Logical term for and.
rules for the
logical term 'and' (see:
Natural Logic) that relate to how
the term 'true' applies or fails to apply to
statements with 'and' are:
If .X and Y. is true, then .X. is true.
If .X and Y. is true, then .Y. is true.
If .X and Y. is not true, then .X. is not true or .Y. is not true.
and-4: If .X. is not true or .Y. is not true, then .X and Y. is not
The last two of these rules lead to what are known as De Morgan's
laws, that relate 'and', 'or' and 'not' to the following effect:
.X and Y. is true IFF .not (not X or not Y). is true
.X or Y. is true IFF .not (not X and not Y). is true
The truth-table for 'and' also written as '&' is
which also conforms to the above rules, supposing that every
statement is true (T) or not (F).
It should be noted that conjunction is more complicated when
probabilities are involved, since then the analysis and definition of
conjunction comes to this, with the help of the notion of conditional
p(X and Y) = p(Y|X)*p(X)
This analysis becomes possible because probabilities are numbers
between 0 and 1 inclusive.