Is the following argument invalid or how do we call this type of argument with contradictory truth values of its premises?

Question

I am not sure if this argument is also called as an invalid argument:

Premis1: $\neg a \to \neg b$ (true), Premis2: $a$ (true), Premis3: $b$ (true)

Answer 1

I assume statement 3) is supposed to be the conclusion?

As such, this is called the Fallacy of Denying the Antecedent

In this fallacy you typically go from $a \rightarrow b$ and $\neg a$ to $\neg b$, but it is easily understood that going from $\neg a \rightarrow \neg b$ and $a$ to $b$ is the same idea: you deny/oppose the antecedent (the 'íf'part of the conditional), in order to deny/oppose the consequent (the 'then' part of the conditional).

A related fallacy is the Fallacy of Affirming the Consequent: going from $a \rightarrow b$ and $b$ to $a$. There, you affirm the consequent in order to affirm the antecedent.

Answer 2

Assuming the argument is actually $\neg a \to \neg b, a \vdash b$, this is a bit obscured version of denying the antecedent. Example: "if something isn't a fruit, then it is not an orange": $a = \text{something is a fruit}$, $b = \text{something is an orange}$. Now, applying this "argument" to apple, we get: apple is a fruit, therefore apple is an orange.