"It is the search for truth that prompts us to advance from meaning to reference" (Frege).
Meaning vs. denotation
Every descriptive linguistic expression has two aspects or dimensions: its meaning and its denotation (or reference). These two aspects are of great importance in logic, mathematics and philosophy (theory of knowledge).
The meaning is the idea associated with the expression, its understanding, its epistemological or cognitive content.
The denotation (or reference) is the object or objects to which the expression refers.
Examples:
"The capital of Spain" denotes or refers to "Madrid".
"The author of Don Quixote" denotes "Cervantes".
"Los Reyes Magos de Oriente" denotes "Melchor, Gaspar and Baltasar".
"The planets of the solar system" denotes "Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune".
"The number of planets in the solar system" denotes the number 8, but the meaning of this expression is not this number. And it is not because the number differs according to the temporal context (today there are 8, but before there were 9, when Pluto was included).
Characteristics:
Meaning is the deep (or internal) dimension of linguistic expression. And denotation is its superficial (or external) dimension.
For some authors, denotation is something that is said of the expression itself. For others, it is something that is said of the meaning. Still others consider the expression as an intermediary between meaning and denotation. This last interpretation is the most widely accepted.
The more general a meaning is, the greater is its denotation. And the more particular a meaning is, the lesser is its denotation. For example, the term "man" denotes more than the term "Spanish", and this has a less general meaning than the term "man".
Denotation is usually contrasted with connotation. Denotation is the reference of the expression. Connotation is the mental (including emotional) ideas or associations that trigger a given expression. Connotations can be ideological, philosophical, political, etc.
Denotation is assumed to be something objective, with the same meaning for all speakers of a language, and independent of any context. Connotation can be a product of a speaker (subjective) or be associated with a culture or spatio-temporal context (intersubjective).
It can be considered that "everything is meaning" and that denotation is a lower-level signification. In fact, one speaks of "denotative meaning". The two dimensions (meaning and denotation) are really part of a continuum. The further one moves toward the denotative aspect, the sharper its meaning is and the more objective it is.
Between expression and denotation there are several possibilities:
Expression denotes an object (or objects).
The expression does not denote any object. For example: the current king of France, the largest prime number, etc. In this case there are two different opinions or positions: 1) the expression is non-denoting (does not denote any object); 2) the expression denotes the empty set.
The expression denotes an object (or objects) of a fuzzy or ambiguous type. For example: all tall men, fast vehicles, etc.
The expression denotes abstract objects. For example: the number 3, the set {a, b, c}, etc.
The expression denotes infinite objects. For example: "the natural numbers", "the prime numbers", "the even numbers", etc.
The expression denotes imaginary objects. For example: Pegasus, Apollo, Santa Claus, Don Quixote, Sherlock Holmes, Hamlet, Snow White, the unicorn, the centaur, etc. Examples of mathematical objects are: the square root of −1, the smallest positive real number greater than zero (the infinitesimal), the largest natural number (infinity), etc. There is a fuzzy boundary between imaginary objects and non-existent objects. But it is not fuzzy if one considers that existence has several levels (physical, logical, mathematical, metaphysical, mythical, abstract, etc.). For example, Don Quixote can be considered to exist at the mental level, and the square root of −1 can be considered to exist at the mathematical level.
Different descriptive expressions can have the same denotation. For example:
The sentences "the morning star" and "the evening star" −Phosphorus and Hesperus, respectively, for the ancient Greeks− are two expressions with different meanings and the same denotation: the planet Venus.
"The capital of France" and "The city of light". Both expressions denote "Paris".
"Meaning" is different from "concept". Meaning is associated with a descriptive linguistic expression, and concept is a mental image or construct.
"Meaning" can be considered equivalent to "semantics", although this concept is often used as the dual or complementary aspect of the syntax of a language. Semantics is the discipline that studies linguistics from the point of view of meaning, but one also speaks of the semantics of a given linguistic expression.
"Denotation" is different from "designation", although they are sometimes equated. If A designates B, A is a term, and B a concept, a property, a proposition, a fact, etc. "Designation" is also often equated with "representation" (A is a label or name representing B). The designation is usually a relation between a linguistic element (A) and a non-linguistic one (B).
The meaning of a term (even of a phrase) can sometimes depend on the context. For example, the word "hen" may refer to the female rooster (the meaning is denotative) or to the qualifier "coward" (connotative meaning). And the word "aurora" can mean the part of the day corresponding to the sunrise, or it can mean hope, the beginning of a new life, a new consciousness, etc.
In linguistics, the term "intension" or "intensive expression" is used to refer to a set of objects, and "extension" to the corresponding denotation. An intensive expression may refer to a finite or infinite number of objects. It is generally admitted that denotation is equivalent to extension, and that meaning is equivalent to intension.
When we have infinite entities, the only way we have to describe them is by an intensive expression. Extensions (finite or infinite) can be, in turn, intensities that can correspond to finite or infinite extensions. For example, the expression "the numbers multiples of 2, multiples of 3, multiples of 4, etc." refers to infinite intensities and where each of them also refers to infinite extensions. In general, what is obtained is a tree structure of intensions from higher to lower level of significance until finally arriving at the extensions, where the root is the initial (intensive) expression and the leaves are the final (extensive) expressions. If we consider that an extension is a terminal or elementary intension, then we can say that, at the last level of the tree (the leaves), intension and extension coincide.
Classical formal logic (propositional and predicative) deals with concrete, extensional terms. It is a superficial logic where intent plays no role. A formal logical system in which intensional features can be represented is often called "intensional logic". In intensional logic, the designated entity is called "extension" (or designation). And the mode of designating it, "intension" (or meaning).
The double dimension meaning-denotation of a linguistic expression makes it possible to clarify or solve important semantic problems, provided that the deep aspect (the meaning) is considered and not the superficial aspect (the denotation).
The problem of identity.
The identity or equality between two expressions a and b (a = b) should not be understood as denotational equality (between objects), but as equality of meanings. The expressions a and b denote the same object, but through different meanings. The identity is, therefore, of meaning, conceptual. For example, "The capital of France" and "The city of light".
The problem of substitutability.
According to the principle of substitutability, one can substitute in any proposition one expression for another whenever both refer to or denote the same object. But this principle is not always applicable to any context. For example, if we substitute in the proposition" "I believed that Barcelona was the capital of Spain" the expression "the capital of Spain" for "Madrid", we obtain "I believed that Barcelona was Madrid". Another example is "Madrid was chosen by Philip II as the capital of Spain in 1561". Substituting "the capital of Spain" for "Madrid", we get "Madrid was chosen by Philip II as Madrid in 1561". Therefore, a substitution of an expression in a proposition must be made by meaning, not by reference, so that its meaning is not altered.
The problem of expressions without references (or with empty references).
In this case, the meaning of these expressions is considered to imply or provide knowledge or information. For example, the expression "Pegasus is a winged horse" is interpreted as carrying information about an entity called Pegasus, regardless of its existential level.
In this issue of the relation between meaning and denotation, different questions can be raised, such as the following:
What is the meaning of "meaning"? What exactly is "meaning"? Is it possible to define it, express it or be denoted by another expression of higher significance?
Is meaning of a subjective (or psychological) kind or of an objective (or logical) kind?
Can a meaning denote another meaning? That is, can a denotation be a meaning?
What is the meaning of a proper name? In a proper noun, is the meaning equal to the denotation?
What is the meaning of a sentence that refers to something that does not exist? The paradigmatic example (proposed by Russell) is "The current king of France is bald." Is this sentence true, false, or has no truth value?
What is the relationship between description, meaning, and denotation?
What is the relation between meaning-denotation and the necessary and contingent properties of objects?
What is the relation between meaning, thought and consciousness?
Frege: Meaning and Reference
The modern understanding of the intensional and extensional aspects of language begins with Gottlob Frege and his famous 1892 article "On Sense and Reference" (Über Sinn und Bedeutung) [Frege, 2003], which had a great impact at the philosophical level. In fact, modern philosophy of language −analytic philosophy− is considered to begin precisely with this article. In it, Frege distinguishes two distinct aspects of linguistic expressions:
The sense of a linguistic expression is the form or manner by which reference is made to an object, the mode of presentation of the reference.
The reference of a linguistic expression is the object to which the expression refers.
Frege's main ideas regarding this subject are:
Meaning is the fundamental element of every expression. It is the cognitive element necessary for referential determination. Meaning is more important than its reference. Meaning is a privileged dimension, since it represents the gnoseological and epistemological.
The sense of a proposition determines the reference (which can be null). On the other hand, the reference does not determine its sense.
There is no direct (dyadic) relation between language and reality, but there is a triadic relation: sense (concept), language (expression) and reality (reference).
A "proper name" is any expression (simple or complex) whose reference is an object. An object is anything designated by a proper noun. A proper noun includes:
Simple nouns, such as "Aristotle, Venus, etc., and numerals such as 3, 7, etc.
Definite descriptions such as "Plato's most eminent disciple," "The second planet of the solar system," "1+2," etc.
After every proper name there is at least one description referring to it. For example, "Aristotle" is a name with a sense equivalent to some description (such as "Alexander's teacher").
The sense of a proper name is the sense of a description associated with it.
Every proper name has a sense, whether it has a reference or not. For example, Pegasus and Vulcan have meaning even if they have no reference.
Truth is indefinable. Nevertheless, a truth value can be associated with any well-formed proposition. The reference is its truth value (true or false). For example, the propositions "Aristotle was a philosopher" and "1+2=3" have as reference "true". And "1+2=7" has as reference "false". Therefore, all true propositions have the same reference: "true". And all false propositions have the same reference: "false".
In the case of proper names, their truth value is their denotation. The denotation plays, in this sense, a role similar to the truth value (true or false) of classical logic.
Equality (or identity) is not a relation between objects but between meanings. The equality expressions a=a and a=b indicate two different things. The first expression is trivial because it provides no information and is cognizable a priori. On the other hand, the second expression has a different meaning because it provides information a posteriori, it has cognitive value: a and b are two different expressions of the same entity. For example, in Hesperus=Phosphorus we have two different senses for the same referent (Venus).
Every thought must be considered as a sense and not as a reference. No truth value can be assigned to a thought, for at the internal (mental) level everything that is thought is always true, whether or not it has reference in the real world. Thoughts are senses without reference, so they do not provide any knowledge if it is not in relation to truth, that is, to propositions whose referent (truth value) is true. "The search for truth is what prompts us to advance from sense to reference" (Frege).
Every linguistic sign (e.g., a proper name) has meaning and reference. Signs are ways of expressing objects. Sign, meaning and reference constitute the triad in which the two basic semantic relations are articulated: the relation of the sign to its meaning and the relation of the sign to its reference. A sign always has meaning, but it may not have reference. The sign-reference relation may not exist in natural languages, but it always exists in scientific languages.
Meaning is not something subjective, that is, it does not belong to the mind of the individual, but is an objective entity, ontologically independent and shared by a community of speakers. Meaning is an abstract notion, not a psychological one. It resides in the "Third World," an objective realm.
This position is called "meaning realism," and it is different from the psychologistic tradition, in which words and their meanings are subjective mental contents. For example, Locke was a psychologist. In his "Essay Concerning Human Understanding", he states: words are signs of ideas (or mental contents); ideas come from our sensible experience; there is no direct relation between world and language; language is only a tool with which we communicate ideas.
Russell: Theory of Descriptions
Bertrand Russell's 1905 article "On Denoting" [Russell, 1981] is one of his major contributions to the philosophy of language and one of the most significant and influential philosophical-linguistic essays of the 20th century. It was published in the journal "Mind" in 1905, reprinted in the same journal on its centenary (2005) and in "Logic and Knowledge" in 1956. This article provoked a great philosophical-linguistic debate that has not ceased until today. In it he expounds his "theory of descriptions".
Russell's theory is based on the following ideas:
A "denotative phrase" is a phrase that expresses a meaning and denotes an object. We only obtain meaning by means of denotative phrases. Meaning is associated with expressions that denote something, whether it exists or not. A sentence is denotative only by virtue of its form. Every well-formed sentence denotes something.
We must distinguish between "immediate knowledge" (of things presented to our senses) and "knowledge about something" (which is obtained by means of denotative sentences).
A denotative sentence can be of two types:
Definite description.
A linguistic expression denoting or referring to a single −existing) or non-existing− object that satisfies a certain description.
All defined descriptions normally begin with "the" or "the" (because they refer to a single object) followed by a description of the object.
Examples in a general context are "The current king of Spain", "The first man to set foot on the moon", "The result of adding 5 and 7", "The richest man in the world", etc. Examples in a particular implied context are: "The captain of the ship", "The highest student in the class", "The head of the family", "My best friend", etc. Examples of definite descriptions that do not denote anything are: "The current king of France", "The philosopher's stone", etc.
Indefinite description.
One that denotes something, but in an ambiguous way (e.g. "A king of Spain"). They begin with "un" or "una".
A definite description does not appear in isolation, but is accompanied by a predicate. The general common form is "He/The F is G", where F is the definite description and G is a predicate. Each particular form is either true or false.
A proper noun is really a definite description disguised as an implied character. For example, the name "Aristotle" is equivalent to some description, such as "Plato's disciple" or "Alexander's teacher" or any other description. A proper name has a meaning, which is an implied description of the object denoted by the name.
A distinction must be made between a non-logical proper noun and a logical proper noun. A logical proper noun is one whose reference exists and whose meaning is its reference, i.e., reference and meaning coincide. For example, if we have the sentences
(1)John is bald.
(2)The tallest man in the world is bald.
both have the same grammatical structure or form (subject - predicate), where the subject in (1) is "John" (which is a proper noun), and the subject in (2) is "The tallest man in the world" (which is a definite description). But they do not have the same logical structure. Therefore, a definite description should not be treated as a proper noun.
Proper names are definite descriptions in disguise. The usual proper names such as Plato, Socrates, Obama, etc. are not "logical proper names". A logical proper name is one whose reference is assured, i.e., we know with certainty that its reference exists, and its meaning is its reference. Non-logical proper names do not guarantee the existence of the referent, even if it belongs to a real person, since we have no direct knowledge of the objects they represent.
When we use a usual (non-logical) proper noun, we are really dealing with a covert definite description because a proper noun can be replaced by a definite description that is different according to the context. The meaning of a proper noun is the meaning of the definite description used. For example, "Socrates" may be shorthand for "Plato's teacher".
A logical proper name is a symbol that represents an object of which we have direct knowledge, and that representation does so without ascribing any property to it. Logical proper names are few. According to Russell, only one English word (this) is used in this way to refer to particular determinate objects.
Logical proper names and definite descriptions correspond respectively to direct knowledge and knowledge by description. To be understood a proper name must have referent, that is, it must be a logical name. On the other hand, any description can be understood (the meaning) without our needing to know its reference.
Contrary to Frege, he holds that definite descriptions (e.g., "The present king of Spain") are neither proper names nor have meaning by themselves (in isolation). They are incomplete sentences that require a context of use that contributes to the meaning of the sentences in which they appear. Any proposition containing a definite description has meaning.
It is impossible to denote a denotative sentence. Meanings cannot be treated as objects; they are abstract entities. If meaning were an object, then it would be the same as denotation. What is meant by denotation cannot be formalized. The relation between meaning and denotation is indefinable and unrealizable. It is only realizable when meaning and denotation are the same thing.
The problem of descriptions without denotation
Russell's theory offers a solution to the problem of the truth value of a sentence that has no denotation (or reference) so as not to violate the principle of the excluded third party. The paradigmatic example given by Russell is "The present king of France is bald". This sentence can be considered from two points of view:
Logical.
The sentence must be either true or false, so as not to violate the principle of the excluded third party. At first it seems that the sentence is false, since France is not a monarchy. But if it is false, its converse ("The current king of France is not bald") should be true, which is also not true. Another way of looking at it is that the sentence is not true because among bald people is not the current king of France. And it is not false because among the non-bald people there is not the current king of France either. We are faced with a logical paradox.
Semantic.
On the one hand, the sentence makes sense (because it is perfectly understood) and on the other hand it does not (because it refers to an entity does not exist (the current king of France).
The solution proposed by Russell to the problem of definite descriptions without reference consists in analyzing the whole sentence ("He/The F is G") and not the definite description in isolation (F). It consists in dividing the sentence "The current king of France is bald" into three components:
There is a x such that x is the current king of France.
There exists no such thing, except x as is the present king of France.
x is bald.
That is, the definite description contains three pieces of information: 1) an implicit existence information; 2) an implicit uniqueness information; 3) a predicate. Then, for the sentence to be true, three conditions must be fulfilled: 1) that the subject of the description exists; 2) that the subject is unique; 3) that the subject has the predicate. In this case, since the first condition is not met, the sentence is false.
And the negation of the sentence affects all of it (broad scope), and not only the predicate (narrow scope). Its negation is not "The current king of France is not bald" but "It is not the case that there is a current king of France and he is bald". Therefore, the sentence is true. In this way, the principle of the excluded third party is not violated. In this case, the negation is included at the beginning of the sentence and its scope is the whole sentence.
Russell's theory also analyzes indefinite descriptions. For example, the sentence "A man is walking" contains the indefinite description "a man", in a place that could be occupied by a proper name. As in the case of definite descriptions, such expressions should not be treated as if they were proper names, but also analyzed logically: "There exists a x such that x is a man and x is walking."
Other Authors
Mill: connotation and denotation
Before Frege, John Stuart Mill in his work "A System of Logic" (1843) [Mill, 2002] discusses denotation, connotation, and other linguistic issues:
He distinguishes between connotative and non-connotative expressions. Connotative expressions are those that have meaning. Non-connotative expressions (such as proper names) are those that lack meaning and have the function of simply pointing to or pointing to an object, without describing it (i.e., without listing any of its properties).
The connotation determines the denotation. It is the same as Frege said: meaning determines reference.
A connotative term is one that denotes a subject and implies an attribute.
A general noun is one that can act as a predicate and denote objects of a given class. The connotation of a general name is a set of attributes. The denotation of a general name is the set of individual elements that have those attributes.
Church: meaning and denotation
Frege provided the basic ideas about in his famous 1892 article, but he did not formalize them into a logical theory. In 1951, Alonzo Church published an article [Church, 1951] in which he described a formal logic (intensional logic) in which every term (including variables) had sense (or meaning) and denotation (or reference).
The logic Church elaborated was very complex and not very general. It was based on Russell's type theory and the lambda calculus (the functional calculus created by Church himself).
Carnap: intension and extension
Carnap −one of the leading exponents of the Vienna Circle− extended and formalized the ideas of Frege and Church. He expounded them mainly in his work "Meaning and Necessity" [Carnap, 2008]. In this work, Carnap expounds a new method for the semantic analysis of meaning, i.e., for analyzing and describing the meaning of linguistic expressions. This method generalizes classical concepts known as class and property.
Every expression, including variables, has intension and extension. An intent can refer to other intents.
The meaning of a word consists in the designation of a concept. A complex concept is a combination of simple concepts. Concepts have extension, i.e. they refer to individual elements.
The concept of verifiability (every sentence to be valid must be experimentally testable) is replaced by that of testability: a statement makes sense if it is translatable into a logical language in which its primitive elements are observational [Carnap, 1936-7].
Carnap's system is not a homogeneous formal system and is very complex. Nevertheless, Carnap's ideas had a great influence later on. They were the origin of the concept of logical grammar (or logical syntax). A grammatical syntax takes into account only the basic syntactic categories and the rules for creating derived categories. But this type of grammar only considers sentences that have a correct grammatical structure, such as "Peter is a prime number". In contrast, a logical syntax goes beyond grammatical syntax, as it establishes the admissible categorial combinations by including semantic compatibility rules. This system avoids erroneous and metaphysical statements and allows scientific knowledge to be expressed in a meaningful language that has a connection or correspondence with reality. In [Carnap, 1993], this issue of meaningful and non-significant language is described.
In "Meaning and Necessity", Carnap makes the observation that Frege's concept of "meaning" should be recursive, that we would need "sense of meaning", etc., i.e., an infinite hierarchy of semantic denotations.
Meinong
Alexius Meinong's object theory is based on the correspondence or identification between objects and thoughts. It is a very simple and straightforward theory:
The existence of entities is not limited to the physical plane. Everything that can be expressed is real, even if it refers to imaginary entities.
Every grammatically correct expression denotes an object with logical existence, a logical object. So "the present king of France," "the golden mountain," and "the round square" are logical objects, although they do not correspond to any real-world entity or object.
Every thought −and, consequently, every expression of language− refers to something, to an object. Every thought is always about something. Every object that we can think, even if it is non-existent, impossible or contradictory, is a genuine object.
Since it is possible to think about impossible objects, these objects must exist for thought to be possible. About what does not exist, one cannot think.
Through thought and language we can describe objects, but objects themselves are independent of thought and language. Objects are not generated by thought and language.
There is a huge realm of objects that have never been the object of any thought. But objects exist, even if we have never thought about them.
Everything is an object, whether it is thinkable or not. If it is thinkable, that object exists. If it is not thinkable, then it has at least one property: that of not being thinkable.
Wittgenstein
Like Frege, the first Wittgenstein (the one of the Tractatus) distinguished between Sinn (sense) and Bedeutung (reference):
Only the proposition has meaning. Every expression only has meaning within a proposition.
Nouns have no meaning; they have a purely referential function.
"Frege says: any correctly formed proposition must have a meaning; and I say: any possible proposition is correctly formed and if it lacks meaning this can only be due to the fact that we have not given meaning to some of its constituent parts. (Even if we believe we have done so.)" (Tractatus, 5.4733).
The second Wittgenstein (that of Philosophical Investigations):
The meaning of an expression is its use. "Ask not for the meaning but for the use."
He doubts that the meaning of a proper name is completely contained in a single description. He gives the example of "Moses did not exist," which can be interpreted in many ways: the leader of the Israelites was not called Moses, that the Israelites had no leader, etc.
Strawson
Peter Strawson, in his essay "On Referring" [1950] criticizes Russell's theory of descriptions, the theory proposed 45 years earlier and which up to that time was considered a paradigm of analytic philosophy.
Human language cannot be identified with a set of propositions with determinate logical forms, as if it were a scientific theory. Natural language does not have an exact logic. One thing is scientific language and another is everyday language. Evaluating one with the canons of the other is a sterile exercise.
Language has no pre-determined meaning, but meaning is acquired through its use. Strawson here echoes the position of the second Wittgenstein. For Russell, language is explained by scientific principles expressed logically. Russell sought an ideal language that would reflect the structure of reality. For Strawson, language is a human activity, and that to seek an ideal language is to promote bad metaphysics.
We must classify linguistic expressions, not by virtue of their form, but according to the use made of them.
We cannot say whether a sentence is true or false, but only whether it is being used to make a true or false assertion.
Definite descriptions do not assert that the object exists, but presuppose its existence. Since "The current king of France" does not exist, there is no reference, so it is meaningless, and the sentence is neither true nor false.
It is necessary to distinguish between sentence and statement. The same sentence can be used in different contexts to make different statements. A sentence has meaning but is neither true nor false. Statements are true or false because they refer to use, and their truth value depends on the context in which they are used. For example, the statement "The king of France is wise" stated during the reign of Louis XIV is true. Declared during the reign of Louis XV is false. Stated today has meaning, but since it has no reference, it is neither true nor false. The same is true of "The present king of France is bald", which has meaning, but no truth value. Instead, for Russell, this statement is false because it has no reference.
Referring is not a kind of sentence. To refer is not to assert, even though an assertion performs a successful act of reference. To affirm involves making reference to something.
The distinction between subject and predicate is one of use (or function). The same expression can be either subject or predicate, depending on usage. Failure to recognize that the distinction between subject and predicate resides in use leads to a bad metaphysics: the assumption that this distinction reflects the structure of reality.
The functional distinction between subject and predicate has "long philosophical shadows" as in the distinction between particular and universal or between substance and quality.
Another bad metaphysics is Russell's claim to "abolish particulars" (proper names and definite descriptions) and replace them with logical structures. Russell intended to "logicize" language by substituting logical formalisms for proper names and definite descriptions.
Proper names are only mere labels whose meanings are the things they designate. Logical proper names constitute bad epistemology.
Strawson advocated a method called "connective analysis": our concepts form a network in which concepts are nodes. An analysis of a concept involves analyzing the nearest concepts in the network. The goal of the analysis is to clearly identify the connections between the most general concepts of everyday language in order to carry out a descriptive metaphysics.
Putnam
For the philosopher Hilary Putnam meanings are not subjective, they are not in the head. To prove it, he proposes the mental experiment of "the twin Earth":
There is another planet Earth (T2) that is exactly like ours (T1). In T2 the liquid of rivers, lakes and seas is apparently identical to water, but that its chemical composition is not H2O but XYZ. A ship arrives at T2 from T1. Its crew members see the liquid and call it "water", because they observe that it is the same as the water they know in T1. Therefore, the term "water" has the same meaning for them on the two planets. When it was subsequently discovered that the composition of the "water" of T2 is different from that of T11, then the meaning would change: TT2 would be passed to mean the liquid whose component is XYZ.
Putnam's conclusion is that the reference of the term "water" is not a function of the speaker's psychological content. Meanings are not subjective and subjective meaning does not determine reference.
Donnellan
Philosopher Keith Donnellan has contributed to the philosophy of language, especially on the topic of the analysis of proper names and definite descriptions. The essay "Reference and Definite Descriptions" [1966] is a critique of Russell and Strawson's theory of definite descriptions. It states that it is necessary to differentiate between the referential use and the attributive use of definite descriptions, which are two uses or ways of referring to an object:
Referential use consists only in referring to an object.
Attributive usage consists of using a description as a reference to an object in a certain context. For example, "The man in the blue suit is tall." In this case, the expression "The man in the blue suit" is a description used to identify or refer to the individual about whom information is then provided (a predicate is applied).
Kripke
Saul Kripke's [1995] book "Naming and Necessity", published in 1980, is considered one of the most important philosophical works of the 20th century. In it he critically examines traditional philosophical problems, and approaches in a new way the meaning-reference issue (as opposed to the prevailing ideas of Frege and Russell), especially the issue of proper names in the philosophy of language. Kripke's theories are more philosophical than linguistic, more oriented to metaphysical theses and less to the explanation of linguistic phenomena. Although his philosophical theses are based on the linguistic.
Metaphysics vs. epistemology.
One must differentiate between metaphysical and epistemological questions. Therefore, we must distinguish between the notions of "necessary" (metaphysical concept) and "a priori" (epistemological concept), concepts that have traditionally been linked. Indeed: 1) there are necessary truths a posteriori (i.e., their truth can only be discovered empirically), as for example "the morning star is the evening star"; 2) there are contingent truths a priori, such as "the standard meter of Paris measures 1 meter". Therefore, the necessary or contingent character of propositions is independent of our knowledge (a priori or a posteriori) of them.
Identity.
An identity statement of the type x=x, e.g. "Aristotle is Aristotle" is an a priori, analytic and necessary truth. An identity statement of the type x=y in general is contingent, e.g. "the morning star is the evening star" (they could have been different celestial bodies). These are a posteriori contingent truths. But there are identities of type x=y which are necessary a posteriori. For example, if gold has atomic number 79 (empirically discovered), then the identity "gold = atomic number 79" is a necessary a posteriori truth. If identity statements are true, then they are necessary, whether a priori or a posteriori.
Names vs. descriptions.
Names are not descriptions. Names have no meaning. Reference is not given by a description. Descriptionist theories −such as those of Frege and Russell in which a proper name is either a description or a more or less disguised description− are false for several reasons, among them: 1) The same name "Aristotle" can be assigned different descriptions, so that the term "Aristotle" is ambiguous; 2) If a concrete meaning is assigned to the name "Aristotle" (e.g., "the teacher of Alexander"), then the term "Aristotle" is not a description. e. "Alexander's teacher"), other meanings that could also be valid (e.g. "Plato's most eminent disciple") are excluded, and furthermore it might not have been so (it is a contingent property and identities cannot be contingent; 3) a definite description may not select a single object; 4) even if no object satisfies a description, it does not thereby cease to refer to something.
Direct reference theory.
Proper names are used to refer to objects that have essential properties, without which they would cease to be what they are. Thus, names are fixed, they are "rigid designators": a proper name refers to the named object in every possible world in which that object exists. Descriptions, on the other hand, designate different objects in different possible worlds. A rigid designator is only a "label" assigned to its referent and does not describe any kind of content. These rigid designators establish the existence of necessary truths a posteriori.
Causal theory of reference.
A proper name refers to a particular object because of a causal connection (or chain) produced by the historical use of the name by a community of speakers. The initial relationship between a name and its reference is produced by "the ceremony of baptism": a name is initially "attached" (implicitly or explicitly) to the object that is its reference. It is the beginning of a causal chain of communication of meanings that are transported from one speaker to another. The name-reference connection is a social phenomenon, not an individual one.
Dummet
Michael Dummet claims that Frege has been misunderstood on the subject of proper names. That Frege admitted that a proper name can have several different senses associated with it by different speakers. And that Frege used definite descriptions as examples of possible senses associated with a proper name.
For Dummet, "sense" is a component of meaning, precisely the one that makes it possible to "grasp" the reference. That is to say, meaning is the factor that makes it possible to find the reference, to understand the words and sentences and to discover whether they are true or false. Something that has meaning is something that makes us think of the reference, even if it does not exist. If sense is the ability to grasp the reference, then the sense of a proper name is the ability to find its reference, without any description of the reference.
According to Dummet, Frege fails in trying to explain the notion of sense because circularity occurs: sense is what makes us understand the expressions of language; but we can only know what the sense of an expression is if we already know the language. If Frege's thesis that the sense of a proper name is the sense of a description associated with it is accepted, two things remain to be explained: 1) the sense of a description; 2) the meaning of "sense".
Meaning and Denotation in MENTAL
In MENTAL we are dealing with an ideal language, not a natural one. Nevertheless, it provides a point of view that clarifies the concepts of meaning and denotation.
In the meaning-denotation issue of descriptive expressions we find ourselves, once again, with the internal-external or generic-specific or universal-particular or deep-surface duality, aspects that in MENTAL are united and harmonized from the perspective of the archetypes of consciousness. The problem of the conceptualization of meaning (or sense) and denotation (or reference), as well as the relationship between the two, is clarified and simplified notably: meaning and denotation are two united and complementary aspects of every expression. Expressions connect on the one hand with the deep (the meaning), and on the other they connect with the superficial (the denotation or reference). This conception coincides with that of Frege, who stated that linguistic signs connect meaning and reference.
The meaning of any expression (whether descriptive or operational) is the interpretation (through the primary archetypes) of a syntactically correct expression, e.g. x+y and even *xy+. In the latter case. the sense is that of a sequence with the specified 4 characters. But there is no meaning, e.g. x(+y nor x}}y, which are syntactically incorrect expressions. All meaning refers back to the primary archetypes.
The denotation or reference of an expression is the result of the evaluation of that expression (which is also another expression) or the expression it represents. Denotation, in this sense, can be considered equivalent to evaluation. Given the dynamic nature of language, denotation/evaluation may vary during a process. Denotation or reference is another expression that is constituted by primary archetypes, so it also has meaning.
Characteristics
Evaluation.
Since evaluation is a cascading process, the meaning-to-reference process may go through a series of intermediate expressions until it reaches the final expression, which is the reference of them all. In self-evaluating expressions, meaning and denotation coincide.
Every expression always has a reference (which can be itself, if the expression is terminal) or it can be the null expression, if so stated. For example,
king(France)/bald // the king of France is bald
If (king(France)=θ, then θ/bald evaluates to θ.
True.
Every expression, without exception, can have a value or degree of truth, depending on the degree of correspondence with reality. "The capital of Spain" and "Madrid", can be assigned an attribute of truth.
In the case that the linguistic expression is a predicate, its extension is the set of objects that have that predicate. For example, the extension of the term or concept "green" would be the set of all green objects. However, there is no clear line of separation between objects that have that predicate and those that do not. In general, the degree of belonging to the set of objects that have the predicate would be represented by a factor f between 0 and 1, and its degree of truth is f*V, being V the magnitude "truth".
The Third World.
Everything that can be expressed exists in the abstract world, an abstract and objective world where all possible expressions "dwell". That world is equivalent to the Fregean Third World. An expression can have its interpretation or manifestation in the physical or mental world.
In MENTAL, every well-formed expression has meaning and exists at an abstract level (in the Third World), regardless of whether that expression is interpreted or related to an entity at a lower level, such as the physical or mental level. All expressions exist in an abstract space or world where the possible is determined by the possible combinations of the primitives.
Levels of signification.
In MENTAL, every expression has meaning of greater or lesser level. Intensive expressions that refer to infinite expressions are the highest level expressions. At the lower end would be extensive expressions based on a single primitive, such as {a b}, a/b, etc. and atomic expressions (those consisting of a single symbol).
Descriptive expressions, which include a pattern, have a higher level of meaning than extensive ones. For example, ( 1…5 ) has a higher level of significance than 12345.
Consciousness.
Meaning and denotation (or reference) can be associated with the concepts of continent and content, general and particular, intension and extension, deep and shallow. In general, meaning is associated with synthetic consciousness, and reference with analytic consciousness.
What connects meaning with denotation is consciousness. And the key to that connection is precisely in the primary archetypes. That is why this relation cannot be formalized, just as the relation between syntax and semantics cannot be formalized.
When Frege states that "The search for truth is what prompts us to advance from sense to reference" it can be interpreted as the search for consciousness, the union of the two poles: the deep (sense) and the superficial (reference).
For Dummet, Fregean "sense" is the component of meaning that makes it possible to connect with reference. But it is really consciousness that is the binding element, that which connects the two.
The formalization of meaning.
Meaning cannot be explained or formalized because it belongs to the internal sphere. Actually, in the initial definition of "meaning" synonyms are used to help understand this concept, but it is not possible to explain it.
The meaning of an expression cannot be formalized because it belongs to the deep world and is inexpressible. Trying to express or explain the meaning is a contradiction because it implies bringing to the surface what is at a deep level. It is the same thing that happens with consciousness, the semantics of a language, truth, information and life. Therefore, a formal connection between meaning and denotation cannot be established either.
Frege uses "sense" in two ways: 1) sense is the meaning of an expression; 2) sense is the mode of presentation of the reference, i.e., the description itself. In the definition of "sense" he uses the second option (the external), but then considers all thought as sense (the internal), the cognitive element necessary to determine the reference. All this confusion stems from the fact that meaning (sense) can neither be explained nor formalized.
Intension - extension.
In MENTAL, meaning and denotation are different concepts of intension and extension, respectively. Intensional in MENTAL is linked to descriptive expressions and parameterized generic expressions that represent other expressions that can be (in turn) intensive or extensive. An extensive expression is an expression that refers to itself.
Universal metaexpressions.
The triad "everything" (Ω), "nothing" (θ) and "something" (α) are the fundamental descriptive expressions.
Ideal language.
The descriptionist theories (Frege's and Russell's) did not establish a formal language, although both were in favor of creating an ideal, logical and philosophical language and to avoid the ambiguity of natural languages. They describe only a part of the language based only on a few concepts (such as proper name, reference, definite description, etc.) but do not explain the relation of these concepts to the world.
In contrast, with MENTAL we have an ideal language that reflects the primary structure of internal (mental) and external (physical) reality, because the language is based on the primary archetypes, which are common to both worlds.
Properties
In the subject of meaning (or sense) and denotation (or reference), immediate substitution, potential substitution (representation), and equivalence play essential roles. The following properties are satisfied:
If (xº = x), that is, if an expression x is self-evaluating, then sense and reference coincide.
If (x ≡ y), that is, if two expressions x and y are equivalent, then x and y have different sense and the same reference.
If (x = y), that is, if x is replaced by y, then x and y have different meaning and the same reference.
If (x =: y), that is, if x represents y, then the reference of x is y.
Examples
((a = c) (b = c)) // a and b have the same reference: c
2+3 and 3+2 have the same reference: 5
a+b and b+a have the same reference (since the sum is commutative), the two expressions are equivalent
( a★3 ) and aaa have the same reference: aaa, since the first expression represents the second one
(a a a a) and aaa have the same reference: aaa, since the second expression is a shortened form of the first one
( 1…4 ) and 1234 have the same reference and different meaning (the first expression is a representation of the second one)
In (x =: (a b c)), the expression x represents and has reference to (a b c).
The expressions ( 1…4 ), 1234 and (1 2 3 4) have different meanings and the same reference: 1234.
The expressions (a b c)/(b=12) and (a 12 c) have the same reference, which is the last expression
If we have the expression ⟨( f(xy) = (x+yx*y) )⟩, then f(r1 r2) has the same reference as f(r2 r1), whichever r1 or r2, since the sum and product of real numbers are commutative.
Russell's theory of descriptions
Russell's theory of descriptions is a restrictive theory:
Russell did not create a formal language for them. He relied exclusively on logic. Russell was a logicist and intended to formalize at the logical level every proposition. But logic is only a semantic dimension, represented in MENTAL by the primitive "Condition". But the issue of descriptions is not a logical issue but a semantic issue.
You are limited to describing a particular object. You cannot refer to more than one object or to a class or category of objects.
The principle of the excluded third party is superseded. A "degree of truth" can be associated with a definite description of the type f*V, which is a magnitude of truth, where V represents "truth" and f is a factor between 0 and 1. For example, "The current king of France" does not refer to any real entity, so it is false (0*V = F), and "The richest man in the world" is true because he exists on the real plane. One can also assign a degree of truth to a sentence of the type "He/The F is G". If F is false, the sentence is false. If F is true, the sentence has as its degree of truth the degree of truth of G.
The meaning and truth value (or degree) of a denotative description depends on the context. For example, "The king of France is wise", which we can encode in MENTAL as ( s = (king(France))/wise ) and two contexts:
Proper names.
In a proper name such as "Cervantes" two levels of significance can be considered:
As "Cervantes", that is, pure text, a sequence of characters. This is its primary meaning.
As an expression formed by a name and an attribute: Cervantes/Name.
A proper name can be a representation of another more complex expression, such as a definite description:
( Cervantes =: author("Don Quixote") )
In general, a description is {⟨x ← pattern(x)⟩}, where pattern(x) is a type of expression. The result is a set of expressions (from abstract space) that satisfy a selection criterion.
The predicate "existence" (E):
⟨( x/E ↔ x≠θ )⟩ // x exists is equivalent to x not being the null expression
⟨( x/E' ↔ x=θ )⟩ // x does not exist is equivalent to x being the null expression
⟨( θ/x = θ )⟩ // a null expression has no attributes
Some examples of descriptions:
(box/blue)/blue // the blue box is blue
father(John) // John's father
( "father" is a function that assigns to each person's name, the name of his father).
king(France) // the king of France
{⟨( x ← (x = king(France) )⟩} // denotation
n/(n = 2+6) // the number resulting from the operation 2+6
Addenda
The descriptions defined in Principia Mathematica
Definite descriptions are described in a more technical way in Russell and Whitehead's Principia Mathematica. They used the notation ιxFx (the object x described by Fx), "ι" being the Greek letter "iota". According to its authors, this expression is not a formula, but a term.
The problem of the existential predicate in Russell
On something that does not exist one cannot assign a predicate because it leads to contradictions. For example, if we say that "The current king of France does not exist" −in traditional notation NoExists(F), being F "The current king of France"), and in MENTAL F/(Exists')− F has the property of "non-existence".
But for an object to have properties it is necessary that it exists (what does not exist cannot have properties) . If an object exists, it has properties, and if it has properties it exists. If F has the property of non-existence, then this means that it exists. Therefore, F exists and does not exist at the same time. Contradiction.
To solve this problem, there are two solutions:
An object can have properties without existing. To have properties it is enough to "be". Being is a deeper ontological level than existing, which is more superficial. F has properties because it "is". Therefore, F does not exist and is, which is not a contradiction. This is the solution given by Russell in "The Principles of Mathematics" (1903).
Every definite description exists. Therefore, F exists and, therefore, properties can be assigned to it, the contradiction disappearing. This is the solution offered by Russell in his 1905 article (On Denoting)
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