Paradoxical Aspects of the Russellian Conception of Existence

Abstract

In this paper, the authors try to clarify the relations between Meinong’s and Russell's thoughts on the ontological ideas of existence. The Meinongian theory on non-existent objects does not in itself violate the principle of non-contradiction, since the problem that this hypothesis offers to the theory of definite descriptions is not so much a logical problem as an ontological problem. To demonstrate this we will establish what we believe are the two main theses basic to the theory of descriptions: the epistemological thesis and logical thesis.

1 Introduction: The Theory of Non-Existent Objects

The first exposition of Russell's theory of descriptions appeared in his On Denoting paper published in the journal Mind in 1905. The theory is regarded (Smith, 2002) as avoiding the difficulties arising from Meinong's ideas, in which among other things that propositions such as "the round square" or "the mountain of gold", against any empiricist prejudice, refer to objects that are not precisely existing objects. In Meinong's thesis, a precise distinction is introduced between the object as an object and as an existing object. The object as such would consist only of its essential determinations, so that existence does not intervene in the least in the definition of what is the object. Such determinations must define the object as an object, if it is to be "something", and not a pure non-being, otherwise it would not constitute any object, although it must be admitted, in Meinong's sense that essential determinations, expressed by the propositions previously alluded to above do not refer to existing objects. These propositions, therefore, would not only have to refer to non-existent objects, but, once it is established that the determinations of roundness and quadrature belong to a round square, and to golden and mountainous belong to the mountain of gold, one can formulate true propositions about such objects. We can then say "the golden mountain is golden and mountainous" and "the round square is round and square". This situation lends itself to painful confusions, because, according to Meinong, the objects named by those propositions cannot be said to exist, but also it cannot be said that they are reduced to mere non-being. They are objects with properties of a well-defined nature, but we cannot affirm their existence. According to Meinong "there are objects about which it is valid that there are no such objects" (Findlay, 1963).

The main objection that Russell makes to Meinong originates in the context of the statements of the kind just quoted. In fact, to maintain that there are objects that do not exist, and yet assume that they are objects, is equivalent to affirming that there are objects for which the principle of contradiction does not hold, or to deny that they are subject to that principle (Russell, 1905). According to Russell, this statement is as unacceptable as the claim "the round square is round and also not round" and furthermore does not exist, assuming that there is an entity or an object with such properties. Russell maintains, not that the objects in question are beyond the principle of non-contradiction, but belong to its domain, that is, that they lend themselves to infringing the principle, so that when we wish to understand them, we find that we cannot understand them “As objects”. The situation that Meinong poses is, according to this, decidedly "intolerable". This is a situation that tends not to solve the theory of descriptions, but to eliminate it.

However, for Meinong's theory of objects, the alleged contradiction of non-existent objects does not seem to represent a problem if one starts from the idea that (1) the thought about an object that does not exist and (2) the thought of an existing object are totally different, and in such a way that the notion of "being" is different in each case. Therefore, when we say that something does not exist, that which we deny or exclude from existence must be something to which we impose this negation, or which we exclude from present existence. Something to which we impose the exclusion of existence is, in this way, something that has no place in the present world of existences. For example, unicorns do not exist, but if I deny that they exist, I do not deny, or contradict myself, about something that is part of the present world of existences, but rather assert the object as a possible mythical object of such or such properties.

We can state that the essential condition under which on object b is a possible non-existent object is represented by the proper use of modal logic in the following formula: $\neg \left( {\exists x} \right)\mathop {}\limits^{{}} \left( {x = b} \right) \to \diamondsuit \left( {\exists x} \right)\mathop {}\limits^{{}} \left( {x = b} \right)$. This formula expresses that, in the sense of the existential quantifier, there is nothing identical to b, although it may exist. Let E be real existence. We observe that in the formula $\left( {\exists y} \right)\mathop {}\limits^{{}} \left( {y = x} \right) \to Ex$, we would not know if the existential quantifier is used to speak of objects that really exist or of objects that only subsist as entia imaginationis. In the first case, the formula would be true and to avoid the mistake $\left( {\exists \mathop {}\limits^{{}} ,y} \right)\mathop {}\limits^{{}} \left( {y = x} \right) \to Ex$ is proposed. In the second case, it would be false, and would be represented by $\left( {\exists \mathop {}\limits^{{}} \circ y} \right)\mathop {}\limits^{{}} \left( {y = x} \right) \to Ex$.

From this point of view, Meinong's idea that there are objects in respect of which it is valid that there are no such objects, is no longer contradictory, if the "there" is meant, in the first place, as mere subsistence, but not as current existence. Secondly, if the "is not there" must mean deprivation of current existence, but not deprivation of subsistence. These would be the reasons why Meinong remained convinced that, even if the gold mountain does not exist, it still refers to an object, that is, it cannot lack meaning and is able to intervene as the subject of a true proposition. Now, since "there are" objects of this nature of which it can be affirmed that "they are not", in the sense of referring to a really existing object, the proposition "the mountain of gold does not exist" not only proves to be significant, but does not imply contradiction from the Meinongian point of view.

We may think that the difficulty, which Russell attributes to the doctrine of “non-existent objects”, is not so much that these objects threaten the principle of non-contradiction, but concerns the circumstance of such objects subsisting. Russell (1959, p.84) confirms that the distinction between the "there" of what subsists, and the "what is not" of what does not exist, from whose union in the affirmation "there is the mountain of gold that does not exist" cannot be a contradiction. However, Meinong's argument is unconvincing in Russell’s view, not because it leads to contradictions, but because it leads to Platonism, that is, to the postulating of subsistent entities in a world different from actual existences.

The theory of non-existent objects is not contradictory within Meinong's own theory. In any case, as soon as Russell stands within Meinong’s domain, it turns out that the real problem is ontological concerning the subsistence of things that do not exist as objects denoted by phrases like "the mountain of gold."

The theory of non-existent objects violates the principle of non-contradiction using the following argument: "Round square which is not round" designates an unreal object. If this proposition is symbolized with the letter "x", then the statements "x is round" and "x is not round" are both analytical, and therefore true, which clearly violates the principle of non-contradiction. Then both statements would be true by virtue of the principle of identity, which insures the truth of every statement of the form "XY is X", but violates the principle of non-contradiction. Given what "x" designates, if both statements are analytically true, then it only means that there is, in the sense of subsisting, the round square that is not round. That is, if both statements are true, it is understood that their truth corresponds to no real existence, but to a subsistence, then the subsistence in question is just that of the impossible object: the round square that is not round. In this sense, the analytical truth of both statements is limited to affirming the existence of a contradictory entity: there is (subsists) a round square that is not round, but this is neither actual nor possible, but impossible. Therefore, we ask ourselves, is it not only a question of asserting the analyticity of the same principle of non-contradiction? (Bunge, 1963). The violation of the principle of non-contradiction by both statements could only arise in the case that there is for both statements, in the sense that there exists, an affirmation of something that corresponds to what is possible, and that the principle of non-contradiction excludes. However, in this sense, it could not be considered a violation, because then it would have to appeal to the values of truth as cases of the replacement of statements, i.e., a case of paraconsistency (Nescolarde-Selva, Usó-Doménech and Alonso-Stenberg, 2015; Usó-Doménech et al., 2014; Usó-Doménech, Nescolarde-Selva and Segura-Abad, 2016; Usó-Doménech, Nescolarde-Selva, Segura-Abad, and Sabán,. 2019). In addition, it is clear that the mentioned statements above are of those that do not have cases of substitution (neither true nor false), because that would require the existence of an object that satisfied a condition that is impossible to fulfill.

Russell's theory of descriptions points out that Meinong's thesis about non-existent objects rests on a double confusion:

• 1. The first is to be confused about the meaning of symbols that stand for themselves independently of any propositional context, and this is just a way of confusing "proper name" with "description".

• 2. The second is to confuse the grammatical object of a sentence in which it appears as the subject in a descriptive phrase, with the logical subject of a proper proposition. That is, in confusing the logical function of the descriptive phrase with the logical function of the proper name.

Both confusions and their corresponding clarifications are closely linked. But Russell's examination of the first confusion attempts to point out that when a descriptive sentence, which has its meaning in the context of the sentence in which it appears, is taken as a symbol with its own meaning. Then the result is a statement that is either false or meaningless, while it makes a distinction that allows formulating statements that may be false or true, but never necessarily false or meaningless. The clarification of this first confusion and the resulting distinction constitute the proof that symbols, and proper names, have an epistemological function very different from that of descriptive sentences. It is proved here, in effect, how these distinctions make it possible to know what they are referring to and how they are referred to, and what we can expect from the point of view of the knowledge of various symbols.

The examination of the second confusion, on the other hand, shows how logically inadequate and misleading statements are made on this basis, while the distinction allows us to formulate statements of a correct logical structure. In fact, the analysis shows that the descriptive sentence is ultimately reduced to an indeterminate argument of a propositional function that could then satisfy a given value, that is, a proper name, whereas its confusion allows giving the formal argument by a certain value.

Then we establish the two main theses, which converges toward and contributes to the theory of definitive descriptions:

• 1. Descriptive statements such as "the mountain of gold", although they may contribute to the meaning of the sentences in which they are presented, have no meaning considered in isolation.

• 2. The same sentences can appear grammatically as subjects of meaningful sentences, and yet, once these sentences are analyzed with all precision, it turns out that they no longer have such subjects.

2 The Epistemological Thesis

In the development of the epistemological thesis, we will note how the concept of existence is a linguistic concept that cannot be applied to the level of the proper name. Instead, we present the interpretation of descriptions defined as incomplete symbols, since the idea of descriptions means nothing independently, and is supported by the epistemological assumption that if an expression is significant then its meaning is a denotation.

Russell sometimes defines the proper name from an exclusively syntactic point of view. The distinction between purely expressive language and purely informative language is considered false, because the subject is expressed in the communication of a message, and conversely, in purely expressive language, as in an exclamation, there is some information. To study the relation between language and reality, the function of ostensibility is defined and propositions are divided into ostensives and estimatives (Russell, 1948). An ostensive definition is the process by which an individual receives instruction to understand a lexeme in a different way than through the use of other lexemes. We can therefore formulate the following law we will call the Law of ostensibility.

Law of ostensibility The ostensibility of a sender is in inverse proportion to the quantity of information being provided by a particular language.

An ostensive definition conveys meaning using examples. In this case a Sender refers to an absolute being or referent (Ogden & Richards, 1989), also known as designatum (Carnap, 1942). In this case the information about the referent is in the foreground. The language in this situation is called an ostensive function (O_F), because the proposition is the translation of an ostension, equivalent to a remark. When the Subject makes a proposition or set of ostensive propositions Subject is operating on Reality. We must remember that for the Subject reality is processed with a system of signs encoded in language. Both signs and language are heterologous systems or related ways of representing reality. This lack of congruence between the system of signs that make up the process of the real and the codified system of linguistic signs has the following implications:

• (a) The semiotic system used to represent reality is a continuous and evolving process, but the coding is discontinuous. This discontinuity requires the perceiving Subject to make an optional decision about which aspect of the process of representing Reality to verify.

• (b) In turn, the coding performed is optional.

• (c) There is one relationship between the systems of the verified coding and the processes of representing the real. In any case, this is like a kind of interpretation of a sector of reality by denotative significance.

Between Reality conceived as system (Nescolarde-Selva & Usó-Doménech, 2014) and the linguistic coding system there is no identity (isology) or isomorphism, but at most, equivalence. In speaking about Reality, a Subject first selects from the perceptual field in a pragmatic way, and then transmits a message even at the expense of substantive categories. For Quine (1960) there is a category of lexemes, and in general terms, units of significance (moneme on) which play an ostensive function. These lexemes would be the extension of demonstrative pronouns "this", "that", etc. We call ostensive those lexemes that function as verbal pointers to references external to the sender that may be real or unreal. These minimal units of significance require an address to a reference that the sender locates outside himself. An ostensive proposition (O) is one in which the sender specifies connotations that apply to the reference, i.e., the nature of the qualitative and quantitative apprehension of reality that at that moment constitutes a referent. An estimative proposition (E) is one which plays an ostensive role not with respect to the referent, but to the sender.

However, at other times he defines it according to the epistemological function that it performs. A proper name is defined syntactically as a word that can only appear in a proposition as a subject, or as a word that can appear in any proposition that does not contain variables, but that does not denote neither a predicate nor a relation (Russell, 1948, p. 73). The epistemological definition of the proper name considers the word that functions as a name in its direct reference to an individual, on whose immediate denotation its meaning rests and only on that, that is to say, independently of what other words may mean with which it eventually appears linked. A proper name is a simple symbol that directly designates an individual which is its signifier, and which has its own meaning regardless of the meaning of the other words (Russell, 1960, p. 174). To the extent that a proper name reduces its significance to this immediate reference to an individual (signifier) of which it is the name, its definition can only be an ostensive definition (Nescolarde-Selva et al., 2014, 2015a, 2015b). On the other hand, insofar as its signification is independent of the other words with which it can be linked, it is not possible to make it manifest by means of a verbal definition. We can say that proper names have a way of signifying autarkic status, because they only represent the individual. But this epistemological definition is not inconsistent, and can be arranged simultaneously with the role assigned to its syntactical definition. This is so since a proper name at the same time that it can only be ordered as a subject of a proposition, can only mean an individual whom it names directly. In other words, if a proper name intervenes as the subject of a proposition that does not contain variables, it is because it denotes something that we know immediately, and vice versa.

It is evident that if a proper name is presented in isolation or in a proposition which does not contain variables, it is always the symbol of an existing individual, since the characteristic of a proper name is that its meaning is exhausted in the direct and immediate reference to the individual whom it names, so that, unless there is an individual whom it denotes, the name cannot have any significance. However, it should be noted that even when a proper name denotes an existing object, it does not manage to be an adequate way to offer us a true or legitimate concept of existence. It is true that a proper name itself designates an existing object, but a proper name never says that the named object exists. It limits itself to designating the immediately present object without implying anything more about it, such as whether it exists or does not exist.

Let x be the symbol of a proper name. The reason why we cannot say whether a named object exists or does not exist is that if we express "x exists" or "x does not exist", these expressions are equivalent to saying that the property of being or existing is applied or is predicated about an individual who is already there before any predication. Moreover, this is because it has been preceded by being named demonstratively, or that an individual, who in turn exists, or since he has been named, predicates the property of not existing. In the first case the predicate of existence says nothing, it is superfluous. In the second case, it is not, it is contradictory. According to this, the concept of existence can only be attained where it is possible to speak of existence, and therefore it is impossible to speak of existence at the level of the proper name and the named. "Existence" and "being", if they have any meaning, have to be linguistic concepts that cannot be applied directly to objects (Russell, 1962, pp. 61–62). This allows us to formulate the following definition:

Definition 1

By proper name should be understood only the subject matter of a proposition without variables, which means designating an individual in an immediate way, independently of what other words mean. That is to say, by designating per se an object about which, nevertheless, it never indicates "that it exists".

We can say that if the proper name only has meaning because it immediately presents an individual object, regardless of the linguistic context in which it may take place, then we cannot analyze it in other words and maintain that the same meaning of the name is preserved in those other symbols in which supposedly we would have analyzed it. The proper name is not only a simple symbol in the obvious sense that the letters of which it is composed are no longer symbols, but also in the sense that their meaning cannot be analyzed in other symbols, however much they appear to have same meaning. We would totally blunder if in wanting to analyze a name, we think that "Miguel de Cervantes", used as a proper name, means the same as "the person named Miguel de Cervantes" or "the person who bears the name Miguel de Cervantes", because while in the first case we simply name an individual, in the second case we "describe" him as the individual who bears that name. In the first case, the name is not part of the affirmed fact, but is the symbol to name something that is a fact. In the second case, the name is not only part of the statement, but also constitutes the fact that we are affirming. When a name is used directly to indicate simply what we are talking about, it is not part of the affirmed fact, or of a falsehood, if it happens that our statement turns out to be false. It is only part of the symbolism, through which we express our thinking. In this way, we cannot displace a proper name by analysis, replace it with a description, and hold that the same meaning is preserved. Nor can it be argued that the proposition in which a particular name is presented remains the same proposition when instead of that name we establish a sentence like "the person (or object) that is called …", or any other descriptive sentence. Because, if name and description do not coincide, its displacement and a substitution in the context of the proposition, carries with it a displacement and a substitution of a proposition by an entirely different one. That is why Russell interprets the proper name as a genuine constituent of the proposition in which it is presented.

Consequence 1 It is impossible that by analyzing a proposition, we can make the symbol disappear that directly designates an existing individual without the same propositional signification also disappearing.

Either the named subject exists, that is, or the name is a proper name, and then it is a logical constituent of the proposition in which it appears as subject, or this is not the case, and then it is not an authentic constituent of the proposition. Only a proper name can have meaning independently of any propositional context, but if it is integrated in a similar context, only the proper name is a logical constituent of the proposition, a logical subject and not just a grammatical subject.

According to Russell, there is a sharp distinction between a proper name and the descriptive sentences that are used to characterize something. For example, "the author of Don Quixote", and in general, all those sentences that begin with the singular definite article "the". The true meaning of these sentences, unlike the proper name, is that when and if they express certain properties about something, they can only establish this reference by virtue of its form. That is, by the meaning of these phrases under the contextual definition of the given article "the", but not by virtue of an immediate and direct presentation of the object. It is clear that what we know about a term, “the one”, implies knowing how many propositions there are that can use it as a subject in a descriptive phrase, and knowing who or what is real as “the one”, that is, knowing a proposition of the form “x is the one”, where “x” is a proper name. The distinction given between a proper name and a description corresponds to the distinction between "direct knowledge" and "knowledge about". From this distinction, it is evident that nothing that has been declared about the proper name can be applied to the descriptions. The descriptive phrases that imply immediate reference to an individual cannot be sustained. Therefore, these phrases do not provide any denotation of existence in the sense that existence is essentially linked to the proper name. Moreover, it is not possible that they mean anything independent of the propositional context in which they may occur, nor do they represent a logical constituent of propositions when they come to form one of their parts. Russell also argues proving that a descriptive phrase is always incomparable with a proper name, which does not mean anything in and of itself, nor can it be a complete constituent of propositions, and can be presented in a concise manner as follows in the following example:

Example 1

If the descriptive phrase "the author of Don Quixote" meant something other than "Miguel de Cervantes," the proposition "Miguel de Cervantes is the author of Don Quixote" would have to be necessarily false, but this would in its turn be false. On the other hand, if the sentence "the author of Don Quixote" only meant "Miguel de Cervantes", it would have to be admitted that the proposition "Miguel de Cervantes is the author of Don Quixote" means the same as "Miguel de Cervantes is Miguel de Cervantes", which is also not accurate because "Miguel de Cervantes is the author of Don Quixote" does not express a tautology. The result is that, if the sentence "the author of Don Quixote" does not mean "Miguel de Cervantes" or anything else, it cannot in any way be said to have any meaning, that is, the sentence "the author of Don Quixote" means nothing on its own.

But if the proper name does not exhaust its meaning in denoting an individual directly, but implicitly or in an unanalyzed form, it has the same meaning explicitly or analytically in the descriptive sentence. And if, in addition, the descriptive phrase means or denotes in an indirect way, the same individual as the proper name denotes directly, in this case the proper name and defined description would mean or would denote the same individual, but in a different way, than in a direct and indirect way. In addition, they would have the same meaning, but in a different way, than in an implicit way that is not analyzed, and this in an explicit or analyzed way. In the previous example, this would mean that "the author of Don Quixote" and "Cervantes" mean or denote the same, but in a different way; and that they have the same meaning, although expressed in a different way. In addition, between both expressions there would be a possibility of a tautology, and that the exclusion between a proper name and a definite description would be suppressed. Moreover, it would be open to the possibility that the proper names can be treated as descriptions, (Quine, 1966, pp. 215 ss.) and that the defined descriptions can be treated as proper names, if the relevant predicate constants are available (Martin, 1958, pp. 54 ss.). Following Russell, we can draw the following conclusions:

• 1. Whatever the meaning of a proposition that contains a description, it can never be equivalent to the proposition that results when that description is replaced by a name, even if this name designates the same as the description describes, because the resulting proposition is necessarily false or is a tautology, whereas the original proposition may be true or false.

• 2. This situation is precisely the proof that the descriptive phrase does not refer, in isolation, to any object.

The descriptive sentence, insofar as it does not refer to or present any object, cannot function as a logical subject of a proposition or a logical constituent of it, although it may happen that it appears in the place of the proposition that is proper to the name. When the latter occurs, any proposition can be analyzed in such a way that, without losing its meaning, the descriptive phrase becomes a subject of the proposition, which shows that it is not a logical subject but a "grammatical subject", to which it can be considered as a genuine constituent of the proposition. This indicates that, in contrast to the proper name, the descriptive sentence can only mean and be defined within the context of the proposition in which it is presented, and not in isolation. It is for this reason that descriptive sentences are incomplete symbols, that is, they can only be defined as they are used in a proposition. Of course, if they are not logical constituents of propositions, it remains to be determined what they mean when they intervene in meaningful propositions.

Let $F(x)$ be a propositional function and x be a term or variable. $\left| {xF(x)} \right.$ is the proper symbol to express a definite description that can be read as "the x that satisfies the function F(x)". Since we can know any number, variable, of propositions relative to the described object without necessarily knowing who or what is such an object, and since the defined description always describes a certain single object and not another, we can say:

• (a) Any description of this class claims a propositional function "F(x)".

• (b) Moreover, at the same time it claims the uniqueness of the value that can satisfy the variable "x", that is, the operator$\left| x \right.$

Any proposition in which the expression $\left| {xF(x)} \right.$ occurs can be expressed in another proposition in which this symbol has disappeared, without the significance being altered in passing of one proposition to another proposition. Only then can it be revealed what the description means in the propositions in which it appears, why the description does not function as a logical constituent of the proposition, and why it is an incomplete symbol. In this way, the confusion between logical subject and grammatical subject has just been disassociated.

3 The Ontological Thesis

In the ontological thesis, it is shown that the logical concept of existence is part of the meaning of the sentences in which a definite description occurs. The ontological sense of existence, linked to the existential quantifier, is fixed as something unique that could exist, and that the description is presented in these sentences only as an indeterminate argument of the propositional functions.

We consider the following proposition P = "The author of Don Quixote was lame". By virtue of the description given therein, and in accordance with the above, this proposition means, first, that "someone" wrote Don Quixote. In fact, this proposition could never be considered true if at least one person had written Don Quixote because nothing is said about who that particular person is. That is to say, the proposition implies a propositional function F(x), of which it is necessary to affirm that it is not always false. At least in one case it is true, and then it implies $\exists x/F(x)$, that is "there is at least one object x, such that F(x)". But the proposition has only reference, it refers to a single person and not to more than one; then, any other person (e.g., "y") who, in addition to x, had written Don Quixote, would have to always be considered as the same person x. Second, what proposition P means is that, given that x and y have written Don Quixote, x and y will always be identical; and about this, it is necessary to affirm that it is always true, because otherwise the proposition could not have a single reference. Thus, the second thing P means is that $F(x),F(y),\mathop {}\limits^{{}} \supset_{xy} \to x = y$, i.e. "if F(x), F(y), then, for all values of x and y, x is equal to y". Then we can deduce that if any other term has the same property F that has x in the propositional function F(x), and in this case it is identical with the latter, it is because it is one and only one, that is, at most, an individual satisfies the function "F(x)".

Now, it is from only this term that the proposition P that is analyzed, affirms, "he was lame". Consequently, what P means in the third place is that "if x wrote Don Quixote, x was lame". In addition, it is no less necessary to assert that it is always true, since the two previous moments of analysis imply the existence of a value of x, and unless it exists, it will not be lawful to ascribe any property to it. If, therefore, it is always true that "if x wrote Don Quixote, x was lame," this meaning can also be expressed by saying, "whoever wrote Don Quixote was lame." Then what implies the integral meaning of P is a dual propositional function in which we find, in the first place "x wrote Don Quixote" and secondly "x was lame". The property consisting of "writing the Quixote" and that which consists in "being lame" are but two cases that illustrate or determine the predicate variables F and f, the dual propositional function that has the analysis of any proposition of the type which we are exposing, and which is framed within the existential qualifier. This will be in symbols: $\exists \alpha /T(F(\alpha )) \equiv_{x} \mathop {}\limits^{{}} x = \alpha \wedge T(f\left( \alpha \right))$ that we can read "there is an object α such that F (α) is true T (F (α)) when and only when x is α, and f(z) is true T(f (α))”. Or "there is an object α such that F(x) is always equivalent to" x is α "and f(α)".

We can ask the following questions: What relationship do the results of this analysis have with the idea of existence? More precisely, what idea of existence is contained in the propositions that accept in their context a descriptive sentence and that the analysis has exposed? In the case of the proper name, existence was data on which nothing could be affirmed or denied. On the other hand, the descriptive phrase, by itself, does not refer to any object, existing or non-existent, which, considered outside of propositional contexts, lacks any designatum capable of being named, but that analyzed in the context of a proposition, "something" would come to mean. We understand that "something" would come to mean the proposition in which it is involved. Therefore, what propositions contain in their context is a description that something must exist, so that these propositions may have meaning, but whose existence is incomparable with that existence that is associated with the significance of the proper name.

We can summarize the two moments of the propositional analysis as follows:

• 1. The first of the two initial moments of proposition P revealed that at least one individual wrote Don Quixote, that is, that the proposition P = "the author of Don Quixote was lame" would always be false if ever there was at least one person who wrote Don Quixote.

• 2. The second moment of the analysis showed that, since the contextual description is a definite description, of unique reference, it was necessary that at most Don Quixote had been written by an individual and only by one. Otherwise, either the proposition is false or "the others" who wrote Don Quixote are the same unique individual, which would confirm the uniqueness of reference and the truth of the proposition.

These two steps help us to understand Russell's concept of existence. Indeed, Russell states that when descriptions are used in a proposition, the result of the first two moments of their analysis is already a concept, and the most consistent concept of existence. Therefore, thanks to these two first steps you can see that the propositions:

• (a) P₁ = "x wrote Don Quixote" is not always false.

• (b) P₂ = "If x and y have written Don Quixote, x and y are always identical," they are equivalent together to the following proposition:

• (c) P₃ = "There is a term α in such a way that" x wrote Don Quixote "is always equivalent to" x is α ", where "α" is a value or a constant that replaces "x" and satisfies "F(x)”. That is, that “$\exists x/F(x)$” and “$F(x),F(y),\mathop {}\limits^{{}} \supset_{xy} \to x = y$”together they are equivalent to "$\left( \alpha \right):F\left( x \right) \equiv_{x} \mathop {}\limits^{{}} x = \alpha$"”.

As we have already indicated, a descriptive sentence can be expressed by the symbol $\left| {xF(x)} \right.$. The examination of the proposition in which this phrase occurs, or the definition of the proposition that uses the symbol $\left| {xF(x)} \right.$, leads, in its first two moments, to extract the meaning that "at least" and "at most" an individual possesses the property of which the description speaks. This is, in our example, that "there exists a term α in such a way that "x wrote Don Quixote" is always equivalent to "x is α"" if it expressed symbolically $\left( \alpha \right):F\left( x \right) \equiv_{x} \mathop {}\limits^{{}} x = \alpha$. But this assertion means nothing other than what we affirm when we say: "the term that satisfies the function "x wrote Don Quixote" exists." Using Russell’s symbols we can write “$E!\left| x \right.F(x)$”. The symbol expresses what is already included in every proposition where a descriptive sentence occurs. However, this is precisely what has been defined as "there is a term α in such a way that "x wrote Don Quixote" is always equivalent to "x is α”". According to this equivalence, we can see that the result of these first two moments of the analysis of proposition P is about a description leading to the following definition:

Definition 2

$E!\left| x \right.F(x). = /\exists \alpha /F(x). \equiv .\mathop {}\limits^{{}} x = \alpha$.

Now, as regards the third of the moments of the analysis of proposition P, "the term that satisfies the function "x wrote Don Quixote", satisfies the function "x was lame", which was defined as "exists an object α such that "F(x)" is always equivalent to "x is α", and f(α)". ($\exists \alpha /F(x). \equiv_{.} \mathop {}\limits^{{}} x = \alpha /f\left( \alpha \right)$), because in every proposition in which a description occurs, the affirmation of existence expressed by the symbol $E!\left| {xF(x)} \right.$ is already contained, the following expression, at the same time as it implies the equivalence of the definition 2, and assumes also in terms of equivalence the integral meaning of the proposition.

Definition 3

$f\left[ {\left| {xF(x)} \right.} \right]. = /\exists \alpha /F(x). \equiv_{x.} \mathop {}\limits^{{}} x = \alpha /f(\alpha )$.

The left-hand end of the equivalence reads "the term satisfying "F (x)", satisfies "f (α)"". The term on the right has already been indicated. We can observe that the affirmation of existence is part of what the propositions in which a descriptive phrase occurs. In fact, this is what the first two moments of the analysis establish what is defined in definition 2, and assumed in definition 3.

Strawson objected to these results (Kirkham, 1992, Chapter 10). According to Strawson to say "the current King of France is bald" is not to assert that there is a person, and only one, who has the property of being King of France and the property of being bald. This would only be true if one made use of that sentence to refer in appropriate circumstances to a particular person. The sentence itself does not state or affirm the existence of such a person. If this sentence is used in any particular context, in a given space–time circumstance, it constitutes an existential assertion. Otherwise, it does not affirm, but merely presupposes, the existence of the person in question. Thus, when a sentence like "the current King of France is bald" is used to refer to a person, this sentence does not constitute an assertion, unless there is a person to whom it refers directly, because when this person does not exist, it cannot be said that the sentence has been used. We find the paradox: How can we say that the individual referred to a sentence in use exists when the same use of the judgment establishes existence?

Since Russell's analysis of these sentences does not take into account the circumstantial use of them, it is also not possible to show any affirmation of existence as part of the sentences that he analyzes, even though they presuppose, for example, the King of France. That is, if someone now pronounces the sentence, she would be making a true assertion if and only if in fact there was a King of France, or if he was bald. Consequently, if one of the functions of the particular article "the" is considered a sign that a single reference is made, a sentence such as “the current King of France is bald" only indicates, but does not assert the existence of this particular individual.

It has already been established that descriptive phrases, and the sentences in which they are presented, express certain properties about something, and that they have this reference only by virtue of their form, that is to say, by virtue of the contextual definition of the article "the”, but not by virtue of an immediate and direct presentation of the object. Therefore, the existence that manifests as part of the sentences that are analyzed, remains "something" unique and indeterminate because of a definition and, therefore, it is a construct of a semantically pure concept that makes an abstraction. According to Russell, the concept of existence is part of what the statements in which the descriptive phrases appear and is a strictly semantic concept. However, Strawson's concept of existence is a pragmatic concept, since it applies only to the object-language designate. Language requires, the time, the place, the situation, the identity of the speaker, the subject, which constitutes the immediate focus of interest, and the personal stories of the speaker, and those to whom he is addressed. Strawson is aware of the difference between one concept and another. Nevertheless, this does not seem to prevent him from incurring a subrogation, rather, a confusion of concepts, in denying the need to carry out the logical analysis of the sentences in question. Russell has been able to reveal as part of them, an affirmation of existence, in the pragmatic sense he espouses, because this supposes, mistakenly, that such was what was intended. Nevertheless, just as Strawson could say that it was absurd to argue that such sentences constituted pragmatic claims of existence, it can be said that it would be absurd to deny that these sentences constitute statements of existence within the framework of a logical definition.

The decisive part of the analysis of the proposition in question, which ends in definitions 2 and 3, is to prove that if the symbol of description “$\left| {xF(x)} \right.$” appears as the subject of a proposition, it is always possible to present that proposition in a form such that the symbol has disappeared. However it is conserving the same meaning by a double propositional function, linked to the existence quantifier. This is possible because the definite description that occurs in the proposition is not a logical constituent of it. Another way of knowing the reason why the symbol of the existence quantifier ∃ is presented in definitions 2 and 3 is to say that the symbol of the description can never be treated as a value of the function in which it is presented. Otherwise, of functions that are always true, we should expect them to be always true, with respect to a description. Thus x = x will always be true, where x can be replaced by a given value or by a proper name. However, this function will not always be true if x is replaced by a description, such as "the author of Don Quixote", because in order for the function to be true in this second case, it is also necessary that the author of Don Quixote exists. Nor can it always be true that $\left( x \right).\mathop {}\limits^{{}} f\left( x \right). \supset .f\left| {xF(x)} \right.$, because while the function f(x) is always true, it may be false regarding the description. That is, it may happen that the description does not describe anything, or what is the same, that $\left| {xF(x)} \right.$ does not exist. The conditional can only be valid when the term satisfying the function "F (x)" exists, that is, when $\exists !\left| {xF(x)} \right.$.

We must ask ourselves the following question: what does it mean to say "the term satisfying the function "F (x)" exists"? For what we have just stated "the term satisfying the function "F(x)" exists" can only indicate that the term has the property "F", otherwise the description does not describe anything, and the term which satisfies the function "F(x)" does not exist. In this way, Russell comes to equate the existence of a term described with the possession of certain properties, as part of that term. Since the current King of France does not exist, we cannot claim he has the property of being bald, or not being bald. When the term satisfying the function "F(x)" has any property, whatever it is, then it exists.

However, we still have to ask ourselves, what does "existence" mean when it comes to the existence of a term insofar as it possesses this or that property? This issue is now presented with all due relevance and substance. In fact, let us assume that the term satisfying the function "F(x)" has a property F. We cannot say that, since such a term exists if it has such a property, it exists in the way the individual named by a proper name exists. This is because the term here is not the nominatum, but the one that owns that property, that is, the descriptum. A term described never implies knowledge about what or who in particular possesses that particular property, and on the other hand, it is necessary to state that such a term exists, since "at least" and "at most", an individual has that property. It must therefore be concluded that the existing one, of which the propositions speak, where a description is used, is a unique being existing that could be designated directly, but which is not really designated in that way (Moore, 1959, pp. 187 and 193). Consequently, this means that when we say "the term satisfying the function "F(x)" exists" ($\exists !\left| {xF(x)} \right.$), the description or its symbol, can only be interpreted as a formal and indeterminate argument of the function in which it is presented. And for which there could be a certain value that makes possible, or a true proposition, in which case the value of the argument requires existing, or a false proposition, and we have the opposite. That is to say, the circumstance that the description or its symbol constitutes only an indeterminate pure argument of a function, leads to interpret "existence" as that which ontologically could happen as a unique value of said argument. This is precisely what the symbol $E!\left| {xF(x)} \right.$ means. This is the notion of existence that was treated when it was said that in every proposition in which a description occurs, content $\exists !\left| {xF(x)} \right.$ already exists as part of what is affirmed.

4 Conclusions

If existence means what could happen as a particular value of the formal argument that constitutes the description, then the descriptive phrase, like the proposition that contains it, can make sense, even if nothing "real" is referred to by the description. Nevertheless, this was not what happened with proper names. Here the immediate reference to the present object is the measure of meaning. There is a contrast both as regards the meaning that a proper name can or cannot have and the meaning a description can or cannot have when they are referred to existence, particularly for the existence of these symbols when they refer to the possibility of speaking or not speaking about the meaning of existence. Because while it was not possible to speak meaningfully that the named person exists or does not exist, although if he is named he exists, it can be meaningful that the descriptum of the corresponding description exists or does not exist. In the latter case the possibility remains open that the value that must satisfy the formal argument of the description is present or not, exists or does not exist, thus leaving open the possibility of affirming or denying it as such. Therefore, only descriptions can be, strictly speaking, affirmed or denied existence. When in ordinary language it is said that something exists, it is always something described, that is to say, not something that appears immediately as a taste to the palate, an odor, a color, but something like "energy", "spirit" or Miguel de Cervantes, insofar as it means to be the author of Don Quixote.

Consider the following reasoning: "My present feeling exists; this is my present feeling; therefore this exists." In agreement with Russell, although the two premises of this reasoning are true, the conclusion makes no sense. Indeed, in a premise I write "my feeling" for a temporary note (present), and as I affirm that it exists, I affirm it as something described. In the second premise, I consider "this" as I just wrote. Nevertheless, in the conclusion, "this" is not taken as a description, but as a designative symbol of an immediate object, as a logical proper name, so that it cannot function as an indeterminate argument for a propositional function, with respect to which it can be presented existence as a case. The "this exists" is an idle addition.

We have explained the reasons why the symbol of the description, which is presented as the subject of the proposition, is only a grammatical subject that easily induces one to confuse it with a logical subject, that is, with a designative symbol of an object, in the strong sense of the term.

Returning to our starting point, when we say "round square does not exist" and we symbolically express the meaning of the sentence in the following way; $\neg \exists !\left| {xF(x)} \right.$, we might think that the "round square" refers to an object. That is to say, it constitutes a logical subject about which we affirm then that "does not exist", when rather we should understand by that sentence that "there is no value that satisfies with character of unique value the function "F(x)"". Therefore, if we take into account that the symbol of the defined description, and the notion of existence that with necessity implies was defined by $\exists /F(x)\mathop {}\limits^{{}} \equiv_{x} \mathop {}\limits^{{}} x = c$, the sentence "the round square does not exist" is a case of a sentence whose symbol "" should be expressed in a precise sense of the following: $\neg \left[ {\left( {\exists c} \right)/F(x)\mathop {}\limits^{{}} \equiv_{x} \mathop {}\limits^{{}} x = c} \right]$. In fact, the definite symbol lends itself to being mistakenly interpreted as "there is a certain object about which we affirm that there is no such object", while the symbol that defines it only expresses: "it is false that there is an object c such that x has a certain property when, and only when x is equal to c."

Thec preceding developments have shown that when Russell defines a proposition in the context of which a description is given as subject, and when, through its analysis, it becomes clear that the concept of existence, the existence here discussed, is not one that is directly linked to the proper name and about which no affirmation is possible. The existence treated is only an existence that can be affirmed of descriptions and that, when these are analyzed, turns out to be a case of propositional function, which is true for at least one value of the variable. Hence the symbol of the description disappears from the proposition that is defined, and that in its equivalent the respective function is preceded by the symbol of the existential operator ∃, which never means actually existing.

Consequently, and according to Russell, to say of something that it exists is to say that there is a propositional function which is sometimes true, that is, that there exists at least one true proposition of the form "x is a such" in which x is a name. The definition of existence thus exposed applies exclusively to indefinite descriptions, where existence is equivalent to declaring that "given an expression f(x) containing a variable x, and which becomes a proposition when a value is assigned to the variable, the expression $(\exists x)\mathop {}\limits^{{}} f(x)$ means that there is at least one value of x for which f(x) is true". Nevertheless, Russell warns that this definition of existence is also valid with respect to defined descriptions, with the proviso that the value of the given variable must in this case be a unique value (Russell, 1969, p. 172). The function here can only be true for value x at most. The form adopted here by the statement of existence has already been exposed with all its implications, and is now simply reiterated as follows: "there is an object (or a value of) c such that "F(x)" is always equivalent to "x is c"".

We must insist on the paradoxical aspect that we have tried to emphasize, when neither applied to definite descriptions nor to indefinite descriptions, the notion of existence alludes, as such, not to an actual existing, but to an object that could exist.