, of the inputs (for x) and of the results (for s).

Generalizing for any dimension d, we define a so-called "dimensional abstract function" (d.a.f.), a function that has the dimension as an additional parameter.

sum.d(x) = s where
  s = x fby.d next.d x + s;
end;

This function can be applied to any dimension, e.g., sum.a(e), where a is a concrete dimension and e is a variable varying in dimension a.

There are also:

• Multidimensional abstract variables. These are variables that vary according to the dimension being specified.
  
  
• Abstract dimensional constants. They correspond to null operations (without arguments). For example, the constant 1 denotes a sequence of infinite equal elements: (1, 1, ...).

Intensional Logic in MENTAL
Sense, Reference and Meaning

Frege did not arrive at a clear definition of "meaning". The only formulation he made of this concept was "mode of expressing or presenting meaning," which is too vague to constitute a definition. There is also a great deal of confusion regarding the concept of "meaning" of a linguistic expression in relation to sense and reference. Here are some opinions:

• The meaning is the sense.
• The meaning is the reference.
• Meaning is double and is in the sense and in the reference (semantic dualism).
• Meaning is the relation or connection between sense and reference.
• Meaning is prelinguistic and has no relation to either sense or reference.

The concept of "reference" is also ambiguous, for doubts arise, e.g., can reference also be an expression or is it related only to the empirical world? And there is also some confusion between "reference" and "denotation". According to some authors, they are different concepts:

• Denotation is more general than reference. Denotation is the general reference, regardless of possible worlds.
  
  
• Reference is the denotation for a particular possible world. So, according to this view, meaning does not determine reference, for it depends on the possible world under consideration.

The principle of downward causality clarifies these concepts and the relation between sense and reference of a linguistic expression. Indeed, a linguistic expression has three aspects:

1. Its meaning, which is subjective, mental.
2. Its representation, which can be oral or written, which is objective.
3. Its denotation, which is the thing designated, which may in turn be another expression.

Representation connects meaning (the upper) and denotation (the lower).

On the other hand, meaning has been tried to be formalized mathematically by a function between linguistic expression and reference. And reference as only two possible types of values: objects (denoted by names) and truth values.

But the meaning (or semantics) of a linguistic expression cannot be formalized because of its deep character. The only way to formalize meaning is through the archetypes of consciousness, which are present in everything: in the meaning, in the reference and in the representation of every expression; in the internal and in the external world. Consciousness connects everything.

Meaning is mental, subjective, so it is not formalizable. But paradoxically it can be formalized with something deeper than the mind and which is not subjective because it is universal: consciousness. Meaning is a pre-linguistic entity and is grounded in consciousness. The formalization of semantics goes beyond mathematics. It is based on the archetypes of consciousness.

In the face of the complex and unnatural attempts to formalize the meaning of linguistic expressions by authors such as Frege, Carnap, Church and Montague, the formalization with MENTAL is the simplest possible and the most powerful.


Sense and reference

In MENTAL a clear distinction is made between the concepts of sense and reference:

• The sense of an expression is the semantic content of the expression, reflected in the semantic primitives used.
  
  
• The reference is the result of the evaluation of this expression or the expressions described by this expression.

The "meaning", defined by Frege as "form of expressing meaning", in MENTAL we replace "form" by "structure", the structure based on the primitives that constitute a universal grammar.

• The reference of an expression can be the expression itself if that expression is self-evaluating. For example, the reference of a name is the expression designated by that name, or the name itself if it does not designate any expression (the name is self-evaluating).
  
  
• The evaluation of an expression is, in general, a recursive process, where there may be several evaluation steps.
  
  
• The reference of a logical expression or relation is its evaluation, which is its truth value.
  
  
• Every well-formed expression has meaning and reference. They can even be imaginary expressions such as (i^2 = −1), whose sense and reference is the expression itself.

It is not enough to consider names and truth values to express the semantics of a linguistic expression. It is necessary to specify its structure. MENTAL is a universal language and a universal grammar that allows expressing the deep structure of sentences. Montague tried to create a universal grammar, but he created a very complex theory. On the other hand, MENTAL is very simple and allows to reflect that structure in a clear and transparent way. In this structure, syntax and semantics are united, they are two aspects of the same thing.


Intensional and extensional expressions

In MENTAL, intensional and extensional expressions are defined as follows:

• An intensional expression is a descriptive expression that refers to one or more expressions, which may themselves be descriptive.
  
  
• An extensional expression is one that refers to itself, i.e., it is self-evaluating.

The intensional is associated with the generic and the plurality of entities (the extensional). The intensive is the descriptive. The extensive are concrete expressions. Intension is the synthetic and compressed. Extension is the analytical and expanded. They are the two modes of consciousness.

Concepts can be considered mental entities that manifest themselves as generic expressions. According to Pavel Materna [2014], concepts are hyperintensional objects.

An intensional expression is generic and can be "manifested" at the surface level as an extensional sentence in a given context. For example, the sentence "It is raining" is intensional because the context has not been specified: neither the place nor the time (the moment). This can be expressed in MENTAL as a generic expression with two parameters:

⟨( it rains/(place/x time/y) )⟩

A particular expression would be obtained by giving values to the parameters.

• There are generic expressions that are neither true nor false. For example, the function ⟨( f(x) = x+5 )⟩.
  
  
• There are expressions that are true in terms of the axioms of the language. For example, ⟨( x+y ≡ y+x )⟩.
  
  
• There are expressions that are true by virtue of their definition. For example, ⟨( x/man → x/mortal )⟩.

Intensional logic

Intensional logic is logic that deals with intensions. But since it has not been defined exactly what intension is and its relation to meaning, it has not been fully and consistently formalized due to the lack of solid and clear foundations, not only logically, but in the other deep dimensions of reality.

But intensional logic can be formalized with MENTAL thanks to generic expressions (parameterized or not), which allow to describe several or infinite expressions, together with the rest of the semantic primitives.

• Intensional logical expressions.
  These are generic expressions in which the primitive "Condition" is involved. Examples:
  
  
  1. The parameterized generic expression
     {⟨( n ← n>5 )⟩}
     describes the infinite natural numbers greater than 5.
     
     
  2. The generic non-parameterized expression
     ⟨( n>5 → (n = 5) )⟩
     represents infinite expressions:
     
     ( n=6 → (n = 5) )
( n=7 → (n = 5) )
( n=8 → (n = 5) )
     ...
  
  
• Quantifiers.
  The traditional quantifiers of first-order predicate logic (those affecting individual elements) and quantifiers affecting the values of properties of those elements are unified. Both are specified by parameters in generic expressions.
  
  
• Possible worlds.
  In MENTAL possible worlds are implicit. Possible worlds are the possible interpretations of expression names. In contrast, the operators of the universal semantic primitives have nothing but one interpretation, their meaning is absolute. MENTAL is the Magna Carta of possible and imaginary worlds. The limits of the possible are determined by language.
  
  
• Logical truths.
  MENTAL axioms and theorems are equivalent to Carnap's logical truths. They are independent of the contents (values) of names (parameters).

Intensional programming

One of the problems of programming languages is the dichotomy between intensional (declarative) and extensional (imperative) languages. In MENTAL, as an operational, descriptive and declarative language (with a common foundation), we can describe infinite expressions, for example, infinite sequences. One of the ways is to use the potential substitution operator (=:), instead of immediate substitution (=). Examples:

1. Sequence of natural numbers:
   
   (n := ( 1 )) // initial value
(n =: (n ∪ (n\(n#) + 1)
   
   More abbreviated, (n = ( 1... ))
   
   
2. Fibonacci sequence.
   
   (f = (0 1)) // initial value
(f =: (f ∪ ( f\(f#) + f\(f# - 1))))

Multidimensional programming

Multidimensional programming is a particular case of intensional programming when the contexts are dimensions, that is, when there are elements that vary according to different dimensions.

Examples:

1. Cumulative sums (of order n) of a sequence:
   
   
   • Operationally and recursively.
     
     ⟨( sum(x n) = (0 ← n=0 →' (sum(x n−1) + xn) )⟩
     
     
   • Descriptively.
     
     ⟨( sum(x n) = (x\1 + ... + x\1;b>n) )⟩
   
   
2. A multidimensional variable that varies along the dimensions a and b can be specified extensively. For example:
   
   
   x/(a=0 b=0) = 1
x/(a=0 b=1) = 3
   ...
   
   Cumulative sums (of order n) along dimension , for context a=k:
   
   (x/{a=k b=1} + x/{a=k b=2} + ... +x/{a=k b=n}) eq.
([x/(a=k b=[1…n]])
   
   This expression can be parameterized to create an abstract dimensional function, where the dimension is just another parameter.

Conclusions

In short, with MENTAL, as a universal language, it allows to formalize sense and reference of the linguistic expressions by means of the archetypes of the consciousness. And it allows to implement intensional logic, intensional programming and multidimensional programming.

According to the principle of generalization, the concepts of "sense" and "reference" as well as the concepts of "intension" and "extension" have to be general, i.e. apply not only to logic.

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Addenda
Theories of meaning

Frege's concepts of meaning and reference have been fundamental in the philosophy of language and have been the origin of investigations and reflections on the question of meaning. Frege is considered the father of analytic philosophy of language and philosophical semantics.

There are different theories about the meaning of a linguistic expression:

• According to conceptualism (or mentalism), the meaning of a linguistic expression is the corresponding mental content.
  
  
• According to RTM (Referential Theory of Meaning), meaning is reference.
  
  
• According to CTM (Correspondent Theory of Meaning), meaning is a relation between expressions of a language and entities that are independent of language.
  
  
• According to behaviorism, the meaning of a linguistic expression is the response produced by that expression when it is perceived as a stimulus.
  
  
• According to internalism (or internalist semantics) whose main representative is Chomsky [2000]: Expressions have no relation to objects in the world. Sentences are neither true nor false. Names are arbitrary and refer to expressions.

And according to authors:

• For Plato, the meaning of a word is what that word designates. The word is like a label assigned to the object.
  
  
• For Saussure, meaning is the concept or content of the linguistic sign.
  
  
• For Paul Grice, the correspondence between linguistic expression and designated entity is only one aspect of meaning. One must take into account the "intention" of the speaker when employing an expression.
  
  
• Max J. Cresswell [1982] defines meaning in the negative sense. "If a sentence A is true and B is false, then A and B do not mean the same thing."
  
  
• David Lewis [1972]: "To say what meaning is, we must first ask what meaning does and then look for something that does that."
  
  
• For Russell, the meaning of an expression consists exclusively in reference to an entity. To signify is to reference. Russell was a referentialist. According to Russell's theory of logical atomism, the world consists of atomic facts which are the referents of atomic statements.
  
  
• According to the theory of language of the first Wittgenstein (the one in the Tractatus), there is an isomorphism between a fact and a proposition. A proposition is a representation (or figure) of a fact. Propositions must only reflect facts; otherwise they are meaningless. The guide to understanding the world lies in logic.
  
  
• According to the second Wittgenstein (the one of "Philosophical Investigations"), the meaning of a linguistic expression is its use.
  
  
• According to the verificationist theory of meaning, the neopositivist theory of the Vienna Circle, every true statement must be verifiable by experience.
  
  
• According to Donald Davidson, at its most basic level, the meanings of words and sentences are derived from perception and interaction with objects in the physical world.
  
  
• Quine [1960] rejected the concept of "meaning" and made a critique of intensional logic. His arguments were:
  
  
  • The meanings of natural language entities and expressions are fuzzy, with no clear boundaries between them. In contrast, individuals, truth values, and sets of individuals, which underlie extensional logic, are the true objects.
    
    
  • There is no rigorous concept of intension, so intensions cannot be included in a rigorous science such as logic.
    
    
  • It is an obscure issue to determine whether or not two natural language sentences have the same intension.
    
    
  • Semantics should be replaced by pragmatics. The meaning of a linguistic expression is its use, as the second Wittgenstein said.
  
  
• Hilary Putnam was a proponent of externalism. In "The Meaning of Meaning" [1975] he claims that meaning is fundamentally derived from the external world from what he calls "natural kinds" (natural kinds). By means of a thought experiment called "the twin Earth," he argues that natural kinds have a hidden structure. There may be one world where the formula for water is H₂O and in another it is XYZ. Therefore, meaning depends on the external world and is not only mental. Mental meaning alone does not determine reference. Meaning is not something absolute, but dependent on the possible world.

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