I'm not very good at believing anything on purpose, but several irrational assumptions have snuck in by accident. One such belief is that reason is the foundation of all mathematical exploration. Without a simple, universal mechanism, built into our being, there is no way we could expect to come to any sort of emotionless concensus.
The following pages explain the tools and notation I have adopted in search of that thought process. I generally resist the stricture of propositional calculus, and first order logic, but barrow heavily from their nomenclature. Some of the notation is mine, and I appologize for any confusion that may come from similarities to that of others. I am powerless to resist using this notation in my writing, and I thought it sporting to give the reader half a chance of knowing what I'm on about.
Generalized Truth Functions
Logic is all about choices. An idea is true or false, a student is present or absent, a switch is on or off. All the truth tables and specialized language of propositional calculus is geared to exploring these choices and their consequences.
In this section I will layout basic concepts and use them to explore logic, in general. With these simple ideas we'll look at topics familiar to logic and some that aren't usually thought of as logic.
It should be noted that some, but not all, notation I use is my own. I have tried to use as much existing notation and vocabulary as I can while still leaving myself free to explore.
Truth Function Algebra
Algebra is the art of moving variables. Commutativity simply shuffles the deck, associativity breaks a problem into more convinient parts, and identities create change while ostensibly doing nothing. No theory containing nexted functions is complete without it.
This section uses generalized truth functions to explore the common tools of algebra and introduces some new techniques not available when only considering two binary functions. My approach to the concepts of commutativity and associativity is more generalized than in abstract algebra. I do away with concern for keeping the same truth functions, so long as the relationship between input variables is maintained.