Beginning Atomic Propositions and Propositional Logic

An intro to the building blocks of logic

In philosophy and mathematics, the study of formal reasoning and the structure of arguments is known as logic. One of the fundamental concepts of logic is the idea of an “atomic proposition.” An atomic proposition is the most basic statement that can be made and it cannot be broken down into simpler components. For example, the statement “the sky is blue” is an atomic proposition, while “the sky is some color” is not.

An atomic proposition is the most basic statement that can be made and it cannot be broken down into simpler components.

“The sky is some color” can be split into many other propositions (colors) and says nothing about the sky (everything has a color!), while “The sky is blue” doesn’t have further break downs. 📷: Shutterstock

Propositional logic (PL) is a type of formal logic that uses symbols and logical connectives to represent and manipulate propositions. It is used to reason about the relationships between propositions, to deduce new propositions from existing ones, and to validate the validity of arguments. In PL, a proposition is represented by a statement that can be either true or false, but not both. Propositions are combined using logical connectives, such as “and,” “or,” and “if…then…” to form more complex statements.

Propositional logic (PL) is a type of formal logic that uses symbols and logical connectives to represent and manipulate propositions.

If peanut butter cups near me, then I will chow down. PBCUPS -> IEAT. 📷: CleverlySimple

Uses

Atomic propositions and PL are useful in many different fields, including computer science, mathematics, philosophy, and artificial intelligence.

• In computer science, PL is used to design algorithms and to reason about the correctness of software.
• In mathematics, PL is used to prove theorems and to study the foundations of mathematics.
• In philosophy, PL is used to analyze arguments and to study the structure of reasoning.
• In artificial intelligence, PL is used to represent knowledge and to perform automated reasoning.

Example of Using Atomic Propositions and PL

Consider the two sentences: “If it is Heads then I win”, “If it is Tails then you lose”. These statements can be represented in PL using the atomic propositions HEADS, TAILS, IWIN, and YOULOSE.

📷: aphablog.com

Let’s start by defining the atomic propositions:

• HEADS represents the statement, “It is heads.”
• TAILS represents the statement, “It is tails.”
• IWIN represents the statement, “I win.”
• YOULOSE represents the statement, “You lose.”

Next, we can use the logical connective “if…then…” (also known as implication) to represent the relationships between the atomic propositions. The first sentence, “If it is Heads then I win,” can be represented as:

HEADS → IWIN

This says that if the statement “it is Heads” is true, then the statement “I win” must also be true.

The second sentence, “If it is Tails then you lose,” can be represented as:

TAILS → YOULOSE

This says that if the statement “it is Tails” is true, then the statement “you lose” must also be true.

Using these two implications, we can now reason about the relationships between the atomic propositions and draw conclusions about what happens in different scenarios. For example, if we know that HEADS is true, then we can deduce that IWIN must also be true. Similarly, if we know that TAILS is true, then we can deduce that YOULOSE must also be true.

In PL, these implications can be combined using other logical connectives, such as “and” and “or,” to form more complex statements. For example, we can combine the two implications above to form the statement:

(HEADS → IWIN) and (TAILS → YOULOSE)

This says that both the statement “If it is Heads then I win” and the statement “If it is Tails then you lose” must be true at the same time.

Can’t live in both Winners Town and Losersville 🤷‍♀. 📷: coachfore.org

In conclusion, atomic propositions and propositional logic are fundamental concepts in mathematics and computer science. By using atomic propositions to represent simple statements of fact, and by using propositional logic to combine and manipulate these propositions, we can create well-formed formulas that can be used to reason about and evaluate the relationships between these statements.

Thank you for reading through my article! If the above helped your understanding, please leave a clap or two or check out other articles I’ve written surrounding the basic blocks of logic in AI

• PEAS: How to Describe Task Environments for Agents in AI
• Describing AI Environments by Type

This was a very beginner article around propositional logic. Stayed tuned for a more advanced walk through!