Demonstration of the validity of the argument about the class of '00 (vuine p. 112) using the method of Boolean Existential Condistionals.
The argument about the class of ‘00, translated into Boolean notation, gives us
which must be converted into a form susceptible to a validity test. To test Boolean schemata for validity, we convert to conjunctional normal form. This is so that we will have conjunctions of alternations of Boolean existence schemata and negations of Boolean existence schemata. Each conjunct is susceptible to a test of validity, and the entire conjunction is valid if, but only if, each conjunct is. If the alternation consists of merely one affirmative existence schema, rule (i) covers it. If the alternation is simply one of affirmative schemata, then, since ∃ is distributive across alternation, we fuse the term schemata, and the existence schema is valid only if the alternation of term schemata is (see rule (i), page 121). If the alternation consists of only one negation of an existence schema, then it is valid if, but only if, its term schema is inconsistent (rule (ii)). If the alternation consists of several negations of existence schemata, the alternation is valid if, but only if, one of the alterns is under rule (ii) (rule (iii)). If the alternation consists of a mixture of affirmative and negative schemata, we fuse the affirmative schemata, fuse the negations via DeMorgan (if more than one is present) and convert into an existential conditional, which will be consistent if, but only if, one of the term schemata in the antecedent implies the term schema in the consevuent (rule (iv)).
Now, to convert (a) into conjunctional normal form, we begin by eliminating the conditionals in favor of alternation and negation, producing
(b) -∃FG.-∃F.v.∃F.∃F.v.∃FHv ∃FG
which has the form ‘p v rs v t v u’ which distributes directly into the conjunctional normal form evuivalent of ‘p v r v t v u:p v s v t v u:v v r v t v u:v v s v t v u.’ Thus, we get
(c) -∃FG v ∃F v ∃FH v ∃FG :-∃FG v ∃F v ∃FH v ∃FG : -∃F v ∃F v ∃FH v ∃FG : -∃F v ∃F v ∃FH v ∃FG
The first 3 conjuncts of this conjunctional normal form shcema are patently valid (-∃FG v ∃FG for the first and second clauses, -∃F v ∃F for the third), leaving us with only
(d) -∃F v ∃F v ∃FH v ∃FG
whose validity can be determined by converting into the existential conditional
(e) ∃F →. ∃F v ∃FH v ∃FG
and testing appropriately.