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, 04:26, 12 January 2008→3.1. Propositional Calculus: cleanup
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=====3.1.1.2 Partitions: Genus and Species=====
=====3.1.1.2. Partitions : Genus and Species=====
Especially useful is the facility this notation provides for expressing partition constraints, or relations of mutual exclusion and exhaustion among logical features. For example,
Especially useful is the facility this notation provides for expressing partition constraints, or relations of mutual exclusion and exhaustion among logical features. For example,
says that the genus g is partitioned into the three species s1, s2, s3. Its Venn diagram looks like a pie chart. This style of expression is also useful in representing the behavior of devices, for example: finite state machines, which must occupy exactly one state at a time; and Turing machines, whose tape head must engage just one tape cell at a time.
says that the genus g is partitioned into the three species s1, s2, s3. Its Venn diagram looks like a pie chart. This style of expression is also useful in representing the behavior of devices, for example: finite state machines, which must occupy exactly one state at a time; and Turing machines, whose tape head must engage just one tape cell at a time.
=====3.1.1.3 Vacuous Connectives and Constant Values=====
=====3.1.1.3. Vacuous Connectives and Constant Values=====
As a consistent downward extension, the nullary (or 0-ary) connectives can be identified with logical constants. That is, blank expressions " " are taken for the value "true" (silence assents), and empty bounds "()" are taken for the value "false". By composing operations, negation and binary conjunction are enough in themselves to obtain all the other boolean functions, but the use of these k-ary connectives lends itself to a flexible and powerful representation as graph-theoretical data-structures in the computer.
As a consistent downward extension, the nullary (or 0-ary) connectives can be identified with logical constants. That is, blank expressions " " are taken for the value "true" (silence assents), and empty bounds "()" are taken for the value "false". By composing operations, negation and binary conjunction are enough in themselves to obtain all the other boolean functions, but the use of these k-ary connectives lends itself to a flexible and powerful representation as graph-theoretical data-structures in the computer.