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| | <pre> | | <pre> |
| − | The whole point of formal logic, the reason for doing logic formally and the measure that determines how far it is possible to reason abstractly, is to discover functions that do not vary as much as their variables do, in other words, to identify forms of logical functions that, though they express a dependence on the values of their constituent arguments, do not vary as much as possible, but approach the way of being a function that constant functions enjoy. Thus, the recognition of a logical law amounts to identifying a logical function, that, though it ostensibly depends on the values of its putative arguments, is not as variable in its values as the values of its variables are allowed to be.
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| − | The "indicator function" or the "characteristic function" of a set Q c X, written "f_Q", is the map from X to the boolean domain %B% = {%0%, %1%} that is defined in the following ways:
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| − | 1. Considered in extensional form, f_Q is the subset of X x %B% that is given by the following formula:
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| − | f_Q = {<x, b> in X x %B% : b = %1% <=> x in Q}.
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| − | 2. Considered in functional form, f_Q is the map from X to %B% that is given by the following condition:
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| − | f_Q (x) = %1% <=> x in Q.
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| − | A "proposition about things in the universe", for short, a "proposition", is the same thing as an indicator function, that is, a function of the form f : X -> %B%. The convenience of this seemingly redundant usage is that it allows one to refer to an indicator function without having to specify right away, as a part of its designated subscript, exactly what set it indicates, even though a proposition always indicates some subset of its designated universe, and even though one will probably or eventually want to know exactly what subset that is.
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| − | According to the stated understandings, a proposition is a function that indicates a set, in the sense that a function associates values with the elements of a domain, some of which values can be interpreted to mark out for special consideration a subset of that domain. The way in which an indicator function is imagined to "indicate" a set can be expressed in terms of the following concepts.
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| | The "fiber" of a codomain element y in Y under a function f : X -> Y is the subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X. In other language that is often used, the fiber of y under f is called the "antecedent set", the "inverse image", the "level set", or the "pre-image" of y under f. All of these equivalent concepts are defined as follows: | | The "fiber" of a codomain element y in Y under a function f : X -> Y is the subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X. In other language that is often used, the fiber of y under f is called the "antecedent set", the "inverse image", the "level set", or the "pre-image" of y under f. All of these equivalent concepts are defined as follows: |
| | Fiber of y under f = f^(-1)(y) = {x in X : f(x) = y}. | | Fiber of y under f = f^(-1)(y) = {x in X : f(x) = y}. |