If X, then Y. Not Y. Therefore, not X.

I don't have the logical or mathematical sophistication to know if this is a correct conclusion or not.


4 Answers 4


Yes, this is Modus Tollens, a Hypothetical that reaches its conclusion by denying the consequent of a conditional statement: If X, then Y. Not Y. Therefore not X. It is deductively valid.

However, If X, then Y. Not X. Therefor not Y. This is deductively invalid. It is a fallacy called "Denying the Antecedent," a hypothetical that reaches its conclusion by denying the antecedent of a conditional statement.

  • 1
    Hypothetical syllogism is not involved... See en.wikipedia.org/wiki/Hypothetical_syllogism
    – virmaior
    Commented Jun 15, 2016 at 15:09
  • 1
    You might want to add to this that the truth of Modus Tollens can easily be checked via a truth table. That's usually the starting point for people when they study logic for the first time, so that might be of some help.
    – MM8
    Commented Jun 17, 2016 at 13:28

The statement (¬Y)→(¬X) is called the contrapositive of the statement XY. A statement and its contrapositive are always logically equivalent.


All this can also nicely be shown by a truth table, like so:

The 5th and 8th column are identical which answers your question.

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    Note that the truth table of the 5th column is precisely the definition of logical implication. As soon we have that we immediately infer by use of negation of p and q the truth table of the last column.
    – Zetaman
    Commented Jun 29, 2016 at 19:54

This causal inference can be applied to invariable observations. For example, because rain falls from clouds, we can posit, 'If it is raining, it must be cloudy.'

This can be represented in symbols as "If A then B", where A=It's raining, and B=It's cloudy. From this, we can deduce that if is not cloudy, then it cannot be raining: "Not B, therefore Not A".

However, we can NOT deduce from "If A then B" that "Not A, therefore Not B": 'It's not raining, so it can't be cloudy'.

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