Question of Validity in syllogisms of deductive argument

Question

I read the following excerpt from my A-Level Philosophy book:

"Two key terms that you need to understand in relation to deductions and other forms of argument are ‘validity’ and ‘soundness’. Validity relates to the form of the argument. Soundness relates to an argument’s premises and its form."

"Deductive arguments have a form which is valid, which just means that if the premises are true, the conclusion must also be true."

Example 1

P1 All bunnies can speak French

P2 Tiggles is a bunny

C Tiggles can speak French

"Although the form of the argument is valid, it is not a sound one."

Here is Example 2:

P1 All bunnies are mammals

P2 Speedy the lizard is not a bunny

C Therefore Speedy is not a mammal

Is this a valid argument? Well, the premises and the conclusion appear to be true. However, this is not enough to make it valid. For, although true, the conclusion does not actually follow from the premises, so, as far as this argument can show, it might have been false. To see this, we can replace some of the terms while keeping the same structure:"

Example 3

P1 All bunnies are mammals

P2 Wilbur the cat is not a bunny

C Therefore Wilbur is not a mammal

As we know that cats are mammals, we can see that the conclusion is false, even though the premises are true, and this shows that this form of argument is invalid."

For Example 2 I understand that, excluding pre-existing knowledge of lizards, that is, as far as the argument is concerned, the conclusion can be true OR false.

My Question: Am I correct in saying, Example 2 has an invalid form, and thus invalid argument according to this excerpt, because by changing the set of animals from Lizards to Cats in example 3, keeping the structure of the syllogism the same, one can clearly see the form of argument is invalid with a false conclusion. (+ and since structure/form is same for example 2, the form is also invalid in 2)

However, I am contradicted by another excerpt I had come across an excerpt LEHMANN ON THE RULES OF THE INVALID SYLLOGISMS that

"A neither valid nor invalid syllogism is one in which the conclusion either can be true or can be false when each of the two premises is true".

So is example 2 an invalid argument OR 'neither valid nor invalid argument' excluding pre-existing knowledge of lizards (as far as the argument shows)?

Answer 1

What you say in bold is almost correct. Example 2 is invalid, because it is possible to create a counterexample by substituting 'cat' for 'lizard'. A counterexample is an instance where the premises of the argument are true and the conclusion false.

I say almost correct, because we have to be careful when talking about invalid forms. An argument is not invalid just because it instantiates an invalid form. A valid form is one such that every argument that is an instance of that form is valid. An invalid form is any form that is not a valid form. Invalid just means not valid. So an invalid form is one such that not every instance of it is valid, or equivalently, that there exists at least one instance of it that is invalid. So to prove an argument is invalid, it is not enough to show that it is an instance of an invalid form: you must be able to show a counterexample. Example 3 does that, so 2 is indeed invalid.

The stuff you quote at the bottom about Lehmann on invalid syllogisms is at best highly idiosyncratic and at worst just wrong. It is inspired by Aristotelian logic in which all syllogisms are categorised in a particular way and supposedly we can tell from a few rules which ones are valid and which ones are invalid. There is no good reason to be using Aristotelian logic: it was superceded by classical logic over 120 years ago.