Mathematical rules are based on the defining limits we place on the particular numerical
quantities dealt with. When we say that 1 + 1 = 2 or 3 + 4 = 7, we are implying the use
of integer quantities: the same types of numbers we all learned to count in elementary
education. What most people assume to be self-evident rules of arithmetic -- valid at all
times and for all purposes -- actually depend on what we define a number to be.
For instance, when calculating quantities in AC circuits, we find that the "real" number
quantities which served us so well in DC circuit analysis are inadequate for the task of
representing AC quantities. We know that voltages add when connected in series, but
we also know that it is possible to connect a 3-volt AC source in series with a 4-volt AC
source and end up with 5 volts total voltage (3 + 4 = 5)! Does this mean the inviolable
and self-evident rules of arithmetic have been violated? No, it just means that the rules
of "real" numbers do not apply to the kinds of quantities encountered in AC circuits,
where every variable has both a magnitude and a phase. Consequently, we must use a
different kind of numerical quantity, or object, for AC circuits (complex numbers, rather
than real numbers), and along with this different system of numbers comes a different
set of rules telling us how they relate to one another.
An expression such as "3 + 4 = 5" is nonsense within the scope and definition of real
numbers, but it fits nicely within the scope and definition of complex numbers (think of a
right triangle with opposite and adjacent sides of 3 and 4, with a hypotenuse of 5).
Because complex numbers are two-dimensional, they are able to "add" with one another
trigonometrically as single-dimension "real" numbers cannot.
Logic is much like mathematics in this respect: the so-called "Laws" of logic depend on
how we define what a proposition is. The Greek philosopher Aristotle founded a system
of logic based on only two types of propositions: true and false. His bivalent (two-mode)
definition of truth led to the four foundational laws of logic: the Law of Identity (A is A);
the Law of Non-contradiction (A is not non-A); the Law of the Excluded Middle (either A
or non-A); and the Law of Rational Inference. These so-called Laws function within the
scope of logic where a proposition is limited to one of two possible values, but may not
apply in cases where propositions can hold values other than "true" or "false." In fact,
much work has been done and continues to be done on "multivalued," or fuzzy logic,
where propositions may be true or false to a limited degree. In such a system of logic,
"Laws" such as the Law of the Excluded Middle simply do not apply, because they are
founded on the assumption of bivalence. Likewise, many premises which would violate
the Law of Non-contradiction in Aristotelian logic have validity in "fuzzy" logic. Again, the
defining limits of propositional values determine the Laws describing their functions and
relations.
The English mathematician George Boole (1815-1864) sought to give symbolic form to
Aristotle's system of logic. Boole wrote a treatise on the subject in 1854, titled An
Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories
of Logic and Probabilities, which codified several rules of relationship between
mathematical quantities limited to one of two possible values: true or false, 1 or 0. His
mathematical system became known as Boolean algebra.
