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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)?

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  • Lehmann makes an idiosyncratic distinction between invalid and "neither valid nor invalid" syllogisms that almost nobody else uses, see Lehmann on the rules of the invalid syllogisms by Turek. What she calls "neither valid nor invalid" is conventionally called "invalid but satisfiable". You can ignore it. Your boldface passage is the conventional analysis. – Conifold Dec 8 '20 at 11:38
  • You can't mix Aristotelian logic with modern language. That was mistake # 1. The formal language of Aristotelian is extremely limited to prevent emotion or psychological content. There are strict rules how to make syllogisms instead of the willy nilly just put any sentences up any kind of way. Aristotelian logic is a method that focuses on SOUNDNESS where Mathematical logic focuses on validity. Soundness means validity must already be present while the premise are true in the real world. This means Aristotelian logic refers to reality more than Mathematical logic. Validly is a lower concept. – Logikal Dec 8 '20 at 19:35
  • People who say that logic is about validity have issues. I can clearly give examples of valid arguments that are practical & apply to reality. At the same time I can give examples of valid arguments that have no real world application or basically false in reality. For those people all about validity how does one know when the reasoning you use apply to reality and when it doesn't? At best your reasoning may be real world applicable 50 percent of the time. Even then you need more knowledge than your premises provide. You would need more than your math logic to resolve real world problems. – Logikal Dec 8 '20 at 19:41
  • There are rules & concepts to Aristotelian logic that people these days seem foreign to about deductive reasoning. One is that propositions must be formed a specific way --not like burger king ,"have it your way." 2nd no false premises are allowed. This ensures soundness is the priority. Soundness is the GOAL because validity is BUILT IN . The same way when you buy a new car it COMES WITH tires, & seats. Cars are not ABOUT TIRES and SEATS. If I get WHAT COMES WITH the purchase why is logic about validity. We KNOW all valid arguments are not true & also not always real world applicable 100 %. – Logikal Dec 8 '20 at 19:48
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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.

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