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For sentences, the signs of equality (<math>=\!</math>) and inequality (<math>\ne\!</math>) are reserved to mean the syntactic identity and non-identity, respectively, of their literal strings of characters, while the signs of equivalence (<math>\Leftrightarrow</math>) and inequivalence (<math>\not\Leftrightarrow</math>) refer to the logical values, if any, of these strings, and signify the equality and inequality, respectively, of their conceivable boolean values.  For the logical values themselves, the two pairs of symbols collapse in their significance to a single pair, signifying a single form of coincidence or a single form of distinction, respectively, between the boolean values of the entities involved.

For sentences, the signs of equality (<math>=\!</math>) and inequality (<math>\ne\!</math>) are reserved to mean the syntactic identity and non-identity, respectively, of their literal strings of characters, while the signs of equivalence (<math>\Leftrightarrow</math>) and inequivalence (<math>\not\Leftrightarrow</math>) refer to the logical values, if any, of these strings, and thus they signify the equality and inequality, respectively, of their conceivable boolean values.  For the logical values themselves, the two pairs of symbols collapse in their significance to a single pair, signifying a single form of coincidence or a single form of distinction, respectively, between the boolean values of the entities involved.

In logical studies one tends to be interested in all of the operations or all of the functions of a given type, at least, to the extent that their totalities and their individualities can be comprehended, not just the specialized collections that define particular algebraic structures.  Although the remainder of the dyadic operations on boolean values, specifically, the rest of the sixteen functions of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> could be presented in the same way as the multiplication and addition tables, it is better to look for a more efficient style of representation, one that treats all of the boolean functions on <math>k\!</math> variables on a roughly equal basis, and with a bit of luck, provides a calculus for computing with these functions.  This involves, among other things, finding their values for given arguments, inverting them &mdash; that is, "finding their fibers" or solving equations expressed in terms of them &mdash; and facilitating the recognition of invariant forms that take them as components.

In logical studies, one tends to be interested in all of the operations or all of the functions of a given type, at least, to the extent that their totalities and their individualities can be comprehended, and not just the specialized collections that define particular algebraic structures.  Although the remainder of the conceivably possible dyadic operations on boolean values, namely, the rest of the sixteen functions f : B?B �> B, could be presented in the same way as the multiplication and addition tables, it is better to look for a more efficient style of representation, one that treats all of the boolean functions of k variables on a roughly equal basis, and with a bit of luck, provides a calculus for computing with these functions.  This involves, among other things, finding their values for given arguments, inverting them, "finding their fibers", or solving equations that are expressed in terms of them, and facilitating the recognition of invariant forms that take them as components.

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 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:

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: