EXISTENTIAL GENERALIZATION

Now we come to Existential Generalization, EG. This rule, unlike EI, needs no restric­tions. If we know Georgie is a space cadet, then we know that someone is a space cadet (UD: people)—i.e., at least one person is.

Existential Generalization (EG)

Infer an existential quantification from any instance of it. From Φn, where n is any individual name, derive ΞxΦx.

Here’s a proof of the argument: Georgie is a SPACE cadet. Therefore someone is a space cadet. (UD: people):

Here’s a more interesting example:

All philosophers ALLOWED in Plato’s Academy had to know GEOMETRY. Some philosophers did not know geometry. So some philosophers were not allowed in Pla­to’s Academy. (UD: philosophers)

Summarizing, UI and EG are generally applicable, with no restrictions. But in order to do UG, we need to have introduced an arbitrary individual u, v, or w (either in a UI step,

or in a supposition for CP or RA), supposed for the sake of example. Thus if the letter u has already occurred in the symbolization of the argument or on a previous line, we should use V or w instead. Likewise when we employ EI on an existential quantification ΞxΦx, we are supposing for the sake of example that the individual i, j, or k, of which Φ is true, is indeed arbitrary. Again, if the letter i has already occurred in the symbolization of the argument or on a previous line, we should use j or k instead. The following table encapsulates this:

Letter designations:

Now with these four quantifier rules in place, we can prove a wide variety of arguments, including categorical syllogisms like the following:

All BATS can FLY.

Another example of the use of EG: the reasoning of an Israeli fighter pilot as he watched a man eject from an Egyptian fighter plane he had just shot down:

The downed pilot who flew the Egyptian FIGHTER had BLOND hair. EGYPTIAN pilots don’t have blond hair. Any non-Egyptian pilot would have to be a RUSSIAN. Hence there are Russians flying Egyptian fighters. [UD: pilots]

Symbolized:

Finally, you’ll remember from the Square of Opposition (p. 237) that an À-statement is the contradictory of an Î-statement, and so entails its negation, and also that an !-statement is the contradictory of an E-statement, and so entails its negation. Here are proofs of those entailments:

17.2.3