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| | <pre> | | <pre> |
| − | The "negation" of x, for x in %B%, written as "(x)"
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| − | and read as "not x", is the boolean value (x) in %B%
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| − | that is %1% when x is %0%, and %0% when x is %1%.
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| − | Thus, negation is a monadic operation on boolean
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| − | values, a function of the form (_) : %B% -> %B%.
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| − | It is convenient to transport the product and the sum operations of !B!
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| − | into the logical setting of %B%, where they can be symbolized by signs
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| − | of the same character, doubly underlined as necessary to avoid confusion.
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| − | This yields the following definitions of a "product" and a "sum" in %B%
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| − | and leads to the following forms of multiplication and addition tables.
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| − | The "product" of x and y, for values x, y in %B%, is given by Table 8.
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| − | Table 8. Product Operation for the Boolean Domain
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| − | o---------o---------o---------o
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| − | | %.% # %0% | %1% |
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| − | o=========o=========o=========o
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| − | | %0% # %0% | %0% |
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| − | o---------o---------o---------o
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| − | | %1% # %0% | %1% |
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| − | o---------o---------o---------o
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| − | Viewed as a function on logical values, %.% : %B% x %B% -> %B%, the product corresponds to the logical operation that is commonly called "conjunction" and that is otherwise expressed as "x and y". In accord with common practice, the raised dot ".", doubly underlined or otherwise, is frequently omitted from written expressions of the product.
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| − | The "sum" of x and y, for values x, y in %B%, is given by Table 9.
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| − | Table 9. Sum Operation for the Boolean Domain
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| − | o---------o---------o---------o
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| − | | %+% # %0% | %1% |
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| − | o=========o=========o=========o
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| − | | %0% # %0% | %1% |
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| − | o---------o---------o---------o
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| − | | %1% # %1% | %0% |
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| − | o---------o---------o---------o
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| − | Viewed as a function on logical values, %+% : %B% x %B% -> %B%, the sum corresponds to the logical operation that is generally called "exclusive disjunction" and that is otherwise expressed as "x or y, but not both". Depending on the context, a couple of other signs and readings that can invoke this operation are:
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| − | 1. "x =/= y", read "x is not equal to y", or "exactly one of x and y".
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| − | 2. "x <=/=> y", read "x is not equivalent to y", or "x opposes y".
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| | For sentences, the signs of equality ("=") and inequality ("=/=") are reserved to signify the syntactic identity and non-identity, respectively, of the literal strings of characters that make up the sentences in question, while the signs of equivalence ("<=>") and inequivalence ("<=/=>") refer to the logical values, if any, of these strings, and serve to signify the equality and inequality, respectively, of their conceivable boolean values. For the logical values themselves, the two pairs of symbols collapse in their senses 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 ("=") and inequality ("=/=") are reserved to signify the syntactic identity and non-identity, respectively, of the literal strings of characters that make up the sentences in question, while the signs of equivalence ("<=>") and inequivalence ("<=/=>") refer to the logical values, if any, of these strings, and serve to signify the equality and inequality, respectively, of their conceivable boolean values. For the logical values themselves, the two pairs of symbols collapse in their senses 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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