Veritam Arcana Mathematica First Ed. By V. Kesten

[xkcdapostle]

*A border of Greek Letters and Latin Characters surround each page, skipping the Contents you begin to read the first page...*

Abstract

Mathematics provides the true constructs to describe and model Magic. To abstract from mere matter and deal with concepts that none can truly experience without understanding, this occurs as much within the School of Mathematics as it does the Schools of the Arcane. Both utilise logical structure to derive new means of exploring knowledge. Through rigour and abstraction, one can precisely and concisely define and create new means for Magic.

This text deals with the language aspects for those taking their first blind steps forth into the realm of Wizardry. Here you will find the concepts for logical thought and methodology in tackling and understanding a general basis for Magic and Knowledge. This text is designed with Novice Wizard's in mind whom have little experience in dealing with the notation.

The Text is structured as follows:

It shall first explore the notation and definition of Mathematical Logic and the Theory of Mathematical objects known as Sets. It will also cover the notion of Functions and their properties. Allowing one to practise at reading and writing in this form of notation. After which it shall proceed to the definition of a Language and other constructs required to formulate a new language, however this will be covered briefly.

This text was compiled for the purpose of instruction for new wizard's and to ensure that a standard of rigour and structure is adhered to. Ensuring that Knowledge can be shared amongst Wizard's without too many difference's in notation to express a Spell and to assist Wizard's in the academic pursuit of discovering new Knowledge in General.

I shall be using the Dwarven Runes for the mathematical examples of Logic's Notation. This is due to my heritage and the consideration that it shall allow you to understand the nature of Abstraction, a section in its own right, however should you require the correct pronunciation of the runes. There is a Symbol's table in the Appendix for your reference.

*This is as much as you can read from the Abstract, better skip to the next page before the dry and concise rhetoric puts you to sleep. Oh, the abstract continues there too; better continue turning those pages...*

[xkcdapostle]

*Escaping the Abstract Section, you reach:*
Chapter One:- Language of Logic and Sets

Introduction

Logic is the formal systematic study of the principles of valid inference and correct reasoning. One may be acquainted with the idea, as rare as it is in these times. Here we deal with the Mathematical techniques to represent logic. Essentially it is the notation for codifying the concepts that is to be expressed.

The Notation used here will be covered in this order:

-Propositional Logic
-Predicate Logic
-Modal Logic

Propositional Logic

Propositional Logic is a system for codifying sentence's from the Natural Languages, or everyday talk into propositions or variables to evaluate their consistency within a Logical Model. One may think of it as an implication arising from certain Abstractions regarding the treatment of number's, called Algebra. Consider the following example:

*All Dwarven runes will be rendered in their phonetic form in speech marks, i.e. the Dwarven Language translator for NWN, so you can assume you've been turning to the appendix and back to refresh your memory on the sounds of the runes, possibly have a good memory for how symbols should sound or just believe that you know Dwarven enough to not bother with the appendix*

'It will rain or snow tomorrow'
'Too warm for snow'
'Therefore it will rain'

These are propositions, however, you may see that they are structured as two premises that lead to a deduced conclusion. In a tongue that has ambiguity in each term's meaning, there can be little precision in the meaning of the sentence. As such one may consider the following translation for the above example:

'It will rain or it will snow'
'It will not snow'
'Therefore it will rain'

This can be further abstracted away, making the original meaning arbitrary and allowing us to see the deduction more clearly and precisely:

"Rh" = 'It will Rain'
"K" = 'It will Snow'

"Rh" Or "K"
Not "K"

"Rh"

Where a premise is always considered true. You see that there is an operator =s which is used as an assignment operator, this is covered in the next Chapter on Functions and need not detain us here.

The use of Or is an operator on two premises, that is True if and only if one of the premises is True. The notation will use + to denote an Or operation applied to two premises. Hence:

"Rh" Or "K" is equivalent to "Rh" + "K".

There are four more operations that can be applied to premises, they are:

-And, represented as @
-Negation or Not, represented as ¬
-Implies or 'If "Q" then "O"', represented as "Q"=>"O"
-Equivalence, represented as <=>

*Bored stiff by this simple stuff, or you simply have no interest, you skip ten pages on Propositional Logic, many of them have Logic Tables and exercises, to then get to a page on Sets*