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Sample sentences
AND operator
IF/THEN operator
NOT operator
OR operator
XOR operator


Chain Rule
Conjunctive Addition
DeMorgan's Law
Disjunctive Addition
Disjunctive Inference
Disjunctive Infer. (XOR)
Double Negation
Modus Ponens
Modus Tollens
Mutual Exclusion


5-step or more
Bad Argument

Modus Tollens

Modus Tollens is a rule of inference pertaining to the IF/THEN operator.

Modus Tollens states that if the consequent of a conditional is false, then the antecedent must also be false.

Imagine we have the following conditional sentence: "If it is raining, then there are clouds in the sky." Formally, we would write:

p -> q: "If it is raining, then there are clouds in the sky."

In this expression, "If it is raining" is the antecedent and "There are clouds in the sky" is the consequent.

Now if we know for a fact that there are no clouds in the sky, we can safely conclude that it is not raining. If the consequent ("There are clouds in the sky") is false, then the antecedent ("It is raining") must also be false, by Modus Tollens. Let's write our steps formally:

p -> q: "If it is raining, then there are clouds in the sky."
~q: "There are no clouds in the sky."
~p: "It is not raining."

The conditional p -> q and the given ~q are above the line of dashes, and the conclusion ~p obtained by applying Modus Tollens is below the line.


Other examples of Modus Tollens

M -> C: If you get enough money, you can buy a new car.
~C: You cannot buy a new car.
~M: You didn't get enough money.


S -> F: If there is smoke, there is a fire.
~F: There is no fire.
~S: There is no smoke.


Links to Relevant Problems

These are links to validity proof problems whose solutions contain Modus Tollens.

2-step problem (A)
2-step problem (B)
3-step problem
5-step problem (A)
5-step problem (B)