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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.

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