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Directory:Jon Awbrey/Papers/Inquiry Driven Systems (view source)
Revision as of 15:38, 12 January 2009
, 15:38, 12 January 2009→1.3.10.3. Propositions and Sentences: markup
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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.
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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.
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.