Learn critical thinking and logical reasoning to create valid arguments, avoid logical fallacies, and solve real-world problems using tools like if-then implications and Aristotle’s logic.

Critical and logical thinking are essential in today’s democratic society and professional organizations. This course lays a foundation for these skills by embedding logic into everyday language and situations.

You will learn to create logically valid arguments, recognize and avoid logical fallacies, and focus on the form of an argument rather than its content. You’ll explore how to prove the validity or invalidity of arguments and understand the historical development of logic from Aristotle’s syllogism to modern predicate logic.

By the end of the course, you’ll be equipped to tackle real-world problems more effectively using logical operators, if-then implications, and other tools in logical reasoning. You’ll gain skills in using logic and language judiciously in both your professional and civic life.

Logical Thinking

# Proposition
$$``\text{Today is Wednesday}"$$

When writing this very lesson, the simple sentence above was true. But is it also true when you're reading this? Is it Wednesday today? Maybe, maybe not. But one thing we can safely say is that no matter which day of the week it is, this sentence is strictly going to be either true or false.

## Predicate and proposition
The simplistic structure of the sentence above points our attention toward something. Its subject ("today") associates a claim about it or assigns a property to it. This property is termed as the predicate ("Wednesday"). All such sentences that a) have a subject and predicate, and b) resolve to either true or false are called **propositions**. Propositions are the fundamental building blocks of logic.

## Exercise
Create a simple sentence that's *not* a proposition.


How about this sentence itself: "Create a simple sentence that is not a proposition." While we can clearly understand its meaning, the meaning isn't strictly reducible to true or false. It's a command and a call for action.

# Simple and compound sentences
"Today is Wednesday" is a simple sentence because it employs a simple proposition, but we can compound propositions through various logical operators, such as AND, OR, NOT, IF-THEN, IF AND ONLY IF, etc.

"Today is Wednesday AND Tomorrow is Thursday" is now one compound sentence, which can only be either true or false based on the truth value of the individual propositions and the rules of truth resolution defined for each logical operator (AND, in this case). **Compound sentences** can have one, two, or however many propositions joined through logical operators, and the whole compound sentence strictly resolves to either true or false in binary logic. Here's another example of a compound sentence using a NOT operator (denoted by $\neg$):
$$
\text{Today is Wednesday} \space\space —— \space A
\newline
\equiv
(\neg(\neg(\neg(\neg(\text{Today is Wednesday}))))) \space\space —— \space B
\newline
\newline

$$
If proposition A is true, the compound sentence B is also true because the NOT of NOT brings the proposition back to the same state where its original truth resolution began.

# Argument

If we combine multiple propositions (simple or compound) in a logical sequence, then we create an argument. In that logical sequence, the final/last proposition is special and is called conclusion. All preceding propositions are called premises. An argument can have one or more premises but a single conclusion.

While all of the individual propositions that make up the premises and conclusion are true or false, the argument, on the whole, doesn't use the currency of truthfulness or falsehood. An argument can be either valid or invalid, depending on the logical sequence. That means if the conclusion logically follows from the set of premises, then the whole argument is termed valid.

Here’s an example:


Learn about the building blocks of logic.

Introduction to Logical Thinking

Fallacies in Logic

Logical Argument

What Is Logic?

First Theory of Logic: Aristotle’s Logic

Second Theory of Logic: Propositional Logic

Third Theory of Logic: Predicate Logic

Logical Reasoning in Action

Wrap Up

Building Blocks of Logic

Proposition

Predicate and proposition

Exercise

Simple and compound sentences