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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.

The Holy Trinity

Core Concepts

$$(\textbf{A}^n),\ (\textbf{A}^n)_\text{CV},\ (\textbf{A}^n)_\alpha$$

At its core, logic has three parts: domains, condition values, and logical functions. With these basic parts one can explore the certainty of orthodox formal logic, the uncertainty of quantifiers, the expanse of sets, and the insight of theory.

More Of The Same, With A Twist

Function Parses and Operators

$$(A,B)_\alpha\big|_{(A,B)_\text{CV}=i}=\alpha_i,\ \vec{\alpha}^4=(\alpha_\mathtt{0x0},\alpha_\mathtt{0x1},\alpha_\mathtt{0x2},\alpha_\mathtt{0x3})$$

Continuing where "The Holy Trinity" left off, this section explores the output of truth functions as individual variables.

I Swear It Was Here A Minute Ago

Term Neglect

$$\big<(\textbf{A}^n,\textbf{B}^m)_\alpha\big>^{(\textbf{A}^n)}_\phi=\big((\textbf{A}^n,\textbf{B}^m)_\alpha\big|_{(\textbf{A}^n)_\text{CV}=0},\ldots,(\textbf{A}^n,\textbf{B}^m)_\alpha\big|_{(\textbf{A}^n)_\text{CV}=2^n-1})_\phi$$

Some times you need to test a truth function's mettle on a domain. A neglect function is just such a test. This special case truth function systematically compares all semi-parses of a function, across a neglected domain, using a specific standard.

In this section neglect is generally defined, and two even more special forms are presented.

Setting Standards

Standard Operator Functions

$$ (\textbf{A}^n)_{\sigma^n_i}\big|_{(\textbf{A}^n)_\text{CV}=j}=A_{i-1}\ \forall j\in[0,2^n-1]\ni i\in[1,n] $$

Among the many strange and complex truth functions that are possible when the concept is generalized, there are several very boring functions that only ever return the value of one operand. This section defines those functions, and provides notation to invoke them.