1. Introduction
Propositional calculus, or propositional logic, is a fundamental branch of logic that focuses on the manipulation and combination of propositionspropositionA statement that is either true or false, and not both.Read the full entry statements that can be definitively declared as true or false. It lays the groundwork for understanding more complex logical systems and finds applications across various disciplines.
2. Propositions
Propositions are declarative sentences that assert a fact about the world, which can either be true or false, such as "It is raining".
3. Truth Values: ⊤ and ⊥
In propositional logic, we use special symbols to represent truth valuestruth valueOne of the two values a proposition can take: true or false.Read the full entry: ⊤ (top) represents TRUE and ⊥ (bottom) represents FALSE. These symbols are standard in formal logic and appear in truth tablestruth tableA row for every assignment of values, with the formula's value in each.Read the full entry throughout this guide.
4. Truth Tables
Truth tables are systematic methods for determining the truth value of logical expressions based on the truth values of their constituent propositions, offering a clear visual representation of logical operations. They can look like the following:
| p | q | p → q |
|---|---|---|
| ⊥ | ⊥ | ⊤ |
| ⊥ | ⊤ | ⊤ |
| ⊤ | ⊥ | ⊥ |
| ⊤ | ⊤ | ⊤ |
5. Logical Operators
Logical operatorslogical connectiveA symbol that builds a compound proposition out of simpler ones.Read the full entry are symbols used to connect propositions or alter their truth values, forming the basis for constructing complex logical expressions. The primary operators include:
OR ∨
True if at least one of the combined propositions is true. ∨
| p | q | p ∨ q |
|---|---|---|
| ⊥ | ⊥ | ⊥ |
| ⊥ | ⊤ | ⊤ |
| ⊤ | ⊥ | ⊤ |
| ⊤ | ⊤ | ⊤ |
IMPLIES →
True except when the first proposition is true and the second is false. →
| p | q | p → q |
|---|---|---|
| ⊥ | ⊥ | ⊤ |
| ⊥ | ⊤ | ⊤ |
| ⊤ | ⊥ | ⊥ |
| ⊤ | ⊤ | ⊤ |
BICONDITIONAL ↔
True if both propositions are equally true or false. ↔
| p | q | p ↔ q |
|---|---|---|
| ⊥ | ⊥ | ⊤ |
| ⊥ | ⊤ | ⊥ |
| ⊤ | ⊥ | ⊥ |
| ⊤ | ⊤ | ⊤ |
6. Expressions
Expressions are more complex statements formed by joining propositions with logical operators, allowing for the representation of nuanced logical relationships.
7. Logical Equivalences
Logical equivalences are expressions that hold the same truth value under all possible conditions. They include fundamental laws like the Law of Identity, the Law of Noncontradiction, and De Morgan's lawsDe Morgan's lawsNegation turns ∧ into ∨ and ∨ into ∧: ¬(p ∧ q) ≡ ¬p ∨ ¬q.Read the full entry.
8. Proofs
Proofs in propositional calculus involve demonstrating the truth of a proposition based on axioms (assumed truths), previously established truths, and rules of inferencerule of inferenceA licensed step from formulas already derived to a new one.Read the full entry. They are crucial for validating logical argumentsargumentA set of premises offered in support of a conclusion.Read the full entry and theorems.
9. Applications
Propositional calculus is not just a theoretical framework but also has practical applications in computer science for software verification, in mathematics for formalizing proofs, and in philosophy for analyzing arguments. Its principles underpin the study of more advanced logical systems, such as predicate logicpredicate logicLogic that looks inside propositions, at objects and their properties.Read the full entry, and play a vital role in the development of logical reasoning and critical thinking skills.