## Symbolic and semantic

Human reason follows some basic patterns.

The truth of one statement is often connected to the truth of another statement. For example, let’s take the following example.

**Example**. If it is cloudy outside, then I don’t need my sunglasses.

To understand the content of this course, the first thing we need to do is disentangle our thought a bit. There are two closely related but distinct facets of this example.

There is the *truth* of the statement. Is it actually cloudy outside or not? Do I actually need sunglasses or not? Whether or not someone actually needs their sunglasses probably depends on the person and the level of cloudiness.

In focusing on the truth of a statement, we care about the *semantics* of the proposition.

Distinctly, we can also view this statement as built up from basic “atoms” using well-defined rules.

In our semantic analysis, we already have broken it down into two separate statements that can either be truth or false in principal:

$A$. It is cloudy outside

$B$. I don’t need my sunglasses

Our example statement is built up from $A$ and $B$ via a if-then construction. It then makes sense to denote it by $A \Rightarrow B$. The fancy arrow $\Rightarrow$ stands in for the if-then.

Thus, *symbolically* we have $A$ and $B$ and $\Rightarrow$ provides a way to connect them to make a new formal symbol \(A \Rightarrow B\)

Symbolically, $A$ and $B$ have no infused meaning so why do we even consider this?

**Because symbolic manipulation can be incredibly powerful.** It can streamline human thought. It can be mechanized in a computer program. It can be converted in a game.

For mathematicians, logical reasoning is especially important. When reasoning about intricate structures, it is incredibly easy to make a mistake. The history of mathematics is rich with such examples.

To help minimize errors, mathematicians grounded their arguments in logical deduction and fostered a culture of *rigorous proofs*. Let’s look at a more mathematical example.

Recall that \(\mathbb{N} = \lbrace 0,1,2,\ldots \rbrace\) is the set of *natural numbers*. A natural number $n$ is *even* if it can be written as $n = 2m$ for some other natural number $m$. A number is *odd* if is not *even*.

**Example**. Let $n$ be a natural number. If $n$ is even, then $n+1$ is odd. t If we denote

$A$. $n$ is even.

$B$. $n+1$ is odd.

then symbolically we can represent the statement as before \(A \Rightarrow B\) But semantically this if very different than our cloudy/sunglasses example above.

We start by introducing the basic structure of logical argument.

## Table of contents

- Propositions and proofs
- And, or, implies
- Not
- Reductio ad absurdum
- Truth tables
- Proof vs truth
- Useful formula