Changes

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.

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.

The ''indicator function'' or the ''characteristic function'' of the set <math>Q \subseteq X,</math> written <math>f_Q,\!</math> is the map from the universe <math>X\!</math> to the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> that is defined in the following ways:

The "indicator function" or the "characteristic function" of a set X ? U, written "fX", is the map from U to the boolean domain B = {0, 1} that is defined in the following ways:

 

 

1. Considered in extensional form, fX is the subset of UxB

<p><math>f_Q \ = \ \{ (x, b) \in X \times \underline\mathbb{B} \ : \ b = \underline{1} \ \Leftrightarrow \ x \in Q \}.</math></p></li>

fX  =  {<u, v> ? UxB : v = 1  <=> u ? X}.

<li>

<p>Considered in functional form, <math>f_Q\!</math> is the map from <math>X\!</math> to <math>\underline\mathbb{B}</math> that is given by the following condition:</p>

2. Considered in functional form, fX is the map from U to B

<p><math>f_Q (x) = \underline{1} \ \Leftrightarrow \ x \in Q.</math></p></li>

fX(u) = 1  <=> u ? X.

</ol>

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 : U �> B.  The convenience of this seemingly redundant usage is that it permits one to refer to an indicator function without having to specify right away, as a part of its only available designation, exactly what set it indicates, even though a proposition is always an indicator function of some subset of the universe, and even though one probably or eventually wants to know which one.

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 : U �> B.  The convenience of this seemingly redundant usage is that it permits one to refer to an indicator function without having to specify right away, as a part of its only available designation, exactly what set it indicates, even though a proposition is always an indicator function of some subset of the universe, and even though one probably or eventually wants to know which one.

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 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.

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 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.