Chris 2021-01-20

Logic 1

Prerequisites

None. But hopefully you are feeling sharp.

Introduction

It is very difficult to start off with rigorous mathematics. When I was younger, I always thought mathematics had some definite beginning. Nothing ever had to be loosely defined or taken for granted. The reality is partially true: the beginning is logic and set theory but they start off very rough.

Now I imagine Mathematics and all its fields as a path. This path comes out from under the dirt in rough patches. The path splits and recombines showing all the fields and its interconnections. There are wider paths (where more people travel) and there are stronger paths (that are more mathematically rigorous).

I will begin as far back as I can. And I will provide my own spin and definitions on these things.

Colloquialisms and Definitions

Sadly, the core of logic and set theory need to start off with things that cannot be rigorously defined. If they were to be defined, they would use terms that are not yet defined (creating circular reasoning). What is circular reasoning? Why is circular reasoning wrong? Um... As you can see, we already hit a snag before we have even defined anything. We need to define terms in order to define circular reasoning. But we also need to know about circular reasoning to know why we need to define terms in a linear order.

We will try to ignore these problems.

• Definition ~ a mathematical description of a term. Words are defined with the notation "[word] - [description]".
• Colloquialism ~ a term that has a generally accepted meaning but cannot be usefully defined. Words are colloquialised with the notation "[word] ~ [description]".
• Good definition ~ a definition is a good definition if it is sensible and useful. That is

From now onward, I will try not to use uncolloquial terms or undefined terms. If I do, please let me know so I can fix the issue. If a term is never defined, you should use a relevant dictionary and treat its explanation as a colloquialism.

Logic

Finally we are able to define the most basic objects in logic. We still need to colloquialise and define things in a circular manner. The specific name for this field is called Boolean Logic.

• True ~ when something is in accordance with everything before it.
• False ~ when something is not in accordance with everything before it.
• Statement ~ a self-contained sentence (or sentences) that is either true or false.

Example 1

1. "The sky is blue" is a statement (and it is true some of the time).
2. "The sky is red" is also a statement (it is just false most of the time).
3. "What is the time?" is not a statement since it is neither true nor false. Most questions and instructions are not statements.
4. "This is good" is not a statement since it is not self-contained. We don't know what "this" is.
5. "Let Bob be a person. Bob has two eyes." This is also a statement even though it consists of two sentences. Most statements however will only consist of one sentence to become clear.