Notes by Lex Toumbourou

A proposition (also called a statement) is a declarative sentence with a truth value of either true or false, but not both.

Every proposition must be:

1. A complete sentence that makes a claim
2. Either true or false (but not both)
3. Definite in its truth value

For example, these are valid propositions:

• "I wrote this article on Thursday" (true)
• "I am 14 years old" (false)
• "1 + 1 = 3" (false)
• "Water boils at 100°C at standard atmospheric pressure" (true)
• "Every even integer is divisible by 2" (true)

These are not propositions:

• "What time is it?" - Questions aren't propositions as they don't make claims
• "Close the door" - Commands don't have truth values
• "What a beautiful day!" - Exclamations and opinions without clear criteria aren't propositions

Also, expressions with variables, where the value of the variable would affect the truth value, aren't propositions:

• $x + 2 = 5$ - Not a proposition, as its truth value depends on the value of $x$.

When a statement contains a variable whose truth value depends on that variable's value, it's called a Predicate.

For example:

• "I am $X$ years old"
• $Y + 1 = 5$
• "The speed of light in a vacuum is approximately $Z$ meters per second."

In propositional logic, we often assign propositions to variables, typically using lowercase letters $p$, $q$, $r$, $s$, or $t$:

• Let $p = \text{"It rained yesterday"}$
• Let $q = \text{"I am happy"}$

These variable assignments allow us to manipulate and analyse propositions using Propositional Logic, where we can combine simple propositions to form more complex ones.

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Backlinks

• Propositional Logic