#logic Propositional logic, also known as [[sentential logic]] or [[statement logic]], is a branch of formal logic that focuses on the study of logical relationships between propositions. A [[proposition]] is a statement that can be either [[true]] or [[false]], and propositional logic deals with the analysis and manipulation of these propositions. In propositional logic, propositions are represented using variables such as p, q, or r. These variables can take on one of two truth values: true (T) or false (F). Logical operators are used to combine these propositions to form more complex expressions. The main logical operators in propositional logic include: 1. Negation (¬): This operator represents the logical negation or denial of a proposition. For example, if p is true, then ¬p is false. 2. Conjunction (∧): The conjunction operator represents logical "and." It is true only when both propositions being connected are true. For example, if p and q are both true, then p ∧ q is also true. 3. Disjunction (∨): The disjunction operator represents logical "or." It is true if at least one of the connected propositions is true. For example, if either p or q (or both) are true, then p ∨ q is also true. 4. Implication (→): The implication operator represents logical "if-then" statements. It states that if the first proposition (the antecedent) is true, then the second proposition (the consequent) must also be true. If the antecedent is false, the implication is considered vacuously true. For example, if p implies q and p is false, then the implication holds regardless of whether q is true or false. 5. Bi-implication (↔): The bi-implication operator represents logical equivalence between two propositions. It states that two propositions have the same truth value - either both are true or both are false. For example, p ↔ q is true if p and q have the same truth value. Propositional logic allows the construction of logical arguments and reasoning by using these operators and their associated rules. It provides a foundation for more advanced logical systems and is widely used in fields such as mathematics, computer science, philosophy, and artificial intelligence. ## Propositional Logic vs. Categorical Logic Category theory is a branch of mathematics that provides a formal framework for studying mathematical structures and their relationships. It abstracts mathematical structures into objects and morphisms, which capture the essential properties and relationships between them. [[Categorical Logic]] is an approach to logic that uses category theory as its foundational framework. It extends propositional logic to categories, which allows reasoning about logical relationships in a more general and abstract setting. In categorical logic, propositions are represented as objects in a category, and logical connectives are represented by morphisms (arrows) between these objects. For example, in the category of sets, the object representing a proposition may be a set of elements, while the morphisms represent logical connectives such as union or intersection. Categorical logic also introduces additional concepts that capture logical properties beyond those found in propositional logic alone. For instance, it introduces concepts like universal quantification (for all) and existential quantification (there exists), which can be represented using categorical constructs like limits and colimits. The advantage of using category theory in conjunction with propositional logic is that it provides a more general framework for reasoning about logical relationships across different mathematical domains. It allows for the study of common structures and patterns across different areas of mathematics through the use of universal properties and functoriality. Overall, propositional logic with respect to category theory and categorical logic provides a powerful tool for studying logical relationships in a wide range of contexts beyond traditional propositional statements. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Propositional logic") or contains(subject, "Proposition") ```