Is this logic correct?

Not enough information:

If A+B=C
then D+D <> C

We don't know what D is.

Hi, Jim!

         The sentence you have presented, which uses the “if…then” connective, is a sometimes called a conditional sentence. A conditional sentence is neither logically valid nor logically invalid because validity and invalidity are properties of arguments, and a conditional sentence is not an argument. When it comes to a conditional sentence, what we may appropriately ask is whether it is true or false, but not whether it is valid or invalid.

         Is the conditional sentence that you presented true? Yes, it is. In truth-functional logic, a conditional sentence is false just in case its antecedent is true and its consequent is false. But the antecedent of your conditional is false. It is false that 5+6=14. So, the conditional is true.  

         Suppose that you did reformulate the sentence into an argument instead of a conditional sentence…

Premise one: 5+6=14

Conclusion: 7+7<>14

         Is this argument valid? Yes, it is. An argument is valid just in case it is impossible that all of the premises are true and the conclusion false. Mathematically, it is impossible for all of the premises of the above argument to be true. 5+6=14 cannot be true. Since it is impossible that all of the premises are true, it follows that it is impossible that all of the premises are true and the conclusion false. Therefore, the argument is valid.  

Outrageous, yes, I know. 

Thank you, Jim

From, Kaiden