Sample Chapter
Well, here is a sample chapter of my little exposition on logic and knowledge that I'm trying to get done this summer. Enjoy. (Note: Some of the symbolic logic will not appear correctly online, particularly in this excerpt the epsilon symbol)
The Theory of Types
It is evident as soon as we begin to proceed down the long and somewhat treacherous road of abstract mathematical logic that certain difficulties present themselves. For example, in set theory, what shall we say of the class of all classes who are not members of themselves? Is this class a member of itself? Let us investigate.
First off, what exactly do we mean by the saying that a class is not a member of itself? Well, we mean, of course, exactly that. For example, the class of all things red. It is obvious that the class of all things red is not itself red and therefore is not a member of itself. However, there are classes such that the class is indeed a member of itself. This is very common of negative classes, i.e. the class of all things not-red. Most certainly the class of all things not-red would include the class itself as it lacks the predicate of redness. With this distinction made, we can move on.
Now let us take class x, the class of all classes not members of themselves. The class x is either a member of itself or not a member of itself by the law of the excluded middle. We shall, to begin, suppose that "x is not a member of itself" is true:
(x Ï x)
Now, if this is the case, viz. that "x is not a member of itself" is true, then x is a class which is not a member of itself and, thus, is in fact a member of itself.
(x Î x) º (x Ï x)
This is plainly contradictory and we shall see that we fare no better by supposing the converse true. If "x is not a member of itself" is false or, in positive terminology, "x is a member of itself" is true, then accordingly by the very nature of the class it is contradictory. It cannot possibly be a member of itself because by fulfilling that property it excludes itself from membership.
(x Ï x) º (x Î x)
This paradox leads us to a supposition not unimportant, for if we cannot dispel this obstinate logical contradiction, we would be compelled to admit flaws within the foundation of logic itself. This paradox, known as Russell's Paradox, named after the eminent logician and philosopher Bertrand Russell who discovered it and later resolved it, is apparent in more forms than this conflict between logical classes.
For example, let us examine the proposition: "The bottle is blue". This is a simple predication of the concept of "blue" to the object "the bottle". The predicate "blue" is a member of the class of all predicates seemingly and thus we can replace "blue" with any other predicate:
"The bottle is green."
"The bottle is large."
"The bottle is open."
"The bottle is cold."
Etc.
We may proceed for some time without any difficulty in meaning, although, there does come a point where there are certain predicates which while fitting within the logical form of the proposition, transform the meaningful proposition into something meaningless and absurd.
"The bottle is a color."
"The bottle is a spatio-temporal property."
"The bottle is a measurement."
Etc.
I think it is evident that prima facie these propositions are of a different nature and that analytical reflection reveals to us that each proposition of this type lacks any substantive meaning. When we have a proposition of the form "x is a color" or "x is a property" we have seen x must be of a certain logical type to fulfill the condition of being meaningful. This directly leads us to Russell's solution, the theory of types.
According to the theory of logical types, there are various levels or orders or propositions and concepts. We shall deal with the first three orders, however there are indeed more than three levels in the hierarchy of types. The first level, known as zero level, or the zero order, consists of entities themselves; the subjects in contradistinction to the predicates or relations; "The bottle" in "The bottle is blue". Next we have the first order which contains the properties and relations directly applicable to objects themselves, i.e. "blue", "yellow", "big", "small", "next to", "above", etc. Then we come upon the second order concepts which are applicable only to first order concepts. For example: "Blue is a color". "Blue", being of the first order is being ascribed the second order property of being a "color". In the proposition, "Being 'next to' is a 'spatial property'", "next to" is a first order relation, whereas "spatial property" is a second order predicate only applicable to first order terms that fulfill the certain condition of being a "spatial property".
We cannot take a zero order concept and ascribe a second order concept to it. For example, to say "The cat is a temporal relation" is senseless and absurd. We can say that "The cat is black" or "The cat is fat" or any other first order concept without any problems, but to predicate a second order concept would result is nonsense.
The reverse also is true as we cannot apply zero order concepts to first order, and we cannot apply first order to second order: "Red is a flower" and "The spatio-temporal property is angry" are both strictly nonsense. Thus, in summation, the theory of types is as follows: given any order n, n can only be applied to the order n-1, and never any other order.
How does the theory of types apply to the logical antimony we encountered earlier with the class of all classes not members of themselves? As with propositions and concepts, there is a hierarchy of types for classes. For any given class, a class has within it other classes if and only if it is of a higher order than the classes contained within. Thus to say that a class is applicable to itself in any sense is nonsense as they are of a different logical type. With this in mind, let us return to our troublesome class of all the classes which are not members of themselves. Let us refer to this class as x. If x contains the classes y which are not members of themselves, the logical type of x is distinct from the logical type of y; the logical type of y if of the order n, and x is of the order n+1.
The Theory of Types
It is evident as soon as we begin to proceed down the long and somewhat treacherous road of abstract mathematical logic that certain difficulties present themselves. For example, in set theory, what shall we say of the class of all classes who are not members of themselves? Is this class a member of itself? Let us investigate.
First off, what exactly do we mean by the saying that a class is not a member of itself? Well, we mean, of course, exactly that. For example, the class of all things red. It is obvious that the class of all things red is not itself red and therefore is not a member of itself. However, there are classes such that the class is indeed a member of itself. This is very common of negative classes, i.e. the class of all things not-red. Most certainly the class of all things not-red would include the class itself as it lacks the predicate of redness. With this distinction made, we can move on.
Now let us take class x, the class of all classes not members of themselves. The class x is either a member of itself or not a member of itself by the law of the excluded middle. We shall, to begin, suppose that "x is not a member of itself" is true:
(x Ï x)
Now, if this is the case, viz. that "x is not a member of itself" is true, then x is a class which is not a member of itself and, thus, is in fact a member of itself.
(x Î x) º (x Ï x)
This is plainly contradictory and we shall see that we fare no better by supposing the converse true. If "x is not a member of itself" is false or, in positive terminology, "x is a member of itself" is true, then accordingly by the very nature of the class it is contradictory. It cannot possibly be a member of itself because by fulfilling that property it excludes itself from membership.
(x Ï x) º (x Î x)
This paradox leads us to a supposition not unimportant, for if we cannot dispel this obstinate logical contradiction, we would be compelled to admit flaws within the foundation of logic itself. This paradox, known as Russell's Paradox, named after the eminent logician and philosopher Bertrand Russell who discovered it and later resolved it, is apparent in more forms than this conflict between logical classes.
For example, let us examine the proposition: "The bottle is blue". This is a simple predication of the concept of "blue" to the object "the bottle". The predicate "blue" is a member of the class of all predicates seemingly and thus we can replace "blue" with any other predicate:
"The bottle is green."
"The bottle is large."
"The bottle is open."
"The bottle is cold."
Etc.
We may proceed for some time without any difficulty in meaning, although, there does come a point where there are certain predicates which while fitting within the logical form of the proposition, transform the meaningful proposition into something meaningless and absurd.
"The bottle is a color."
"The bottle is a spatio-temporal property."
"The bottle is a measurement."
Etc.
I think it is evident that prima facie these propositions are of a different nature and that analytical reflection reveals to us that each proposition of this type lacks any substantive meaning. When we have a proposition of the form "x is a color" or "x is a property" we have seen x must be of a certain logical type to fulfill the condition of being meaningful. This directly leads us to Russell's solution, the theory of types.
According to the theory of logical types, there are various levels or orders or propositions and concepts. We shall deal with the first three orders, however there are indeed more than three levels in the hierarchy of types. The first level, known as zero level, or the zero order, consists of entities themselves; the subjects in contradistinction to the predicates or relations; "The bottle" in "The bottle is blue". Next we have the first order which contains the properties and relations directly applicable to objects themselves, i.e. "blue", "yellow", "big", "small", "next to", "above", etc. Then we come upon the second order concepts which are applicable only to first order concepts. For example: "Blue is a color". "Blue", being of the first order is being ascribed the second order property of being a "color". In the proposition, "Being 'next to' is a 'spatial property'", "next to" is a first order relation, whereas "spatial property" is a second order predicate only applicable to first order terms that fulfill the certain condition of being a "spatial property".
We cannot take a zero order concept and ascribe a second order concept to it. For example, to say "The cat is a temporal relation" is senseless and absurd. We can say that "The cat is black" or "The cat is fat" or any other first order concept without any problems, but to predicate a second order concept would result is nonsense.
The reverse also is true as we cannot apply zero order concepts to first order, and we cannot apply first order to second order: "Red is a flower" and "The spatio-temporal property is angry" are both strictly nonsense. Thus, in summation, the theory of types is as follows: given any order n, n can only be applied to the order n-1, and never any other order.
How does the theory of types apply to the logical antimony we encountered earlier with the class of all classes not members of themselves? As with propositions and concepts, there is a hierarchy of types for classes. For any given class, a class has within it other classes if and only if it is of a higher order than the classes contained within. Thus to say that a class is applicable to itself in any sense is nonsense as they are of a different logical type. With this in mind, let us return to our troublesome class of all the classes which are not members of themselves. Let us refer to this class as x. If x contains the classes y which are not members of themselves, the logical type of x is distinct from the logical type of y; the logical type of y if of the order n, and x is of the order n+1.