Making valid argument invalid and vice verse

Question

I'm going through a logic textbook and there is an exercise of assessing truth or falsity of following claim: You can make a valid argument invalid by adding extra premisses

At first, I had reasoning similar to the one included in the answers: the claim is false, because adding premiss to already valid argument will be redundant. My made-up example:

All mermaids are mysterious. Ariel is a mermaid. So Ariel is mysterious.

And added premiss: Elsa is a blonde haired girl, either true or false, won't change a thing in valid argument. But then I thought about premiss:

Ariel is not a mermaid.

Will adding it won't change above argument into invalid? Adding contradiction seems to make things broken, but since the answer is "false" I guess this "counterexample" isn't a good one.

I have the same problem with the opposite: apparently, making any invalid argument into valid one is trivial: just add conclusion to the premisses. But is invalid argument

Ariel is a mermaid. So Ariel is not a mermaid.

will be "fixed" by including conclusion in premisses?

Can anybody explain how to deal with these contradictions and how this doesn't change the conclusions that you can't make valid argument invalid and that you absolutely can make invalid one valid.

Answer 1

There is nothing contradictory about your assessment. In classical logic, and many other logics, you cannot make a valid argument invalid by adding premises, but you can make an invalid argument valid by adding premises.

If you want a somewhat imprecise way of thinking about it that might be helpful, try this. Imagine that the premises of your argument contain a quantity of information. If the argument is valid, the premises entail the conclusion, so all of the information in the conclusion is present in the premises. Adding extra premises adds extra information, but since you already have all the information you need to get to the conclusion, it is redundant. So, adding premises to a valid argument cannot turn it invalid. Removing premises can. On the other hand, if you have an invalid argument, there is insufficient information in the premises to entail the conclusion. But if you add extra premises you can supply the missing information and make the argument valid. As you say, the simplest way to do this is just to add the conclusion itself to the premises.

Adding a contradiction to the premises of a valid argument does not make it invalid, since by the principle of explosion, any conclusion at all follows from inconsistent premises. For an explanation of this, see my answer to this question.

A more technical way to state all of this is to say that classical logic is monotonic, i.e. it has the property of monotonicity of entailment. Most commonly studied logics are monotonic, though there are some non-monotonic logics, such as default logics.

Answer 2

You need a clear definition of 'valid' to answer this, and you probably ought to check your book on logic to see which one they're using.

A common one is something like "A deductive step is 'valid' if it is impossible for the premisses to all be true and the conclusion false." In symbols, this is something like ~(P1 Λ P2 Λ ... Pn Λ ~C) for all combinations of true/false assignments to the free variables in all the formulas. A contradictory set of premises therefore makes an argument 'valid' automatically. However, such an argument cannot ever be 'sound'. It is impossible for all the premisses to also be true.