1

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.

5

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 Jun 15 '16 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 Jun 17 '16 at 13:28
3

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

2

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

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

  • 1
    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 Jun 29 '16 at 19:54
1

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'.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.