Validity

But what exactly does it mean to say the argument form is valid?

An argument form is valid if it is impossible to replace its variables with words that make the premises all true while it makes the conclusion false. (Note: each variable occurs twice, so to replace a variable with a word or phrase is to replace it with the same word or phrase in both of its occurrences.)

On the other hand, if the variables can be replaced with content that makes true premises and a false conclusion, then the argument form is invalid. This is the difference between valid and invalid forms. Accordingly, valid argument forms can accommodate truth value combinations of premises and conclusion that correspond to lines 1, 3, and 4 below but they cannot accommodate the combination given in line 2.

That is, line 2 is the telltale line because valid forms can never have this combination while invalid forms always can have it.

Put another way, a valid form is "truth-preserving" in that if you put nothing but truth into the premises, the form necessarily preserves that truth for the conclusion

Of course, we may not know whether a premise is true. For example, if we let A=(pet owners in New York City over 100 years old), B=(millionaires who owe back income taxes), and C=(citizens who will never be arrested for tax fraud) we can get:

Now the well-known syllogistic form of that argument, viz.,

is valid, but probably no one knows whether these sentences are true. But since it is valid we do know that IF the first two sentences (the premises) are true THEN the last one (the conclusion) is true also.

Perhaps most people can see intuitively that this form is valid; that is, most anyone can just see that "if All A are B and all B are C, then it has to be that all A are C."

However, a majority of the remaining syllogisms are not valid, and ordinary intuition is not so clear on them all. For example, it may seem on first glance that

is valid, and examples of true premises with a true conclusion easily come to mind and may seem to confirm it, such as:

However, as is noted above, invalid forms as well as valid ones can accommodate true premises and a true conclusion so this example doesn't prove the case one way or the other. However, since valid forms do not allow any instances of true premises with a false conclusion just one example of such is all it takes to show the form to be invalid. One such example is:

Of course, there are innumerable other examples as well. So this is not a "truth-preserving" form; it is invalid, although it may seem otherwise at first. Likewise, it may be thought on first glance that the following forms are valid

when, in fact, they are not.

On the other hand, people often find many of the forms puzzling on first glance and can't easily decide what to think about them. Some such instances may be:

And surely no one can tell on first glance whether long and complicated, nonstandard form sorites are valid.

In every age, from Aristotle to the present, teaching logic has meant teaching a way to distinguish valid from invalid argument forms. Alternative ways exist, but I propose the way of the schematics to be most effective technique ever!