Tell me about what is Falsum in Logic # ChatGPT Answer Falsum, also known as "[[absurdity]]" or "[[contradiction]]", is a concept in logic that represents a proposition that is always false. In other words, it is a contradiction that cannot possibly be true under any circumstances. The symbol for Falsum is often represented as ⊥ or '⊢ ¬', which is the negation of the tautology symbol. It's used in various fields such as propositional logic, proof theory and philosophy to denote an impossible or contradictory statement. In formal systems, Falsum can be used to represent the result of a contradiction derived from an assumption. This can be useful in proof by contradiction (also known as reductio ad absurdum), where an assumption is shown to lead to a contradiction, therefore proving that the assumption must be false. Falsum plays a crucial role in certain logical principles such as "ex falso quodlibet" (from falsehood, anything follows) and "non-contradiction" (nothing can both be and not be). These principles form the foundation of classical logic and many other logical systems. # Bard Answer In formal logic, **falsum**, also known as **[[Bottom]]** or sometimes **contradiction**, refers to a specific proposition that is always false. It serves as a crucial concept in various logic systems, particularly in proving contradictions and demonstrating the truth of other propositions. Here are some key points about falsum: - **Symbol:** Typically represented by the symbol ⊥ (bot) or sometimes ¬⊤ (not true). - **Meaning:** Denotes a statement that is inherently untrue, regardless of any context or circumstances. - **Properties:** - **Negation:** ¬⊥ (not falsum) is always true, as negating something always false results in truth. - **Implication:** From falsum, you can infer any proposition (⊥ ⊢ φ). This follows the principle of "ex falso quodlibet" (from falsity, anything follows). - **Proofs:** Demonstrating a statement leads to falsum (⊢ ⊥) implies that the statement is a contradiction and therefore false. - **Role in Proofs:** - **Proof by contradiction:** In this technique, you assume the opposite of the statement you want to prove. If this leads to falsum, the original statement must be true. - **Reductio ad absurdum:** Similar to proof by contradiction, but focuses on showing that the opposite of the statement leads to nonsensical or impossible conclusions, implying its falsity. Understanding falsum is essential for grasping the nuances of formal [[logic]], particularly when dealing with proofs and inconsistencies. Its unique properties allow for powerful reasoning techniques and contribute to the rigorous nature of logical systems. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Falsum" ) or contains(subject, "falsum" ) or contains(subject, "Bottom" ) or contains(subject, "bottom" ) ```