A switch statement is a trinary truth function that returns the value of a consequent, , if the antecedent, , is one and that of an anti-consequent, , when the antecedent is zero. My convention is for this function’s domain to list the antecedent first, followed by the anti-consequent and finally the consequent.
As stated above, this function will return the value of the anti-consequent when the antecedent is zero, and disregard the value of the consequent.
Similarly, when the antecedent is one, the function returns the value of the consequent.
These two tables succinctly assign unique values to all eight parses of . This allows us to calculate the value of .
Thus the switch statement is written:
Truth Functions and Their Parses
To this point I have discussed generalized truth functions and their parses separately. I have explained how the values of parses define the numerical value of operators, but thus far, there hasn’t been any logical connection between a truth function and its parses.
Consider the generalized truth function,
According to the page on semi-parses, setting the value of the first term, , to zero, would produce the semi-parse,
And, similarly, setting the value of that same term to one, would yield,
Now, consider a switch statement where the antecedent is , the anti-consequent is and the consequent is .
As defined, this switch will return the semi-parse when and the semi-parse when , which is identical to what does. Therefore,
That being said, I can continue this madness and rewrite the anti-consequent and consequent as switch statements.
This can continue until all variables in are fixed. At which point the semi-parses of will all be true parses of , and the truth function will be in a form where all variables are inputs and the operator is fixed. Such a function is in standard form, that is without any variable operators.