set+theory

=Set Theory=

toc Read here about set theory. When we consider a scale as a set, set theory's langauge and methods are very useful to discussing and understanding the differences between scales.

=Overview= Mathematicians have codified an "intuitive" or "naive" set theory, where a set is a collection of distinct objects, considered as an object in its own right. Georg Cantor, a founder of set theory, defined a set as, “A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] and of our thought - which are called elements of the set.”

The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.

=Extension And Intension= A list of the reasons we deprecate the usage of this terminology in scale studies.

1. Extension and intension are not well known terms 2. Intension resembles intention too closely and is confusing 3. The oppositional symmetry of in- and ex- is reserved in scale studies as referring to pheonomenon that are internal to a given scale and external to that scale, whereas in set theory and logic intensional refers to the definition and extensional refers to the reality of which items are included.

About Intensional Logic
Intensional Logic: Stanford, First published Thu Jul 6, 2006; substantive revision Thu Jan 27, 2011

If you are not skilled in colloquial astronomy, and I tell you that the morning star is the evening star, I have given you information—your knowledge has changed. If I tell you the morning star is the morning star, you might feel I was wasting your time. Yet in both cases I have told you the planet Venus was self-identical. There must be more to it than this. Naively, we might say the morning star and the evening star are the same in one way, and not the same in another. The two phrases, “morning star” and “evening star” may designate the same object, but they do not have the same meaning. Meanings, in this sense, are often callled intensions, and things designated, extensions. Contexts in which extension is all that matters are, naturally, called extensional, while contexts in which extension is not enough are intensional. Mathematics is typically extensional throughout—we happily write “3 + 2 = 2 + 3” even though the two terms involved may differ in meaning (more about this later). “It is known that…” is a typical intensional context—“it is known that 3 + 2 = 2 + 3” may not be correct when the knowledge of small children is involved. Thus mathematical pedagogy differs from mathematics proper. Other examples of intensional contexts are “it is believed that…”, “it is necessary that…”, “it is informative that…”, “it is said that…”, “it is astonishing that…”, and so on. Typically a context that is intensional can be recognized by a failure of the substitutivity of equality when naively applied. Thus, the morning star equals the evening star; you know the morning star equals the morning star; then on substituting equals for equals, you know the morning star equals the evening star. Note that this knowledge arises from purely logical reasoning, and does not involve any investigation of the sky, which should arouse some suspicion. Substitution of co-referring terms in a knowledge context is the problematic move—such a context is intensional, after all. Admittedly this is somewhat circular. We should not make use of equality of extensions in an intensional context, and an intensional context is one in which such substitutivity does not work. The examples used above involve complex terms, disguised definite descriptions. But the same issues come up elsewhere as well, often in ways that are harder to deal with formally. Proper names constitute one well-known area of difficulties. The name “Cicero” and the name “Tully” denote the same person, so “Cicero is Tully” is true. Proper names are generally considered to be rigid, once a designation has been specified it does not change. This, in effect, makes “Cicero is Tully” into a necessary truth. How, then, could someone not know it? “Superman is Clark Kent” is even more difficult to deal with, since there is no actual person the names refer to. Thus while the sentence is true, not only might one not know it, but one might perfectly well believe Clark Kent exists, that is “Clark Kent” designates something, while not believing Superman exists. Existence issues are intertwined, in complex ways, with intensional matters. Further, the problems just sketched at the ground level continue up the type heirarchy. The property of being an equilateral triangle is coextensive with the property of being an equiangular triangle, though clearly meanings differ. Then one might say, “it is trivial that an equilateral triangle is an equilateral triangle,” yet one might deny that “it is trivial that an equilateral triangle is an equiangular triangle”. In classical first-order logic intension plays no role. It is extensional by design since primarily it evolved to model the reasoning needed in mathematics. Formalizing aspects of natural langauge or everyday reasoning needs something richer. Formal systems in which intensional features can be represented are generally referred to as intensional logics. This article discusses something of the history and evolution of intensional logics. The aim is to find logics that can formally represent the issues sketched above. This is not simple and probably no proposed logic has been entirely successful. A relatively simple intensional logic that can be used to illustrate several major points will be discussed in some detail, difficulties will be pointed out, and pointers to other, more complex, approaches will be given. There is an obvious difference between what a term designates and what it means. At least it is obvious that there is a difference. In some way, meaning determines designation, but is not synonymous with it. After all, “the morning star” and “the evening star” both designate the planet Venus, but don't have the same meaning. Intensional logic attempts to study both designation and meaning and investigate the relationships between them. Recognition that designating terms have a dual nature is far from recent. The Port-Royal Logic used terminology that translates as “comprehension” and “denotation” for this. John Stuart Mill used “connotation” and “denotation.” Frege famously used “Sinn” and “Bedeutung,” often left untranslated, but when translated, these usually become “sense” and “reference.” Carnap settled on “intension” and “extension.” However expressed, and with variation from author to author, the essential dichotomy is that between what a term means, and what it denotes. “The number of the planets” denotes the number 9 (ignoring recent disputes about the status of bodies at the outer edges of the solar system), but it does not have the number 9 as its meaning, or else in earlier times scientists might have determined that the number of planets was 9 through a process of linguistic analysis, and not through astronomical observation. Of the many people who have contributed to the analysis of intensional problems several stand out. At the head of the list is Gotlob Frege.

Definitions
Sets can be defined by extension or intension. An extensional definition of a concept or term formulates its meaning by specifying its extension, that is, every object that falls under the definition of the concept or term in question. For example, an extensional definition of the term "nation of the world" might be given by listing all of the nations of the world, or by giving some other means of recognizing the members of the corresponding class. An explicit listing of the extension, which is only possible for finite sets and only practical for relatively small sets, is called an enumerative definition. Extensional definitions are used when listing examples would give more applicable information than other types of definition, and where listing the members of a set tells the questioner enough about the nature of that set. This is similar to an ostensive definition, in which one or more members of a set (but not necessarily all) are pointed out as examples. The opposite approach is the intensional definition. An intensional (intension, not intention) definition defines by listing properties that a thing must have in order to be part of the set captured by the definition. An intensional definition gives the meaning of a term by specifying all the properties required to come to that definition, that is, the necessary and sufficient conditions for belonging to the set being defined. For example, an intensional definition of "bachelor" is "unmarried man." Being an unmarried man is an essential property of something referred to as a bachelor. It is a necessary condition: one cannot be a bachelor without being an unmarried man. It is also a sufficient condition: any unmarried man is a bachelor. As becomes clear, intensional definitions are best used when something has a clearly defined set of properties, and it works well for sets that are too large to list in an extensional definition. It is impossible to give an extensional definition for an infinite set, but an intensional one can often be stated concisely — there is an infinite number of even numbers, impossible to list, but they can be defined by saying that even numbers are integer multiples of two. Definition by genus and difference, in which something is defined by first stating the broad category it belongs to and then distinguished by specific properties, is a type of intensional definition. As the name might suggest, this is the type of definition used in Linnaean taxonomy to categorize living things, but is by no means restricted to biology. Suppose we define a miniskirt as "a skirt with a hemline above the knee." We've assigned it to a genus, or larger class of items: it is a type of skirt. Then, we've described the differentia, the specific properties that make it its own sub-type: it has a hemline above the knee. Intensional definition also applies to rules or sets of axioms that generate all members of the set being defined. For example, an intensional definition of "square number" can be "any number that can be expressed as some integer multiplied by itself." The rule — "take an integer and multiply it by itself" — always generates members of the set of square numbers, no matter which integer one chooses, and for any square number, there is an integer that was multiplied by itself to get it. Similarly, an intensional definition of a game, such as chess, would be the rules of the game; any game played by those rules must be a game of chess, and any game properly called a game of chess must have been played by those rules.

Overview 1
In mathematics, the extension of a mathematical concept is the set that is specified by that concept. For example, the extension of a function is a set of ordered pairs that pair up the arguments and values of the function; in other words, the function's graph. The extension of an object in abstract algebra, such as a group, is the underlying set of the object. The extension of a set is the set itself. That a set can capture the notion of the extension of anything is the idea behind the axiom of extensionality in axiomatic set theory. This kind of extension is used so constantly in contemporary mathematics based on set theory that it can be called an implicit assumption. It can mean different things in different cases, and there is no universal definition of the term "extension." Russell's paradox, discovered by Bertrand Russell, specified as "the set of all sets that are not members of themselves," was an interesting case of a specification of a set (a supposed intension) that could not be satisfied—it could have no extension—because the intension specified in the definition of that set led to a contradiction. The result of the discovery of Russell's paradox was to show that the so-called naive set theory of Gottlob Frege required revision because Frege had thought that any specifiable condition (intension) should be able to define a set (extension), but this assumption was shown by Russell to be false. This required a revision of the axioms of set theory so that they would not permit such contradictory membership conditions (such contradictory intensions) to be specified within the system. Russell and Whitehead's solution (in their work Principia Mathematica) was to set up a theory of types, in which membership was restricted to a given type, and there were different levels (or types) of membership. Other set theories have coped with the problem in different ways.

Overview 2
An extensional definition of a concept or term formulates its meaning by specifying its extension, that is, every object that falls under the definition of the concept or term in question. An intensional (intension, not intention) definition defines by listing properties that a thing must have in order to be part of the set captured by the definition. In linguistics, logic, philosophy, and other fields, an intension is any property or quality or state of affairs connoted by a word, phrase or other symbol. In case of a word, it is often implied by its definition. Intension and intensionality (the state of having intension) should not be confused with intention and intentionality, which are pronounced the same and occasionally arise in the same philosophical context. Where this happens, the letter 's' or 't' is sometimes italicized to emphasize the distinction. In the context of formal logic, the extension of a whole statement, as opposed to a word or phrase, is sometimes defined (arguably by convention) as its logical value. So, in that view, the extension of "Lassie is famous" is the logical value true, since Lassie is famous. In computer science, some database textbooks use the term intension to refer to the schema of a database, and extension to refer to particular instances of a database. The distinction, however, is the same: intension is the logical specification of something, whereas extension is the set of objects or other things that satisfy the conditions of the logical specification given in the intension.

Overview 3
A fundamental distinction about sets is intension vs. extension. The intension of a set is its description or defining properties, i.e., what is true about members of a set. The extension of a set is its members or contents. The intension of a set may appear to be more important than the extension. The extension of a set may change without changing its intension, and a given set of members may satisfy many intensions. But intensions are hard to pin down, so extensional definitions are often preferred. In mathematics, a set is its extension. Types are a form of intentional definition. A type defines the possible extensions of a set. We think of sufficiently different types as being incompatible. But in traditional set theory, every set induces union and intersection sets. This indicates that the extensional definition of set doesn't quite capture the idea of a collection or set of things. Frege's distinction between sense (meaning) and reference, and Mill's distinction between connotation and denotation are similar. Intensions and extensions are also related to the distinction between analytic and synthetic statements. An analytic statement follows from intensions while a synthetic statement is verified by extensions. Kripke specifies a partially defined statement by two sets of members: its extension and its anti-extension. The anti-extension is those members that clearly do not satisfy the predicate.

Defining Intension
In linguistics, logic, philosophy, and other fields, an intension is any property or quality or state of affairs connoted by a word, phrase or other symbol. In case of a word, it is often implied by its definition. The term may also refer to the complete set of meanings or properties that are implied by a concept, although the term comprehension is technically more correct for this. Intension is generally discussed with regard to extension (or denotation). For example, the intension of a car is the all-inclusive concept of a car, including, for example, mile-long cars made of chocolate that may not actually exist. But the extension of a car is all actual instances of cars (past, present, and future), which will amount to millions or billions of cars, but probably does not include any mile-long cars made of chocolate. The meaning of a word can be thought of as the bond between the idea or thing the word refers to and the word itself. Swiss linguist Ferdinand de Saussure contrasts three concepts: * the signified—the concept or idea that a sign evokes. * the signifier—the "sound image" or string of letters on a page that one recognizes as a sign. * the referent—the actual thing or set of things a sign refers to. Intension is analogous to the signified extension to the referent. The intension thus links the signifier to the sign's extension. Without intension of some sort, words can have no meaning. Intension and intensionality (the state of having intension) should not be confused with intention and intentionality, which are pronounced the same and occasionally arise in the same philosophical context. Where this happens, the letter 's' or 't' is sometimes italicized to emphasize the distinction.

Defining Extension
In any of several studies that treat the use of signs, for example, linguistics, logic, mathematics, semantics, and semiotics, the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension or intension, which consists very roughly of the ideas, properties, or corresponding signs that are implied or suggested by the concept in question. In philosophical semantics or the philosophy of langauge, the extension of a concept or expression is the set of things it extends to, or applies to, if it is the sort of concept or expression that a single object by itself can satisfy. (Concepts and expressions of this sort are monadic or "one-place" concepts and expressions.) So the extension of the word "dog" is the set of all (past, present and future) dogs in the world: the set includes Fido, Rover, Lassie, Rex, and so on. The extension of the phrase "Wikipedia reader" includes each person who has ever read Wikipedia, including you. In the context of formal logic, the extension of a whole statement, as opposed to a word or phrase, is sometimes defined (arguably by convention) as its logical value. So, in that view, the extension of "Lassie is famous" is the logical value true, since Lassie is famous. Some concepts and expressions are such that they do not apply to objects individually, but rather serve to relate objects to objects. For example, the words "before" and "after" do not apply to objects individually—it makes no sense to say "Jim is before" or "Jim is after"—but to one thing in relation to another, as in "The wedding is before the reception" and "The reception is after the wedding." Such "relational" or "polyadic" ("many-place") concepts and expressions have, for their extension, the set of all sequences of objects that satisfy the concept or expression in question. So the extension of "before" is the set of all (ordered) pairs of objects such that the first one is before (precedes) the second one.

Computer Science And Intension
In computer science, some database textbooks use the term intension to refer to the schema of a database, and extension to refer to particular instances of a database. The distinction is the same: intension is the logical specification of something, whereas extension is the set of objects or other things that satisfy the conditions of the logical specification given in the intension.