The Frege-Russell Classification of ‘is.’

One: Introduction.
One of the basic assumptions of the first-order logic of Gottlob Frege, reproduced in the notation of Bertrand Russell, is that the ‘be’ conjugation, usually represented in the form of the third-person singular ‘is,’ is semantically ambiguous between:

1. The ‘is’ of predication, as in ‘Socrates is wise.’
2. The ‘is’ of identity, as in ‘Mark Twain is Samuel Clemens.’
3. The ‘is’ of class-inclusion, as in ‘Man is an animal.’
4. The ‘is’ of existence, which can include statements such as ‘Socrates is,’ and the more common type of ‘there is’ locutions.

The two variants of ‘the ‘is’ of existence,’ which Peter Geach called the ‘actuality’ and ‘there-is’ uses, are often themselves regarded as separable. The notation of both Frege and Russell, does not strictly distinguish between them but is more usually concerned with the ‘there is’ variant.

Nevertheless, the fact that these different uses of ‘is’ are expressed by means of three different and irreducible symbols in Frege’s notation, means that in at least three of these four different uses of the word ‘is’ are taken as having a logically different meanings, independent of the syntactic environment in which they occur. The distinctive character of identity statements is particularly important within Frege’s philosophy of language, since it underpins the distinction between sense and denotation which he introduces to solve paradoxes regarding identity statements and propositional attitude reports.

This notion of ‘is’ as having a number of different meanings, which is here given to a formal treatment, is the culmination of a long tradition in philosophy. With Aristotle for example, there are indications that he held to a distinction between the predicative and the existential uses of ‘is.’

For it is not the same thing not to be something and not to be simpliciter, though owing to the similarity of language to be something appears to differ only a little from to be, and not to be something from not to be.
On Sophistical Refutations. 167a4-6

Since the ‘there is’ use does not occur in Attic Greek, Aristotle himself does not make this distinction. However, with identity, class-inclusion and predication, these are regarded by Aristotle as cases of ‘the way in which ‘to be’ (ousia) is stated. All are thus regarded as cases of predication of what is to be regarded as essential or accidental, with ‘Socrates is a man’ and ‘Socrates is pale’ taken as examples of the first and the second respectively. It is upon this basis that Aristotle then groups accidental predications into those particular classes of essential predication that form the basis of his categories.

At the other end of the historical scale, with Ludwig Wittgenstein, the question of whether this particular aspect of Frege and Russell’s logic was rejected in his later philosophy is perhaps more difficult. The following quotes seem to indicate that his position on this point was somewhat ambiguous.

What does it mean to say that the "is" in "the rose is red" has a different meaning from the "is" in "twice two is four"? If it is answered that it means that different rules are valid for these two words, we can say that we have only one word here. –And if all I am attending to is grammatical rules, these do allow the use of the word "is" in both connections.—But the rule which shews that the word "is" has different meanings in these sentences is the one allowing us to replace "is" in the second sentence with the sign of equality, and forbidding this substitution in the first sentence.
Philosophical Investigations I, 558.

Now isn’t it queer that I say the word "is" is used with two different meanings (as the copula and as the sign of equality), and should not care to say that its meaning is its use; its use, that is, as the copula and the sign of identity?
One would like to say that these two kinds of use do not yield a single meaning; the union under one head is an accident, a mere inessential.
Philosophical Investigations I, 561.

Nevertheless, the question remains as to whether these two different uses highlighted by Wittgenstein, in addition to those identified by Frege and Russell, are in fact aspects of the same use. And this indeed is what the AIT analysis shows: That each of these meanings of 'is' can be explained in terms of the relation between the Processant function and that of the Extantal Imbuant.

Two: The ‘is’ of Predication.
Sentences like ‘Socrates is wise,’ attribute a particular quality or characteristic to a subject. It is clear that this notion of predication lies closest to the conception of the Processant function identified by AIT.

The main difference lies within the emphasis placed by AIT upon Extantal Imbuancy, and its linking of the function of the 'be' conjugation to this linguistic phenomenon. Thus a predicate can be seen as a separable mode of existence of the subject, which means the attribution of a quality or characteristic in such a way that the attribution itself becomes that which allows a statement to have truth or falsehood. In other words, the utterance ‘The tall man’ is no different from the sentence ‘The man is tall’ in terms of the qualities attributed to it. The difference lies within the fact that the ‘is’ here allows the quality of ‘tallness’ to be separable from ‘The man.’ This mode of existence is exhibited in such a way that it can be examined and hence challenged.

The standard notion of ‘predication’ is strongly linked to that of the ‘copula.’ But from the above example it can be seen that the so-called ‘copuletic function is inferred from the fact that where a predicate is displayed as a quality of the subject, it can be effectively challenged as to whether it does indeed pertain to the subject. Thus it is assumed that the possibility of this disjunction arises out of a previous ‘conjoining’ of the predicate to the subject. Yet since this quality stated in the predicate can equally be stated as part of the subject, in such a way as not to allow its disjunction, the notion of ‘joining’ is at best misleading.

The notion of Extantal Imbuancy allows this to be made clearer, since "The tall man" announces the existence of this particular entity as an inseparable whole, whereas "The man is tall" allows the announcement of the same entity in terms of separable, and thus challengeable, elements. We can then ‘test’ this proposition, as to whether 'tall' does indeed correspond to 'the man,' but only because this has been initiated into a separable pertaining correspondence an explicit Processant. Without a Processant, there would be no predicate to ‘join’ onto the subject.

Thus the ‘is’ here allows modes of existence to be displayed in terms of a possible correspondence wherein, the predicate pertains to the subject. It should of course be noted that there are in fact two types of ‘correspondence’ simultaneously at work here. The first is we can call a ‘semantic correspondence,’ and the second, directly attributable to the Processant function, we can call ‘propositional correspondence.’ is Here, the semantic reference of the predicate is separated from that of the subject in order to be related back to it. This propositional correspondence itself depends upon a semantic correspondence between the predicate and the referent of the subject. If both these semantic correspondences hold, then the proposition itself can be said to be true. Thus the Processant allows correspondence/non-correspondence to be introduced into the formal structure of language, in a way that goes beyond the correct/incorrect use of words upon which semantic correspondence depends.

It is this notion of the Processant as a correspondence function that forms the connection between ‘the ‘is’ of predication,’ ‘the ‘is’ of identity,’ ‘the ‘is’ of class-inclusion,’ and ‘the ‘is’ of existence.’

Three: The ‘is’ of Identity.
This refers to sentences of the type "The Morning Star is the Evening Star," "Mark Twain is Samuel Clemens," or "Carl XVI Gustaf is the King of Sweden." But it can also be seen within sentences like "Seeing is believing," where both the verbs are gerundialised.

Since the ‘is’ in such sentences can be taken as stating ‘is the same as’ in way that is not the case with predication, this ‘is’ is usually interpreted as having an entirely different function here. This aspect of identity statements means they have a ‘substitution property’ which, again, distinguishes them from predication statements.

For example, the sentence "Carl XVI Gustaf is the King of Sweden," can be interchanged to give "The King of Sweden is Carl XVI Gustaf." The resulting sentence, whilst not identical to the first, has a definite and strongly related sense. This ‘substitution property’ cannot be seen as operating within a sentence like "Carl XVI Gustaf is eating," and it is upon this fact that the distinction between ‘the 'is' of predication’ and ‘the 'is' of identity’ is based.

In order to resolve this apparent distinction, two key elements of the AIT analysis have to be seen as operative here. The first is propositional correspondence, and the second is the Modal Switch. This is seen within a sentence like "Carl XVI Gustaf is eating the apple," which would be analysed as follows:

"Carl XVI Gustaf [Extantal Imbuant] is [Modal Processant] eating {the apple [Extantal Objectant]} [Modal Informant].."

Here, the Extantal Objectant serves to establish that the apple is the object to which the Imbuant is referred, in such a way that the mode of existence of the Imbuant is also that of the Objectant. This means that a ‘substitution’ can also occur in this case, to give "The apple is being eaten by Carl XVI Gustaf." Here, the mode of existence is transformed from active to passive with regard to the switched Imbuant, in order to preserve the original relation wherein the Objectant is referred to the Imbuant.

Thus, the ‘substitution’ at work within the identity statement can be seen as a particular form of Modal Switch. The difference is that with the substitution of "Carl XVI Gustaf is the King of Sweden," for "The King of Sweden is Carl XVI Gustaf," the original relation is preserved with no resultant change from active to passive tense. It is this difference between these two forms of Modal Switch that is important here. This is because, in any form of identity statement, the Modal Informant consists of an Extantal Objectant alone. The Processant, in this case 'is,' simply has to perform its standard function of initiating a separable pertaining correspondence between the Imbuant and the Informant.

In its standard function, the Objectant is extantally subordinated to the Imbuant, as the object to which its mode of action is directed, and is at the same time bound to the mode of action. The Objectant is related to the Imbuant through a shared mode of action. The mode of action hence corresponds to both the Imbuant and the Objectant, with the difference that with the Objectant this is a bound correspondence which can only be made separable by reversing the sentence by means of the Modal Switch.

This relation is maintained in the identity statement, but here the correspondence function of the Processant is focused directly onto the Objectant, such that a correspondence obtains directly between them, as opposed to the correspondence of both to a shared mode of action of one upon the other. Hence the relation between Imbuant and Objectant necessarily has a compressed and more limited scope than when mediated through a shared modal aspect. Consequently, the Modal Switch has the same compressed form, with no scope for tense adjustment, and is manifested as the ‘substitution property.’

Therefore, we can say that identity statements depend upon three elements:

1. The Processant function, which operates in exactly the same way as within predication statements.
2. They must have a Modal Informant which consists of an Extantal Objectant alone.
3. There must be an equivalence of singularity between Imbuant and Informant/Objectant, which allows them to display the ‘substitution property.’

The Processant function allows for two semantic elements to have reference to the same entity in such a way that the reference of the second element is independent of the reference of the first. And it is the fact that these two elements always refer to the same referent that is crucial to the Processant function. But beyond its reliance on this basic correspondence, all other characteristics specific to identity statements can be accounted for by reference to semantic/syntactic factors other than the Processant function. With identity statements, it is the correspondence function of the Processant ensures that ensures that an Extantal Objectant must function simultaneously as a Modal Informant. In other words, whatever is marked out in the Objectant phrase refers to whatever is extantialised by the Imbuant.

This can be seen most clearly within direct statements of non-identity. For example, in the sentence "Carl XVI Gustaf is not the King of France," the Objectant, which is stated as not corresponding to the Imbuant, has a far more ambiguous extantialisation, inasmuch as it may not be an Objectant that corresponds to any Imbuant. Nevertheless, statements of non-identity can still be said to display the ‘substitution property.’ For although "The King of France is not Carl XVI Gustaf," extantialises a different entity, the same non-correspondence is stated here as in "Carl XVI Gustaf is not the King of France."

The equivalent singularity of reference within identity statements depends upon the equivalence between proper nouns and definite article phrases. This allows the correspondence between the Imbuant and the Informant/Objectant to be one of ‘identity.’ In other words, the Processant function maintains the two particularities in such a way that the second refers to the same entity as the first. On the other hand, a statement of non-correspondence between two particularities is interpreted as concerning two particularities, even where the second only has an ambiguous extantialisation.

That the identity statement depends upon this equivalent singularity of reference can be seen shown within such sentences as "The Munsters are the Adams Family" or "Stan and Ollie are Laurel and Hardy." Again, the way in which these kinds of sentences depend upon the Processant function is such that they either display or do not display correspondence. But within a sentence like "Stan is the Adams Family," this correspondence is rendered semantically impossible by the lack of equivalent singularity. This again derives from the limitation placed upon the correspondence function, which has a ‘compressed form’ wherever the Informant consists of an Objectant alone.

Four: The ‘is’ of Class-Inclusion.
This refers to sentences of the type ‘Man is an animal,’ ‘Fido is a dog, ‘The Whale is a mammal,’ Mice are rodents,’ etc.

Class-inclusion statements, like identity statements, have a Modal Informant that consists of an Extantal Objectant alone. Furthermore, in line with the AIT argument, the function of the ‘is’ within class-inclusion statements is no different to that of predication statements. Thus the distinction between class-inclusion and identity statements lies solely within the fact that the former do not display an equivalent singularity of reference between the Extantal Imbuant and Informant/Objectant. It is for this reason that they lack the ‘substitution property.’

The equivalent singularity of reference within identity statements depends upon the equivalence between proper nouns and definite article phrases. With class-inclusion statements on the other hand, whilst the Informant/Objectant must have an indefinite reference, either in terms of an indefinite article phrase or an indefinite plural, the Imbuant does not require this. Thus class-inclusion statements must be seen in terms of the relation between the Processant function and indefinite reference, and this resolves itself into the question of the distinction between definite and indefinite reference.

In the opening paragraph of On Denoting, Bertrand Russell points out that the definite article involves uniqueness. However, all that he has to say about the indefinite article is:

A phrase may denote ambiguously; e.g., ‘a man’ denotes not many men, but an ambiguous man.

which in effect, says no more than that the indefinite article creates an indefinite denotation.

But when looking at the difference between ‘the man’ and ‘a man,’ the most striking feature is that the uniqueness and ambiguity by which Russell distinguishes them is that they can be seen as two different implications in each case. They do not indicate singularity in an equivalent manner precisely because, in each case, there is a different implication as to how the entity named relates to other entities to which the word ‘man’ also applies.

When we say ‘the man,’ the implication of uniqueness means that the man spoken of is unlike any other man we might speak of. He is not only distinguished from any other entity to which the word ‘man’ does not apply, but also from all other entities to which the word ‘man’ does apply. This is implication inasmuch as the manner in which ‘the man’ differs from other entities which can be named in the same way is not specifically stated. When, on the other hand, we say ‘a man,’ he too is distinguished from any other entity to which the word ‘man’ does not apply, but not from all other entities to which the word ‘man’ does apply.

Thus the singularity which is implied by ‘the man’ is indeed that of uniqueness, inasmuch as it states that no other man could be spoken of. On the other hand, the singularity stated by ‘a man’ implies that other men could be spoken of but are not. Hence the singularity implied by the indefinite article does not extend beyond the simple fact that he is spoken of. Thus when we say ‘a man,’ we are distinguishing what we are talking about from all other entities that are not men, but not necessarily also from all other entities that are men.

In this way, the distinction between definite and indefinite articles rests upon the different way each depends upon the classification function of the regular noun. The noun function itself is not of course explicitly singular; the noun is applied to particular entity according to the identification of the particular distinctions commensurate with its use. Definite and indefinite articles both singularise the applicability of a regular noun; but the distinction between them rests upon the different implied relation of similarity that the singularised entity in each case has to other entities to which the noun also applies.

Therefore, with class-inclusion statements, the correspondence function of the Processant again maintains the two elements of the statement in such a way that the second refers to the same entity as the first. But where the Informant/Objectant is an indefinite article phrase, the Imbuant is stated as corresponding to any entity which falls under what is stated in the Informant/Objectant. Thus to say Socrates is a man,’ implies a correspondence between Socrates and any entity that is a man.

With class-inclusion statements however, the implication of singularity that an indefinite article usually carries is lost when it functions within an Informant/Objectant phrase. In other words, here the Imbuant is precisely not distinguished from other entities to which the noun of the Informant/Objectant indefinite article phrase also applies. This is because the singularity implied by an indefinite article is such that the entity named by the noun is distinguished from all other entities to which the noun applies only to the extent that they are not spoken of. But with class-inclusion statements, the Processant function ensures that the indefinite article phrase refers to what is already spoken of in the Imbuant. In this case therefore, it is the Imbuant, whether definite or indefinite, singular or plural, which is distinguished from all other entities to which the Informant/Objectant refers, only to the extent that it is spoken of.

With ‘Socrates is a man’ therefore, Socrates is distinguished from all other entities to which the noun ‘man’ applies only to the extent that he is mentioned and they are not, but with the implication that he is distinguished from all other entities that are not men, but not from all other entities that are men. Thus we say that Socrates is ‘included’ in the ‘class’ of all entities to which the noun ‘man’ applies.

But in order to be a class-inclusion statement, another purely semantic constraint, similar in a way to the equivalence of singularity that characterises identity statements, must be present. Namely, the indefinite article phrase that functions as the Informant/Objectant, must have a wider range of reference than the Imbuant. In this respect, the difference between ‘A whale is a mammal’ and ‘A mammal is a whale,’ assuming the meaning of both terms is understood, would mean that the second sentence would be interpreted as a statement concerning some particular mammal. This in turn brings out more clearly the ambiguity of the indefinite article, inasmuch as it can mean either ‘any’ or ‘at least one’ depending upon the semantic context in which it occurs.

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