Logic Assertion & Proposition
What Exactly is a Proposition?
Imagine you are playing a game where every sentence you say must be judged as either correct or incorrect. You cannot say things like "Wow!" or "Please close the door," because they cannot be assigned a truth value. In logic, we call a sentence that can be judged as true or false a proposition (or a logical assertion).
For example, the statement "Colombo is further north than Singapore" is a proposition. We may not know the answer immediately, but it is a fact that is either true or false (in reality, Colombo is slightly north of Singapore, so it is TRUE). Another example: "The number 2 is greater than 5" is definitely a proposition (it is FALSE).
Simple vs. Compound Propositions
Propositions can be basic or they can be built from smaller ones. A simple proposition contains no logical connectives. A compound proposition joins two or more simple propositions using words like "and", "or", or "if...then".
| Type | Example | Truth Value |
|---|---|---|
| Simple | The sky is blue. | TRUE (on a clear day) |
| Simple | 5 + 3 = 10 | FALSE |
| Compound | The sky is blue AND 5 + 3 = 10 | FALSE (one part is false) |
Logical Connectives: The Glue of Logic
When we build compound propositions, we use special symbols called logical connectives. The most common ones are:
- Negation (NOT): It flips the truth value. If P is true, then Ā¬P is false. In MathJax: $\neg P$.
- Conjunction (AND): True only if both parts are true. Notation: $P \land Q$.
- Disjunction (OR): True if at least one part is true. Notation: $P \lor Q$.
- Implication (IF...THEN): Notation: $P \rightarrow Q$. It is false only when $P$ is true and $Q$ is false.
These symbols help us write logical statements in a compact, mathematical way.
Truth Tables: Mapping Every Possibility
A truth table is a chart that shows the truth value of a compound proposition for every possible combination of its simple parts. It is like a scorecard for logic. Let's look at the truth table for $P \land Q$ (AND):
| P | Q | P ā§ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
As you can see, the only time $P \land Q$ is true is when both P and Q are true.
Real-Life Logic: The Geography Example
Let's return to our opening example: "Colombo is further north than Singapore." We can treat this as a simple proposition and call it C. How do we determine its truth value? We check the facts. The latitude of Colombo is about 6.9Ā° N, and Singapore is about 1.3Ā° N. Since 6.9 is greater than 1.3, Colombo is indeed further north. Therefore, C = TRUE.
Now imagine a compound statement: "Colombo is further north than Singapore AND 2 + 2 = 4." This becomes $C \land T$ (where $T$ stands for the true math fact). Since both parts are true, the whole statement is TRUE.
Important Questions About Propositions
ā A: Not always. For example, $x + 2 = 5$ is not a proposition because we don't know what $x$ is. It becomes a proposition only when $x$ is given a specific value (like $x = 3$ making it true, or $x = 1$ making it false). These are called open statements or predicates.
ā A: No, that is the most important rule! A proposition must have a definite truth valueāit cannot be ambiguous. This is called the law of non-contradiction. A statement like "This sentence is false" creates a paradox and is not considered a valid proposition in classical logic.
ā A: Symbols allow us to focus on the form of the argument rather than the content. For example, whether we talk about geography or math, the structure $P \land Q$ works the same way. This is the power of logicāit is a universal language.
Why This Matters: The Foundation of Computing
Understanding propositions is not just for philosophy class. Every computer program you use relies on logical propositions. When you write if (age >= 18) in code, you are checking if the proposition "age is greater than or equal to 18" is TRUE. If it is, the computer performs an action. If it is FALSE, it does something else. This is the heart of decision-making in technology.
Footnote
[1] Truth Value: The attribute assigned to a proposition, either true (T) or false (F).
[2] Logical Connectives: Operators like AND ($\land$), OR ($\lor$), and NOT ($\neg$) that combine one or more propositions.
[3] Truth Table: A mathematical table used in logic to compute the truth values of logical expressions.