Logical Variables

Logical variables are letters used to represent logical propositions.

They play a central role in logical reasoning and mathematical logic.

A Practical Example

Consider the proposition "all men are mortal."

Rather than writing out the full sentence every time we refer to it, we assign it a letter for simplicity and clarity.

For instance, we might use the letter P:

$$ P: "all \ men \ are \ mortal" $$

This allows us to write:

$$ P \ is \ true $$

or

$$ P \ is \ false $$

without repeating the entire proposition.

Domain of a Variable

The domain of a variable is the set of all values it can take.

In other words, it's the universe of discourse within which we interpret the open statement.

Example

The proposition "all men are mortal" can be either true or false.

$$ P: "all \ men \ are \ mortal" $$

Therefore, the logical variable \( P \) has the domain $ \{ T, F \} $, meaning it can assume the logical values true or false.

Example 2

Now consider the open statement:

$$ x^2 - 4 = 0 $$

In this case, the domain of the variable $ x $ is the set $ \mathbb{R} $, that is, all real numbers.

The distinction between the domain and the truth set. The domain and the truth set are not the same and should not be conflated. The domain encompasses all the possible values a variable can take. The truth set, by contrast, is a subset of the domain consisting only of those values that make an open statement true.

Difference Between Statements and Propositions

Not every statement qualifies as a proposition.

Only those statements that are unambiguously either true or false are considered propositions.

Example

The statement x + 1 > 0 is not a proposition because it depends on the value of x: it’s true when x > -1 and false when x < -1.

Such a statement is referred to as an open statement.

Note. To turn a statement into a proposition, specific numerical values must be assigned to the variables.