The 'is' of identity 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" or, the Wittgensteinian's favourite, "meaning is use."
This is usually interpreted as meaning that, since the *is* can be taken as stating
*is the same as, * in way that wouldn't be possible for predication, the *is* here has an entirely different function. At the same time, this aspect of identity statements means it also has a *substitution property* which, again, distinguishes it from predication.
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, 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 largely this fact upon which the distinction between *the 'is' of predication* and *the 'is' of identity* is based.
BUT, and this is the crucial point here, once we introduce the AIT notion of the Modal Switch, this distinction proves not to be cut and dried. This is seen when you introduce a sentence like "Carl XVI Gustaf is eating the apple. " This would be analysed as follows:
"Carl XVI Gustaf' <Extantal Imbuant 'is' <Modal Processant 'eating' <Modal Informant 'the apple' <Extantal Objectant"
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, " to "The King of Sweden is Carl XVI Gustaf," the original relation is preserved with no resultant change from active to passive tense.
It is the 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,' only has to perform its standard function of initiating a separable pertaining correspondence between the Imbuant and the Informant.


In its more standard function, the Objectant is a secondary sentential instantiation. On the one hand, it extantialises something separate from the Imbuant. But on the other hand, in terms of the way it is related to the Imbuant through a mode of the Imbuant's existence that is also that of the Objectant.
This relation is maintained in the identity statement, but here the Informant function, created by the Processant, is focused directly onto the Objectant. Thus the relation between Imbuant and Objectant is necessarily compressed, and is far more limited in scope than it would be if mediated through a distinct mode of existence. Consequently, the Modal Switch has the same compressed form, appearing as the *substitution property.*
However, it should be noted that not all sentences where the Modal Informant consists of an Extantal Objectant alone display this *substitution property.* An example of this would be a *generic implication* type sentence like: "The whale is a mammal." So, where the Modal Informant consists of an Extantal Objectant, the correspondence function of the Processant has a *compressed form, * where there is no scope for tense adjustment. The Modal Switch either produces sentences that make sense, and thus have a *substitution property,* or do not.
Therefore, identity statements depend upon three things.

1) They depend upon the Processant, which is only required to work 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 a SEMANTIC EQUIVALENCE between Imbuant and Informant/Objectant, which allows them to display the *substitution property. *


The Processant function itself has to do with *correspondence* which has nothing to do with identity, rather, it allows for two semantic correspondences to refer to the same thing (whatever is extantialised within the Imbuant) in such a way as to be separable.


What is important as far as AIT is concerned, is to show that 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, and, more generally, any kind of sentence where the Informant consists solely of an Objectant, the correspondence function of the Processant

ensures that the Objectant functions as an Informant, and hence limits its capacity to be a secondary Extantal Imbuant. In other words, whatever is marked out in the  Objectant phrase applies to whatever is extantialised by the Imbuant.

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, " to "The King of Sweden is Carl XVI Gustaf, " the original relation is preserved with no resultant change from active to passive tense.
It is the 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, ' only has to perform its standard function of initiating a separable pertaining correspondence between the Imbuant and the Informant. In its more standard function, the Objectant is a secondary sentential instantiation. On the one hand, it extantialises something separate from the Imbuant. But on the other hand, in terms of the way it is related to the Imbuant through a mode of the Imbuant's existence that is also that of the Objectant.
This relation is maintained in the identity statement, but here the Informant function, created by the Processant, is focused directly onto the Objectant. Thus the relation between Imbuant and Objectant is necessarily compressed, and is far more limited in scope than it would be if mediated through a distinct mode of existence. Consequently, the Modal Switch has the same compressed form, appearing as the *substitution property. * However, it should be noted that not all sentences that where the Modal Informant consists of an Extantal Objectant alone display this *substitution property. * An example of this would be a *generic implication* type sentence like "The whale is a mammal. " So, where the Modal Informant consists of an Extantal Objectant, the correspondence function of the Processant has a *compressed form, * where there is no scope for tense adjustment. The Modal Switch either produces sentences that make sense, and thus have a *substitution property, * or do not. Therefore, identity statements depend upon three things.

1) They depend upon the Processant, which is only required to work 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 a SEMANTIC EQUIVALENCE between Imbuant and Informant/Objectant, which allows them to display the *substitution property*. The Processant function itself has to do with *correspondence* which (as I hopefully made clear in Part One) has nothing to do with identity. Rather, it allows for two semantic correspondences to refer to the same thing (whatever is extantialised within the Imbuant) in such a way as to be separable.

But what is important as far as AIT is concerned, is to show that 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, and, more generally, any kind of sentence where the Informant consists solely of an Objectant, the correspondence function of the Processant ensures that the Objectant functions as an Informant, and hence limits its capacity to be a secondary Extantal Imbuant. In other words, whatever is marked out in the Objectant phrase applies 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 does not correspond to the Imbuant, has a far more explicit extantialisation, and this becomes a statement concerning two definite entities, as is not the case with "Carl XVI Gustaf is the King of Sweden. " Nevertheless, the statement of non-identity still displays the *substitution property. * In fact, here, the *substitution property is more explicit, * inasmuch as the relation between "Carl XVI Gustaf is not the King of France, " and "The King of France is not Carl XVI Gustaf, " since it pertains to two separate entities, says substantially the same thing in a way which the two possible statements of identity, "Carl XVI Gustaf is the King of Sweden, " and "The King of Sweden is Carl XVI Gustaf, " does not. But, to reiterate, it is the Processant function which allows the identity statement to be about one entity, and not two.
Nevertheless, not all sentences where the Informant consists of an Imbuant are identity statements. In order for this to be the case, the Objectant/Informant must have an equivalent particularity to the Extantal Imbuant, either through the use of a definite article, or of proper nouns.
A proper noun can have a definite article phrase as its equivalent, and vice versa, because the definite article always extantialises a particularity, in such a way as to allow the subsequent nouns or adjectives that are syntactically linked to it to define this particularity, and not each to have their own referent. (I'll say more about the distinction between the definite and indefinite articles in the next section. )
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. Again, this has to do with the limitation placed upon the correspondence function, which has a *compressed form, * wherever the Informant consists of an Objectant alone.