Penevalid arguments

Let’s now have a look at such cases, where the validity of an argument depends upon an existential import that is implicit in the context. These arguments are sometimes called penevalid (from the Latin for “almost valid”), although we shall be treating them as valid enthymemes.

This form was known as Darapti in medieval logic. To Lewis Carroll its validity was obvious, and he used this as a main argument against rival interpretations of existential import. On our interpretation the form Darapti is invalid, but certain arguments appar­ently having that form are valid when an obvious implicit premise is made explicit. A specific example:

All VERMONT-made beers taste GREAT.

All Vermont-made beers are MICRO-BREWED.

Therefore at least some micro-brewed beers taste great.

The obvious validity of this argument depends on the implicit premise that “there are Vermont-made beers.” Thus when we diagram the argument, we see that when this premise is added, we get an x in the V-region. But the only part of V that is not empty is VGM, so we must insert the x there. That is, the implicit premise also guarantees that there are great tasting micro-brewed beers (and the world is the better for it!):

Now we symbolize the argument and prove its validity as follows:

Perhaps you are not convinced, and think that the traditional approach of interpreting À-statements as having existential import might be the correct one, so that Darapti is always valid. If so, consider the following argument:

All ALIEN spacecraft can CLOAK themselves to avoid detection.

All alien spacecraft are PILOTED by aliens.

Therefore there are spacecraft piloted by aliens that can cloak themselves to avoid detection.

This also has the form Darapti: All A are C. All A are P. Therefore some P are C; with UD: spacecraft, A := is alien, C := can cloak itself to avoid detection, P := is piloted by aliens. But this argument is not valid! For it is possible to accept the truth of both prem­ises, and yet deny the truth of the conclusion: one only needs to deny that there are any alien spacecraft! This should help to convince you that Darapti cannot be regarded as gen­erally valid; and that valid arguments having this form are better regarded as enthymemes with an implicit existential premise.

18.2.3