The study of propositions and their logical relationships through connectives: conjunction (AND), disjunction (OR), negation (NOT), and implication (IF-THEN).
P → Q, P ⊢ Q
If P implies Q, and P is true, then Q must be true. The fundamental rule of detachment.
P → Q, P ⊢ Q
If P implies Q, and Q is false, then P must be false. The contrapositive argument.
P → Q, ¬Q ⊢ ¬P
A proposition cannot be both true and false at the same time and in the same respect. The bedrock of classical logic upon which all valid reasoning depends.
¬(P ∧ ¬P)
A proposition that is true under every possible interpretation. A logical truth independent of content.
P ∨ ¬P
A proposition that is false under every possible interpretation. The negation of a tautology.
P ∧ ¬P
For all x in a domain, a property holds. The assertion of generality.
∀x P(x)
There exists at least one x in a domain for which a property holds.
∃x P(x)