First-order logic
Introduction
First-order logic is a useful notation for dealing with logical reasoning. Using this notation and the axioms and laws, you should be able to reduce complicated logical statements to simpler ones, or gain insight into the implications of them.
Symbols
The basic symbols are:
- ∧ Logical AND. If the left side is true, and the right side is true, then it is true. Otherwise, false.
- ∨ Logical OR. If either the left side or the right side or both are true, then true. Otherwise, false.
- → Logical "if". If the left side is true, then the right side is true. However, the right side may be true if the left side is false.
- ↔ Logical "iff" (if and only if). If the left side is true, then the right side is true. Also, if the right side is true, then the left side is true.
- - Logical NOT. "It is not the case that X". If true, then false. If false, then true.
- ∀ "For all, it is true that...". For instance, ∀(X→Y) means "For all values, X implies Y." Or, "Y is true if X is true for everything."
- ∃ "For some, it is true that..." or "there exists such that...". For instance, ∃(X→Y) means "For some values, X implies Y."
