"Arithmetic has been the starting point of the train of thought which has led me to my Conceptography" (Frege).
"The primitive components must be taken as simple as possible if order and clarity are to be produced" (Frege).
"When Frege introduced quantification, he illuminated three subjects: logic, language, and ontology" (Quine).
Frege's Conceptography
Gottlob Frege −considered the greatest logician since Aristotle− is the father of modern logic (mathematical logic) and analytic philosophy (philosophy based on the logical analysis of language). In 1879 (at the age of 31) he published a revolutionary work entitled "Begriffsschrift" (Conceptography, Ideography, conceptual notation or conceptual writing), subtitled "A language in formulas of pure thought, in imitation of arithmetic" [Frege, 1972]. In it he presented a new logical notation for representing concepts and propositions, as well as new concepts, including predicates and logical quantifiers. It was his first revolutionary text on mathematical logic, ushering in a new era in logic, a discipline that had remained virtually unchanged since Aristotle. In turn, this work was an attempt to create a universal, precise and formalized language for application in science and philosophy.
In the preface to this work, Frege points out that the system presented in it was the realization of Leibniz's dream of a universal language for science, a language in which the concepts and truths of science could be precisely expressed symbolically. But in this work he only sketched his system. A more elaborate system appeared in the two volumes of Grundgesetze der Arithmetik (The Basic Laws of Arithmetic), published, respectively, in 1893 and 1903.
Frege's philosophy
Logicism.
Frege was an advocate of logicism, the idea that mathematics is reducible to logic, because logic is the most abstract thing that exists and connects with the deepest laws of thought. But his defense of logicism was limited to arithmetic. His work "The Basic Laws of Arithmetic" was the record of the intent of his logicist project.
To carry out his logicist project, Frege developed as a foundation a semantic doctrine, with a number of key notions and which he reflected in three articles, "Function and Concept" (1891), "On Sense and Reference" (1892) and "On Concept and Object" (1892).
His logicist project finally failed because of its inconsistency, because of Russell's famous paradox. During the last stage of his life, Frege abandoned logicism and made a Kantian turn, trying to show that arithmetic (and mathematics in general) should be based on geometry and not on logic. His attempt was left unfinished due to his death in 1925.
Rejection of psychologism.
Frege rejected psychologism in logic and mathematics. Psychologism (or subjectivism) is the view that all knowledge is just subjective ideas abstracted from individual experience. Mathematical psychologism holds that mathematical objects (such as numbers) are abstractions from sensible experience. Logical psychologism is only laws of "right thinking". The empiricist philosopher John Stuart Mill was the most prominent among the psychologists for taking his conception to the extreme of considering mathematics and logic as disciplines subordinate to psychology.
Frege distinguished between the laws of psychological thinking and the laws of logic. The object of the latter is to investigate truth, true being.
The Third World.
Logicism and anti-psychologism are framed in Frege's theory of the three worlds. According to Frege, there are three worlds or realms: 1) the objective, sensible, physical world; 2) the subjective world of ideas, mental processes, and psychological representations; 3) the world of logical concepts and entities, an objective, timeless world that is independent of individual subjective minds. For example, numbers are logical-type entities residing in the Third World, which are independent of all psychological processes and representations.
For Frege, abstract formal expressions have meaning, are objective, and belong to the Third World. He rejected the formalism of arithmetic, according to which number and arithmetical expressions are a mere set of meaningless, meaningless symbols. With the idea of the Third World, Frege aligned himself with Platonist realism.
General characteristics of Conceptography
Universal language.
Frege intended to create, in the spirit of Leibniz, a Lingua Characteristica Universalis and not a mere Calculus Ratiotinator. The language should be universal in the sense that the universe of discourse would be all things, a language that would make it possible to connect in a single domain fields that were previously separated and that could be extended to fields that needed a formal language.
Frege was the first author in history to attempt to create a formalized universal language for science. Previously, Descartes (with his Mathesis Universalis) and Leibniz (with his Lingua Characteristica) had intuited it, but failed to make it effective. Frege was the first to attempt it seriously. Following Leibniz's philosophy, he used symbols to express elementary concepts and combinations of these symbols to express compound concepts.
Ideal logical language.
Frege was a universalist who sought through logic the unification of mathematics, linguistics, philosophy, and the structure of reality. For Frege, the purpose of logic is not to calculate (make inferences), but to model and understand reality. For this purpose he created an ideal logical language, the Conceptography, a formal language for "pure thought". Pure thought is based exclusively on general forms, dispensing with any particular content.
Natural language is ambiguous, unsuitable for science and philosophy, so Frege was encouraged to create a precise language, a language of purely logical foundation, a perfect formal language. His first approach to this ideal formal language was to attempt to formalize arithmetic by means of logic, because he believed that arithmetical entities were logical entities. In doing so, he realized that the concepts he handled were general, that they went beyond arithmetic, so he extended his system to the representation of concepts of all kinds and not only mathematical, so he finally created a language applicable to the formal sciences in general. "Arithmetic was the starting point of the train of thought that led me to Conceptography." Frege was a logician who saw in logic the universal paradigm that would allow him to create a universal language. This idea of logic as the foundation of perfect language would later be followed by Russell in an attempt to found mathematics as well.
Analytic language.
Kant maintained that geometry and arithmetic are a priori synthetic sciences, requiring intuition. Frege wanted to show that arithmetic is analytic, that is, that its fundamental concepts are exclusively logical, so that all arithmetic could be defined in purely logical terms, reduce it to logical laws.
Axiomatic language.
Frege wanted to establish a formal axiomatic system for arithmetic, as in Euclid's geometry. From a reduced set of axioms one could derive the totality of the truths (theorems) of arithmetic. Later this system would be extended to all mathematics.
Frege formalized the notion of demonstration in terms that are still valid today: a demonstration is a finite sequence of sentences such that each of them is either an axiom or follows from the previous ones by a valid rule of inference.
Scientific language.
In the preface to his Conceptography, Frege states that this language can be extended to other sciences that would require a formal language, adding domain-specific concepts and axioms. The language would be especially applicable in particular to geometry and physics. "The passage to pure motion theory, and even to mechanics and physics, could follow from here" [Frege, 1972].
Plain language.
Frege sought primitive logical concepts of the greatest simplicity and in the smallest possible number. From these fundamental primitive concepts, by combinatorics, the derived, compound, and complex concepts would be constructed. "The primitive components must be taken as simple as possible if order and clarity are to be produced" [Frege, 1972]. "... our effort must be practically to go back to the logically simple, which as such is not properly definable" [Frege, 2003].
Conceptography as science.
Frege held that his system of Conceptography was more than a Lingua Characteristica and a Calculus Ratiotinator. That it was a true science in itself, where every theorem of this science could be proved by its primary concepts and its fundamental logical axioms
Creative language.
Frege not only intended to create a symbolic language that would reflect already existing concepts (and their relations), but should also allow for the formation of new concepts and new relations. Frege believed that his conceptual notation could help philosophy break the stranglehold of the word over the mind in an attempt to capture the deep, the hidden in natural language.
Analytic philosophy.
Frege wanted his language to be not merely a descriptive tool, but a tool of scientific-philosophical analysis of ordinary language, to help detect what is hidden or implicit, to overcome its deficiencies and ambiguities, and to search for its essential concepts and their relations. This line of inquiry would later be called "analytic philosophy".
A language distinct from Boolean algebra.
Conceptography differs from Boolean algebra in three respects:
Language.
Boolean algebra functions only as a Calculus Ratiotinator. Frege's language was further intended to be a Lingua Characteristica Universalis in Leibniz's sense, a universal language.
Relation between mathematics and logic.
Boole performed a mathematical analysis of logic. Frege proceeded in reverse: he pretended to base all mathematics on logic, for he held that logic does not depend on mathematics, but that it is mathematics that derives from logic.
Semantics.
Boole's logical calculus is formal, syntactic. The proposition has no content, it is empty, and its semantics is reduced to a mere truth value. On the other hand, with Frege, every proposition has philosophical roots: it has structure and conceptual content, that is, it has a sense, a meaning associated with it. Boolean algebra created a symbolism for already existing concepts. Conceptography created new concepts and a new symbology.
Therefore, the expressive power of Conceptography is superior to Boolean algebra.
General principles
The principle of compositionality.
The meaning of a compound expression is determined by the meaning of its components and the relationships between them. If one component is exchanged for another with a different meaning, the meaning of the sentence changes.
Meaning and reference.
These are the two most important concepts in Frege's philosophy, although they are still debated topics today, especially the concept of "sense". Sense and reference are two distinct aspects of meaning. The meaning of an expression is its form and content. The reference is the object to which the expression refers.
The reference of a proposition is its truth value. The reference of all true propositions is the True, and the reference of all false propositions is the False. Two different expressions have different meanings, but they can have the same reference. For example, 22 and 1+3, have different senses and the same reference, 4. Not every sense has reference.
The principle of context.
Only in the context of a proposition as a whole do we know the meanings of its component words. Isolated words have no reference, complete statements do.
Connection between thought and language.
According to Frege, there is a close connection between the elements of a thought and the sentence that expresses it, although two different sentences can express the same thought. At the linguistic level, it is a correspondence between syntax and semantics.
Usage and mention.
Frege distinguishes between "use" and "mention" of an expression. Use refers to the designated object. Mention refers to the expression itself (the material or formal expression).
Detailed characteristics of Conceptography
Functions and objects.
Frege's ontology consists of two fundamental types of entities: functions and objects. A function is an "unsaturated" entity, that is, incomplete, with one or more variables that have to be completed with their corresponding arguments (objects) for their "saturation". An object is a "saturated" (complete) entity. Within the objects are included the logical values True) and False.
According to Frege, ordinary grammar has hindered scientific development. In this sense, he replaces the grammatical concepts of subject and predicate by those of argument and function, since these concepts are more general and flexible.
Frege took the concepts of function and argument from mathematics, but generalized them to apply to statements: a function is a statement in which one or more variables appear. When values are assigned to the variables (the arguments), there is a resulting statement, which can be true or false. An example of a function is "x is a planet in the solar system", where x is the variable that can take different arguments. If x=Earth, the statement is true. If x=Moon, the statement is false.
The set of arguments (objects) that makes a function true is called the "path" or "course" of that function. In the example function above, its path is "Mercury, Venus, Earth, Mars, Jupiter, Saturn, Neptune, and Uranus". If two functions have the same path they are equal to each other.
In an expression, whether one part is an argument and the other a function depends on the point of view you choose. For example, in "Peter hit Marco", you can consider it as the function "x hit Marco" with x=Peter, or as the function "Peter hit x" with x=Marco. One could even consider the two-variable function "x hit y", with x=Pedro and y=Marco.
Variables and constants.
Frege distinguishes between variables and constants. Variables (symbolized by letters) can represent different contents at different times. This makes it possible to express general laws, such as the commutative law of addition: x+y = y+x. There are object-related variables and function-related variables.
Constants (which are special symbols) have a completely determinate meaning such as the number 12 or the arithmetic operators "+" and "−".
Concepts.
A concept is a special type of function. A concept is a function that maps all arguments to a truth value. For example, x>>2 defines the concept to be greater than 2. If x=5, the statement is true and if x=1, the statement is false.
Another example is the function "x conquered Gaul", which is a concept because you can assign values to x and the result is a true or false statement. If x=Caesar, the sentence is true , and if x=Galileus, the sentence is false.
An object that makes the concept true is said to "fall" under that concept. For example, object 3 falls under concept x>>2, object 2 falls under concept x 2=4, and object 5 falls under the concept P(x) which indicates "x is prime".
Extension of a concept.
The extension of a concept is the set of objects that fall under that concept. For example, the extension of the concept "blue" is the set of all blue things, the extension of the concept "3" is the set of objects consisting of 3 elements, the extension of the concept "truth" is the set of all true objects, the extension of the set "prime" is the set of prime numbers, the extension of the concept ">2" are the numbers 3, 4, 5, etc.
Predication as a functional application.
Predicate is a particular case of functional application. That is why functional notation is used, as in P(x) (x is prime). So a predicate is a function that maps its arguments to a truth value. For example if M(x, y) represents the predicate "x is greater than y", then M(5, 3) is true and M(3, 5) is false. Therefore, a predicate is a concept, and this is formalized as a function.
Logical concepts.
Frege uses the following logical concepts:
Judgment.
A judgment (assertion, assertion or assertion) is expressed by the symbol "⊢" to the left of an expression. Judgments are assertions or theorems (deductions). The symbol consists of a vertical and a horizontal bar. The horizontal bar (content bar) indicates a content: it is a mere combination of concepts by way of representation, but not a judgment. The vertical bar (judgment bar) indicates a judgment, a statement. Judgments can be particular or universal. An example of a particular judgment is ⊢3×7 = 21. An example of a general judgment is ⊢a+b = b+a.
All judgments (which are represented by the symbol ⊢) indicate the same predicate: "it is the case", "it is a fact" or "it is true". For example, "Archimedes perished at the capture of Syracuse" is transformed (by including the judgment symbol) into "It is a fact that Archimedes died at the capture of Syracuse."
Condition.
Use the following symbol:
The horizontal bar is the content bar, and the vertical bar is the condition bar. Material implication is defined as "If B then A", with B being the antecedent and A being the consequent, and with no need for causal relationship between A and B. It is equivalent in modern notation to A→B. This is the only rule of inference (the modus ponens). The other modes of inference are reduced to the modus ponens. It is the only rule that is used to derive all arithmetical truths. "Since it is possible to get by with only one means of inference, then it is a precept of clarity to do so" [Frege, 1972].
Negation.
It is symbolized by a small bar perpendicular (negation bar) to the content bar at its bottom, indicating that the judgment does not take place:
Equivalent in modern notation to ¬A.
Function.
The notation ⊢Φ(A) is used for a one-argument function (A) and ⊢Φ(A, B) for a two-argument function (A and B). The expression Φ(A) can also be thought of as a function of argument Φ, since Φ could be replaced by another symbol.
Equality of contents.
Symbolized by: ⊢A≡B, it is an identity of reference and not an equality between symbols, i.e., it is "use" and not "mention". It indicates that the content of B can replace A.
Generality.
A symbol formed by a horizontal line with a concavity in whose interior a letter is placed) is used. It is used to quantify individual variables and functions. For example,
is the law of commutativity of addition.
Another example is
which is a second-order function that maps G to V if G maps every object to V. Otherwise, it maps to F.
A Latin letter indicates an unquantified variable and has as its domain the content of the entire judgment. If a Latin letter appears in an expression that is not preceded by a judgment slash, then the expression is meaningless.
Definition.
The notation : ||&—A≡B is used. It is used to establish a definition where the symbol A represents the content of B.
Logical axioms.
Frege declared 9 of his propositions to be axioms. He justified this because they express intuitive truths. All other properties follow from these 9 axioms and a single rule of inference (the modus ponens). Axiom 7 is from second-order predicate logic. In modern notation, these axioms are:
Nº
Axiom
1
A→(B→A)
2
[A→(B→C)] → [(A→B)→(A→C)]
3
[D→(B→A)] → [B→(D→A)]
4
(B→A) → (¬A→¬B)
5
¬¬A → A
6
A → ¬¬A
7
((c≡d) → ∀f ( f(c)→f(d) ))
8
c≡c
9
∀a(f(a)) → f(c)
MENTAL vs. Conceptography
Between Conceptography and MENTAL there are more than analogies; there are coincidences, although there are also differences:
General characteristics
Motivation and objectives.
Conceptography was initially born as a means to clarify and formalize arithmetical concepts by means of logical concepts. Later Frege believed he saw universal features in his language.
MENTAL was initially created with the goal of creating a unified language for computer science. But it was soon realized that primitives, because of their high level of abstraction, had a universal application, serving as a foundation (at a theoretical and practical level), not only for computer science, but also for mathematics and the formal sciences in general.
Universality.
Frege does not make clear what is the basis of his language to be considered universal. Moreover, Frege's language is not general enough, it is not universal, because it is restricted to second-order predicate logic, and a universal language should have no restrictions.
MENTAL bases its universality on the primary archetypes, the archetypes of consciousness. It is a language without restrictions, for the primary archetypes are degrees of freedom.
Foundation.
Conceptography is based on logic exclusively. Frege justified this on the grounds that since the purpose of science is the search for truth, logic had to be the foundation of the universal language for mathematics and for science in general. The firmest way to support a scientific truth is its logical demonstration. All knowledge must be based on logic. "The task of logic is to discover the laws of truth, not the laws of assertion or thought." Paradoxically, he claimed that truth is indefinable.
Frege makes a hodgepodge of semantic doctrine and logical definitions. MENTAL uses semantic primitives, only one of which (the Condition) is of a logical type, which serves as the foundation of decision and implication logic.
Ideal language.
Conceptography was intended to be an ideal language to overcome the limitations of ordinary language, a language based on "pure thought".
MENTAL is in the same sense, for what Frege calls "pure thought" can be interpreted as archetypes, since according to Jung, archetypes are forms without content. We could say that Frege was searching, more or less consciously, for the primary archetypes, the deepest part of reality, and he believed that these archetypes were of a logical type. But logic is not the deepest foundation of reality. It is only one of the dimensions of reality.
Predicate logic.
Frege regards predicates as functions that map to truth values. Frege's predicate logic is "second-order" in that it allows for the quantification of functions. For example, in modern notation, the axiom (a≡b → ∀F(Fa→Fb)).
In MENTAL, predicates are of a general nature, not exclusively linked to logic. The predication is a particularization of an expression, which includes in particular the concept of attribute or predicate, of notation x/y (y particularizes x). Predicate is a different concept from function; it is a relationship between objects, although it can also be used as a function. For example,
(3*x + y)/(y=2) ev. (3*x + 2).
Predicates can be applied to any type of expression. And it can be of any order, there is no limit.
Philosophy.
Conceptography is considered to have marked the beginning of analytic philosophy, for Frege conceived of a philosophical language. Frege wanted to make logic the foundational discipline of philosophy.
The primary concepts of MENTAL are, in addition to primary archetypes, philosophical categories. MENTAL is a philosophical language.
Psychology.
For Frege, the task of mathematics and logic is not to investigate individual subjective mental contents, but he admitted that perhaps the task of both disciplines is to investigate the mind in general.
The primary archetypes of MENTAL constitute a model of mind and consciousness.
Simplicity.
Frege sought the logical concepts of the greatest simplicity and in the smallest possible number.
MENTAL is based on simple concepts of supreme abstraction: the universal semantic primitives, of which only one is a logical primitive (the Condition).
Science.
Frege held that Conceptography was a science in itself.
MENTAL is also a science in itself, but not just another science, but a meta-science, or universal science, for it is the foundation of the formal sciences.
Axioms.
Frege uses 9 logical axioms, formalized in the Conceptography. In "The Basic Laws of Arithmetic" he used an axiom −V namely, the axiom of double negation−, which deals with the extension of a concept, an axiom that was shown to be inconsistent. Axiom V implies a contradiction, as Russell pointed out in a letter to Frege: there are concepts whose extensions include the concepts themselves.
Today it is recognized that Frege proved that arithmetic is reducible to the second-order logic extended by Hume's principle. This principle says: "Given two concepts F and G, the number of Fs is equal to the number of Gs if and only if there is one-to-one correspondence between the Fs and the Gs. This is expressed by the so-called "Frege's Theorem": all fundamental principles of arithmetic are derivable from Hume's principle as a single axiom, along with the definitions of "0", "successor" and "natural number" within second-order logic in which variables varying over concepts, not just elements of a set, are admitted.
MENTAL is based on 12 semantic axioms, which are universal semantic primitives or primary archetypes. It also uses general axioms that relate primitives. The question of consistency is not considered, since this issue is associated with truth and falsity, and in MENTAL truth is not intrinsic, but extrinsic: true (V) and false (F) are constants that can be used in expressions (as an attribute, as a logical magnitude, etc.). Truth is indefinable, as Frege claimed. MENTAL is not based on the notion of truth, but on the notion of existence of an expression in abstract space.
Type of language.
Frege's language is descriptive and normative (about how to make inferences). But it is not operational; no processes can be defined.
MENTAL is a descriptive and operational language. And inferences are automatic.
Principle of compositionality.
Frege defined the principle of compositionality.
In MENTAL, compositionality is based exclusively on primitives. Structural semantics is the same as lexical semantics.
Notation.
Conceptography is a two-dimensional graphical language. MENTAL is linear, one-dimensional.
Frege introduced a new original notation, although it was very complex. Today an easier and more understandable one is used, developed by Peano and adopted by Russell (it is called "Peano-Russell notation") which was used in the work Principia Mathematica (by Russell and Whitehead), inspired by Frege's logicism. MENTAL, in turn, simplifies and generalizes the Peano-Russell notation.
Analytic vs. synthetic.
Ever since Kant formulated the distinction between analytic and synthetic judgments, philosophers have tried to clarify the distinction in general between the concepts "analytic" and "synthetic", without a general consensus having been reached. Here we will define these two concepts in the simplest way (following the principle of Occam's razor), on the basis of the two modes of consciousness. The synthetic is the intuitive, necessary, universal and a priori. The analytic is the rational, the contingent, particular and a posteriori.
In this sense, the principles of mathematics (including arithmetic) are synthetic: they are intuitive, universal, necessary and a priori. All primitives of MENTAL are synthetic (they are based on intuition), a priori (prior to experience), universal and necessary. Everything else (derivatives and expressions) are analytic (based on reasoning), a posteriori, particular and contingent.
Third World.
Frege was a Platonist. He said that abstract entities and logical and mathematical truths have a real existence, independent of our mental contents, and that they inhabit a "Third World," an objective world that transcends the physical and the mental.
The primitives of MENTAL are primary archetypes, a concept equivalent to Platonic Ideas. Primary archetypes are inexpressible and transcendent, of which we can only see their concrete manifestations. The expressions of MENTAL, which are manifestations of the primary archetypes, can be considered to reside in a third world of abstract entities, for they are neither physical nor mental.
Other logics.
Frege did not explore other logics, such as modal logic (that which deals with questions of necessity and possibility) and temporal logic (the logic in which verbal tenses are involved). MENTAL is a language in which all logical concepts (the modal, the temporal, the fuzzy, etc.) are generalized, i.e., they are not exclusively tied to logic.
Application to other sciences.
Frege said that his language could be extended to other sciences, such as geometry and physics, by adding concepts and axioms specific to these domains.
MENTAL can be applied directly to geometry as analytic geometry and geometric algebra, without adding new concepts. And MENTAL is a language applicable to classical, modern and computational (or digital) physics.
Differential and integral calculus.
Frege was confident that Conceptography would serve as a foundation for differential and integral calculus. It did not.
MENTAL allows to define in an incredibly simple way the concept of infinitesimal (ε*ε = 0) and, starting from it, the differential and integral calculus.
Paradoxes.
Frege's formal system of arithmetic turned out in the end to be inconsistent, with the appearance of Russell's paradox.
In MENTAL there are no paradoxes. There are fractal expressions.
Detail characteristics
Function and argument.
In Conceptography, function is a fundamental concept. But functions are only patterns of propositions and cannot perform processes. They are only descriptive.
In MENTAL, function is a derived concept. Although there are many forms of implementation, it is usually implemented by a generic expression with parameters. In MENTAL, a clear distinction is made between parameters and arguments (the concrete values of the parameters). Functions can perform processes.
Number.
According to Frege, numbers concern concepts and not things. He defined the number of a concept F as the set of all sets that are equinumerable with F. The number 0 is defined as the number associated with the concept of "not being identical to itself".
Despite all Frege's efforts, the concept of number is indefinable. In MENTAL, number is based on the semantic primitive "Sum", which is indefinable (like all primitives); we can only see its manifestations.
Sense and reference.
These two concepts have in MENTAL a simple interpretation: the sense of an expression is the expression itself, which has a meaning based on semantic primitives. The reference is the result of its evaluation, which may correspond to an abstract mental entity or to a physical object. When an expression is self-evaluating, meaning and reference coincide.
Use and mention.
"Use" and "mention" in Frege have their correspondence in MENTAL:
Usage: x, which refers to the content of x, to its evaluation.
Mention: x°, which refers to the expression x itself.
For example, a+b may refer to 7 (if a=3 and b=4). And (a+b)° refers to
(a+b).
Logical primitives
Condition.
Frege uses the modus ponens as the only law for deriving inferences.
In MENTAL, the modus ponens is used as either implication logic or decision logic. The distinction is based on whether or not it is used within a generic expression, respectively. In its generic form it allows conclusions to be derived automatically. The MENTAL notation is linear and simpler: (A→B), which indicates "If A, then B".
Negation.
Logical negation in MENTAL is a derived operation, based on the "Contrary" metaoperator.
Equality.
Frege uses the symbol "≡".
In MENTAL, the symbol "≡" indicates equivalence, which is a concept identical to Frege's concept of equality. The symbol "=" indicates substitution.
Definition.
The definition in Conceptography is a primitive: ||—A≡B
In MENTAL, this is interpreted as A representing B and expressed as
(A =: B).
Generality.
Frege defined the universal quantifier. The scope of a universal quantifier is all objects, the universal class. The existential quantifier is derived from the universal and negation. For Frege, the existential quantifier is a second-level concept.
In MENTAL, generality is not only associated with logical expressions. Universal quantification is expressed in a generic expression, which can be of any type. There is no specific operator for quantification. Generic variables are expressed in bold type.
In MENTAL, the universal quantifier is associated with the parameters of a generic expression. The scope of a universal quantifier is the generic expression in which it is included. The existential quantifier is a derived expression, as Frege stated. ∃xPx is defined as follows:
({⟨(x ← x/P)⟩}# > 0)
(the number of elements having property P is greater than zero). This expression has the advantage that the number of elements satisfying the property P can be specified. And the non-existence ¬∃xPx is expressed as.
({⟨(x ← x/P)⟩}# = 0)
This common position aligns with Kant's view that "existence" is not a real predicate. Indeed, existence is the basis of every entity, of which it does predicate.
Examples of logical expressions
Frege's notation
Modern notation MENTAL
¬(B → A) (B → A)'
(B → ¬A) (B → A')
¬B → A (B' → A)
A → (B→ C) A→ (B→ C)
∀A ∀B (A → (B → A)) ⟨A → (B → A)⟩
Quantified expressions
The following table compares general sentences in Frege's notation, in modern predicate calculus notation, and in MENTAL notation.
Example
Frege's notation
Modern notation MENTAL
Everything is deadly
∀xMx ⟨x/M⟩
Something is deadly
¬∀x¬Mx eq. ∃xMx {⟨(x ← x/M)⟩}#>0
Nothing is deadly
∀x¬Mx eq. ¬∀xMx {⟨(x → x/M)⟩}#=0
Every person is mortal
∀x(Px → Mx) ⟨( x/P → x/M )⟩
Some person is deadly
¬∀x(Px → ¬Mx) eq. ∃x(Px ∧ Mx) {⟨( x ← x/P ← x/M )⟩}#>0
No person is mortal
∀x(Px → ¬Mx) eq. ¬∀x(Px ∧ Mx) {⟨( x ← x/P ← x/M )⟩}#=0
All and only people are mortal
∀x(Px ↔ Mx) ⟨( x/P ↔ x/M )⟩
Conclusions
The author of this work considers that in order to conceive MENTAL he has made a "journey" similar to that made by Frege, and that there are remarkable parallels between Conceptography and MENTAL:
Frege created Conceptography by searching for the logical foundations of arithmetic. He later realized that the language he had devised was universal, that it was valid for all sciences, especially the formal sciences.
MENTAL was initially conceived as a computer language, more specifically as a specification language and a multi-paradigm programming language. Later, because of its supreme level of abstraction, the language was applicable to mathematics and all formal sciences in general. And finally that it was a "theory (and practice) of everything".
Both are Platonic languages that attempt to realize Leibniz's dream of creating a universal language for science (Lingua Charasteristica Universalis) and that serves to perform reasoning as calculus (Calculus Ratiotinator).
Frege tried to rely on logic, since he considered it to be the most fundamental science. He failed because logic is only a deep dimension of reality.
MENTAL is based on the primary archetypes, of which logic is a part.
For Frege, Conceptography was the formalization of the Third World, the world of logical (objective and timeless) concepts and entities.
MENTAL is the universal language for the formalization of the Fregean Third World, but conceived as the world of expressions of primary archetypes.
Frege conceived of Conceptography as an analytic language. MENTAL is a synthetic and analytic language, the synthetic being the foundation of the analytic.
Frege used as the only rule of inference the modus ponens. MENTAL only uses the primitive "Condition", which is also modus ponens, but not the sense of truth values, but of existential values.
Frege used logical axioms. MENTAL uses semantic axioms (the universal semantic primitives) and general axioms relating the primitives.
Addenda
More on Frege and Conceptography
The term "Begriffsschrift" is not original to Frege. It was first used by Franz B. Květ in 1857 and by Friedrich Adolf Trendelenburg in 1867. These two German philosophers thought, like Leibniz, that natural language was imprecise, inadequate and insufficient for the analysis of philosophical and scientific subjects, so it was necessary to create a new sign language −following the idea of Leibniz's Lingua Characteristica Universalis of Leibniz−, free from psychological influences, in which the structures of signs would reflect the structures of concepts, in which the signs would be independent of the sensible and would link directly and systematically with the objective contents of concepts, and not with subjective contents. Nevertheless, the term "Begriffsschrift" is associated with Frege, because of the impact on logic and philosophy of the 1879 publication of his Conceptography.
Frege was accused in his time by Ernst Schröder of having created, not a universal language (as his title promised) but only a Calculus Ratiotinator, whose task had already been accomplished by Boole. Frege retorted that his language was a true Lingua Characteristica and that the logical calculus was a necessary component of that language.
In his time, Frege's work in logic had hardly any repercussions, and he even had to publish his last work "The Basic Laws of Arithmetic" at his own expense. At the University of Jena - where he taught mathematics - it was even stated about his work that "it was of no interest to the university".
But in 1903, Russell included an appendix in his work "The Principles of Mathematics" in which he compared his ideas with those of Frege. Frege's ideas also spread through the writings of his student Rudolf Carnap and other scholars of his work.
Wittgenstein recognized "Frege's great work" and his influence on the style of his logical sentences. The Tractatus contains 17 mentions of Frege, some of a critical nature.
Frege's work had a great influence on later work, including Russell and Whitehead's formalization of Principia Mathematica, Russell's theory of descriptions, Tarski's theory of truth, and Gödel's incompleteness theorem.
Bibliography
Dummett, Michael. Frege: Philosophy of Language. Harvard University Press, 1993.
Dummett, Michael. The Interpretation of Frege´s Philosophy. Harvard University Press, 1987.
Dummett, Michael. Frege and Other Philosophers. Oxford University Press, USA, 1993.
Frege, Gottlob. Conceptografía. Los Fundamentos de la Aritmética, Otros Estudios Filosóficos. UNAM (Universidad Nacional Autónoma de Mexico), 1972. Disponible en Internet.
Frege, Gottlob. Estudios sobre semántica. Ediciones Folio, 2003.
Frege, Gottlob. Ensayos de semántica y filosofía de la lógica. Tecnos, 1998.
Frege, Gottlob. Escritos filosóficos. Crítica, 1996. (Incluye Los Fundamentos de la Aritmética.)
