## 29 Dec 2012

### Pt1.Ch2.Sb6 Somers-Hall’s Hegel, Deleuze, and the Critique of Representation. ‘Symbolic Logic.’ summary

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[Note: All boldface and underlining is my own. It is intended for skimming purposes. Bracketed comments are also my own explanations or interpretations.]

Henry Somers-Hall

Hegel, Deleuze, and the Critique of Representation.

Dialectics of Negation and Difference

Part 1: The Problem of Representation

Chapter 2: Difference and Identity

Subdivision 6: Symbolic Logic

Very Brief Summary:

Russell deals with a paradox similar to Aristotle’s problem of defining the highest category being. Russell notes that we cannot determine whether or not the class of all non-self-including classes is included in itself, because either way, the classification contradicts its definition. Russell’s solution is to say that one level of the hierarchy refers to its lower level and not to itself. This has many problems, for example, we cannot make universal statements or predications like ‘truth’ that range over all levels.

Brief Summary:

Aristotle’s problem of defining the highest category is related to Russell’s solution to the paradox of the class of all classes that do not include themselves. In Russell’s set theory, something is included in a class if it has the property shared by all things in that set. One property could be ‘being a set that does not include itself as a member’. But then it becomes impossible to determine whether the set of all non-self-inclusive sets belongs to itself as a member. For, if it did, then it includes itself as a member, which makes it ineligible to be included in the class of non-self-inclusive classes. If it did not belong in itself, then it fits the criteria to be in the class of non-self-inclusive sets, which contradicts its status of not belong in itself. Russell’s solution is to divide levels in the hierarchy of classification, such that the properties shared at one level do not apply to the things of the higher level. So no set has self-referential properties, like ‘belonging to itself’. This is similar to Aristotle’s rule that differentiae differ from genera. In both Aristotle’s and Russell’s systems, we cannot define a highest class, because all classes need a higher one in order to be defined. And in Russell’s system, we cannot make universal statements about all levels of things; for we can only refer to one level at a time.

Summary

Previously we saw Aquinas solution to Aristotle’s problem of defining the highest category, being. Aquinas uses the concept of analogous isomorphic relations. We cannot say what God is, but we can say that the relation between the highest category and its species is like the relation between some other genus and its species. They are proportional relations.

Now Somers-Hall examines a similar sort of problem and solution in Russell’s and Whitehead’s Principia Mathematica. Here we deal symbolic logic, which examines pure syntactical relations between entities. This “creates a sharp divide between the form of thought and its subject matter.” (57) The hope is that the formal analysis does not involve some particular metaphysical view.

It also allows the characterization of phenomena in an artificial language, formulated to allow precision in the expression of propositions and to allow a purely mechanical movement between propositions, allowing the rules of derivation to be formally specified. (57)

In formal logic there are symbols for both variables and constants, constants like &, ∨, and ¬.
One of formal logic’s most fundamental formalizations is the logic of classes.

The logic of classes deals with the relations between sets of things (in the loosest possible terms), such as the inclusion or exclusion of both individuals and classes of things within classes of things. (57)

According to this view, Aristotle’s hierarchy is a theory of classes that provides “rules whereby different individuals and species can count as being included in or excluded from genera, which in turn are either included or excluded under their aspect as species”. (57) Recall Aristotle’s system. What makes a species belong to a particular genera is that the genera is predicated of the species (man is an animal that thinks), and what makes a species be a certain species to its genus is that it has a specific, essential difference from the genus [taken in its pure generality] and from the other species in that genus (man is a thinking animal [man is not just an animal, so it is different from the genus, and man is not like the other animals, so it is different from the other species.]) The modern theory of classes defines membership in a class using two methods.

The two methods for defining membership in the modern theory of classes:

[1] by extension: giving a list of all the members in the class, and

[2] by intension: giving a particular property that may identify the members of the class. This provides the criterion for membership.

Although we may theoretically use the extensional method, it could perhaps take too much time. So practically speaking we define membership intensionally by means of their common property. So in more formal terms, something is a member of a given class if and only if it has the property shared by all members of that class. In terms of Aristotle’s classification, a member of a species is as well a member of that species’ genus.

Russell therefore originally put forward the intuitive axiom that given a class of things with a property F, x will be a member of this class if and only if it has the property F. On this basis, we are therefore able to formalize aspects of the hierarchy of Aristotle. Thus all mammals are animals is rendered as ∀x(x∈a → x∈b) . Literally, this means that for any arbitrary object (x), if that object is a member of the class of mammals (a), then it is a member of the class of animals (b). We could further define the set of men as a subset of mammals and in this way allow the mechanical derivation of the animality of man from his status as a mammal. This process could be repeated to give a complete set-theoretic representation of Aristotle's arborescent hierarchy. (57)

[Recall Aristotle’s problem of the equivocity of the highest category, being. All other things falling under it are beings. So they have the same name as being. However, as they are species of being, they must have a different definition. And also recall, being itself cannot be defined, because definition requires differentiation from a genus, but there is no higher genus than being.] We may formulate in set theory Aristotle’s problem of the equivocity of being: is it possible to specify the class of all class (the class to which all classes belong)? In Aristotle’s system, differentiae must differ from genera, so the highest category ‘being’ cannot be included as a member of itself. In Russell’s system, however, there can be the self-referentiality of saying that if something is a class, then it is a member of the class of all classes (thus classifying the class of all classes within itself).

In the case of Russell's system, however, at first glance it is possible to specify a class that is a member of itself, as, given the class of things that have the property of being classes, x will be a member of this class only if it is a class. The class of all classes meets this definition and is therefore a member of itself. (57)

Let’s first note one of Russell’s earlier paradoxes, the paradox of the barber. There is a barber who only shaves men who do not shave themselves. But then, who shaves the barber? If he shaves himself, then he is not in the class of men who do not shave themselves, and thus he should not be shaving himself in the first place. But if he does not shave himself, then he is in the class of men who do not shave themselves, and thus should be shaving himself. “In this case, therefore, the barber is both in a state and not in that same state simultaneously, which, within a representational system, will lead to the collapse of all states within that system.” (58) The solution in this paradox is simple. We merely say that there can be no such barber.
But in class theory, we can have empty sets, like ‘descendents of the last Tsar of Russia’. We need not affirm that there are any members in the set. [So we can say there is the class of barbers who always shave those who don’t shave themselves, and then also say that there are no members in this class.] So we can say there is a class whose members are classes that are not members of themselves. If that class itself is a member, then it is not included in itself, which means it could not have been a member to begin with. But if that class is not a member, then it is a class that is not included in itself, which should have made it a member in the first place. [Now, because our thinking is fundamentally based on such classification principles, we can say that] the paradox is an inherent part of the structure of our thinking.

One of the fundamental principles of the theory of classes is that a class can be defined intensionally by any clearly defined property that belongs to all members. Such a definition of class carries with it no existential statements, so we could define the class of, for example, the descendents of the last Tsar of Russia, without having to affirm that this class contains any members. It could instead simply be an empty class. Given this, we can specify the following class: "Let w be the class of all those classes which are not members of themselves." If we take a class, x, and say of it that it is a (member of) w, then this is equivalent of stating that "x is not an x" (as we might say that the class of all chairs is not itself a chair) . If we give x the value of w, however, then the statement "w is a w" becomes equivalent to the statement "w is not a w." This is because if w is a member of itself (w is a w), then according to the definition of what it is to be a w, w is not a member of itself. On the other hand, if it is not a member of itself, then it meets the definition for membership of itself. It is therefore the case that w both is and is not a w. Clearly this particular case parallels the result of the barber's paradox, but whereas in that case, the paradox relied on the subject matter being discussed, the barber, this second paradox instead is derived from the fundamental rules of the logical system itself. The paradox is therefore implicit in the very structure of thought. (58)

So we see that this problem mirrors Aristotle’s problem. Later we will examine Russell's solution, “Whatever contains an apparent variable must not be a possible value of that variable”. (58) But already we can see how it is similar to Aristotle’s stipulation. “This principle is analogous to Aristotle's requirement that the differentiae must differ from the genus under | consideration, which also prevented genera from referring to themselves”. (58-59) For Russell, the individual is absolutely simple, and then there is a hierarchy building up from it. So in this hierarchy, we classify something as belonging to higher class. The class on the higher level is on an ordered different from the lower level, so it is meaningless to have a class that includes itself. At best, we can have a class of all other lower-order classes that do not contain themselves.

Russell's own implementation of this principle is to propose a hierarchy of types, meaning that an apparent variable must be of a different type to its possible values, known as Russell's theory of types This works by constructing a hierarchy from the most basic individuals to the highest classes. The individual in Russell's system functions as the anchor for the whole and is considered to be absolutely simple. By means of this, Russell hopes to remove the possibility of any self-reflexivity at the base of the hierarchy. Taking absolutely simple, nonreflexive elements as our starting point means that, provided we are rigorous in our development of the system from these points, the problem of self-reflexivity may not arise. Designating these simple individuals using the subscript 1, as they form the base of the hierarchy, we can formulate simple propositions of the form, a1 is F, or this individual has a specified property (F) . This proposition, as the highest type it contains is of the first order, we can call a first order proposition. We can then generalize this proposition into the form "All a1’s are F." Here, a1 is no longer taken as an individual, but instead as a variable ranging over all individuals. The statement "All a1’s are F" will be a second-order proposition; it will be of a different type to the previous statement, a1 is F. This process can be repeated indefinitely, thus generating a hierarchy of classes, each level of which is of a different type . As statements refer to the level below themselves, reflexivity is blocked. If we return to the issue of Russell's paradox, it is now clear how this difficulty has been solved. The class w now must be of a different level to the classes over which it ranges, so if its members are classes of type n, w will be itself a class of type n + 1. This means that the question of w's membership of itself becomes not even false, but meaningless, as it is impossible within the system to formulate the function w being applied to itself. Instead, w is a type n + 1 class, which refers only to type n classes that are not a member of themselves. Types thus parallel the function of the differentiae in Aristotle's system by creating a hierarchy of terms, and the problem that emerges is in turn similar. (59)

One problem regards the impossibility of univocity in the system: we cannot have a highest type, because every type requires a higher one, and also, there cannot be simple universal statements [because any statement must refer specifically to one level and not to the others], and furthermore, there cannot be central notions like truth as it would apply to all levels, for it cannot refer beyond a particular type. [Also, properties have levels too] so we would need to say ‘all n-th level properties of a’, because we cannot say ‘all properties of a’. If we wanted to speak of all properties of a, we would need really to be implying that we are making just as many statements as levels of properties. [But this means we are making use of the parallel in structure between one order to the next, so we are using analogy as with Aquinas’ solution. In both cases, there is an ambiguity: what makes one level higher to another is the same relation as some other level to another. But then in the end we would not be able to say something about a highest class, at best only the sort of relations to it.]

As "it will be possible, sometimes, to combine into a single verbal statement what are really a number of different statements, corresponding to different orders in the hierarchy," it becomes clear that what is being used to overcome the limitations of the symbolic system is once again a form of analogy, and in particular a form of isomorphy, as that which grounds the single verbal usage are the structural parallels between the various statements. Thus, the Principia Mathematica ultimately suffers from the same limitations as the scholastic system, systematic ambiguity being synonymous with analogy. (60)

[Recall how Aristotle’s theory of essence was unable to explain the nature of the transition from state-to-state involved in change. It can tell us about the state in one moment and the state in the next moment, but not the in between when the thing takes on two incompatible states at the same time.] For Russell, we cannot give an account of a flux of change. We can only speak of the different moments in a change, and consider time to be the differences in status between something at time 1 and that something at time 2. So like Aristotle, Russell does not give a satisfactory account of the change from state-to-state. For Zeno, we may infinitely divide time or space. His

view derives from his belief that a stretch of time merely involves a particular kind of infinity, namely, that between any two points of time, there is another point of time. This means that time appears as a continuum, while each moment is instead a fixed state. The stability of the system is therefore once again purchased at the cost of excising all traces of motion. The problem of the nature of change itself is still present, however, as is clear from Russell's definition: "Change is the difference, in respect to truth or falsehood, between a proposition concerning an entity and a time T and a proposition concerning the same entity and another time T', provided that the propositions differ only by the fact that T occurs in the one where T' occurs in the other." This definition suffers from the difficulty that although it gives us a criterion to show that a change has taken place, it in no way defines the nature of that change itself. The same explanatory gap that occurred through Aristotle's use of analogy again occurs here. (60)

Somers-Hall, Henry (2012) Hegel, Deleuze, and the Critique of Representation. Dialectics of Negation and Difference. Albany: SUNY.