👉 Derived from the philosophy of mathematics, particularly from the works of mathematicians like Immanuel Kant and Bertrand Russell, "therefore" (often symbolized as ⊢) denotes logical implication in mathematical reasoning. It signifies that if the premises (the statements leading up to the conclusion) are true, then the conclusion must also be true. In formal logic and proof theory, if a series of logical steps (proofs) starting from axioms or previously proven theorems lead to a statement, we write "therefore" after that statement to indicate that the conclusion logically follows. For example, if we have a series of steps: \( P \rightarrow Q \), \( P \), therefore \( Q \), then \( Q \) is derived from the premises, and we write this as \( \text{⊢} P \rightarrow Q \text{, } P \text{, } \vdash Q \). This formalizes the idea that conclusions are not just assumed but are rigorously derived from given assumptions.
