Modus ponens

• In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP):

• If P, then Q. • P. • Therefore, Q. • or in logical operator notation:• P → Q • P • ⊢ Q • where ⊢ represents the logical assertion.• The argument form has two premises. The first premise is the "if-

then" or conditional claim, namely that P implies Q. The second premise is that P, the antecedent of the conditional claim, is true. From these two premises it can be logically concluded that Q, the consequent of the conditional claim, must be true as well.

Modus ponens

• Here is an example of an argument that fits the form modus ponens:

• If democracy is the best system of government, then everyone should vote.

• Democracy is the best system of government.

• Therefore, everyone should vote.

Modus tollens• Modus tollens (Latin: mode that denies) is the formal name for

indirect proof or proof by contrapositive (contrapositive inference), often abbreviated to MT.

• Modus tollens has the following argument form:• If P, then Q. • Q is false. • Therefore, P is false. • In logical operator notation:• P → Q • ¬Q • ⊢ ¬P • where ⊢ represents the logical assertion.• Or in set-theoretic form:• P ⊆ Q , x ∉ Q ∴x∉ P • ("P is a subset of Q. x is not in Q. Therefore, x is not in P.")• .

Modus tollens• The argument has two premises. The first premise is the conditional

"if-then" statement, namely that P implies Q. The second premise is that Q is false. From these two premises, it can be logically concluded that P must be false. (Why? If P were true, then Q would be true, by premise 1, but it isn't, by premise 2.)

• Consider an example:• If there is fire here, then there is oxygen here. • There is no oxygen here. • Therefore, there is no fire here. • Another example:• If Lizzy was the murderer, then she owns an axe. • Lizzy does not own an axe. • Therefore, Lizzy was not the murderer