User:AP295/Logic

This is a work in progress. It will be moved out of my userspace when finished.

Logic is my favorite subject. For some time I've wanted to write up some material for an introductory course on logic (propositional and FOL). Why? Most undergraduate textbooks that cover logic have a lot of room for improvement. Many of them somehow manage to be both overly verbose and simultaneously loose and imprecise. I think I can do it better.

One does not need any mathematical background in order to understand formal logic, but this material will assume the reader has a basic understanding of algebra. Logic forms the basis for mathematics. Yet what exactly are we going to be studying here? In particular, the object of study will be a set of rules and definitions that together comprise a formal system of logic. It will allow us to work with logic formulas and make logical deductions. You already have an informal understanding of many ideas that we will cover. The purpose of this is to have a precise, unambiguous system of logic. All important concepts will be defined precisely and concisely. The reader will not be expected to glean important concepts from examples or analogies alone, (which is a sloppy teaching style, no doubt the reason I failed my math classes in high school) though examples will be used for illustrative purposes.

First we shall define the idea of a logic formula. This is entirely analogous to the idea of an algebraic formula, which you already understand how to construct: You know that x+3, 10, y^2, (x+y)+z are all valid formulas in algebra. You also know that something like x++^y is not a valid formula, even though x,+,^,y are all symbols that are used in algebra. You know that symbols like @,#,& are generally not used in algebra at all. You know what is and isn't an algebraic formula because algebraic formulas have a specific structure. Let's try to define the set of algebraic formula more precisely. For simplicity we'll only consider a subset.