Modus Tollens
If P implies Q, and Q is false, then P must be false
P→Q, ¬Q ⊢ ¬PSymbolic Logic Examples
P→Q, ¬Q ⊢ ¬PBasic Modus Tollens
formal
Conditional:
P→Q
Negated Consequent:
¬Q
Negated Antecedent:
¬P
Complex Example
demonstration
If it rains:
R→W
Ground is not wet:
¬W
Therefore:
¬R (It did not rain)
Formal Proof Step
advanced
Given:
(P∧Q)→R
Fact:
¬R
Conclusion:
¬(P∧Q)
Key Point: Modus Tollens allows us to reason backwards from the negation of the consequent to the negation of the antecedent.
Examples & Applications
Example 1: Logical Deduction
beginner
If it rains (Premise):
Then the ground gets wet
The ground is not wet (Fact):
We observe the ground is dry
Therefore (Conclusion):
It did not rain
Explanation: This is a classic example of Modus Tollens. Since the consequent (ground wet) is false, we can validly conclude the antecedent (raining) is also false.
Example 2: Academic Context
intermediate
If you study consistently (Premise):
Then you will pass the exam
You did not pass the exam (Fact):
The exam result shows failure
Therefore (Conclusion):
You did not study consistently
Explanation: While this may seem harsh, Modus Tollens allows us to make this logical inference. If the studying was truly sufficient, then passing should have followed.
Key Insights
Contrapositive Foundation: Modus Tollens relies on the fact that P→Q is equivalent to ¬Q→¬P. Understanding contrapositives is key to mastering this rule.
Debugging and Diagnosis: Essential for troubleshooting and error detection. When something that should happen doesn't happen, Modus Tollens helps identify the cause.
Practical Application
Proof by Contradiction Setup: Often used in proof by contradiction - assume the opposite of what you want to prove and derive a contradiction.
Common Errors: Don't confuse with denying the antecedent (P→Q, ¬P ⊢ ¬Q) which is invalid. Only denying the consequent works.
Scientific Method: Fundamental to hypothesis testing - when predicted outcomes don't occur, we reject the hypothesis.