8.4 Conjunctive Syllogisms - My Notes from "The Reasonable Person" by Mark Grannis

Conjunctive Syllogism

"a compound syllogism in which a negated conjunctive major premise and a categorical minor premise affirming or denying one of the conjuncts lead to a categorical conclusion." 

Example:

Not both P and Q;                 We cannot both visit the museum and visit the zoo; 

P;                                           We are visiting the museum; 

Therefore ~Q.                       Therefore we cannot visit the zoo.

Ponendo Tollens 

Remember, ponendo tollens is a mood which "affirms one element in order to deny another" and as such here for conjunctive syllogisms it is valid. 

Tollendo Ponens 

Remember, tollendo ponens is a mood which denies one element in order to affirm another, and as such here they are basically all invalid for conjunctive syllogisms. This is because in denying one conjunct in no way implies the truth of the other conjunct. 

There is one exception. 

Formally Perfect Conjunctive Syllogism

"a conjunctive syllogism in which, necessarily, only one conjunct can be false." 

Here we can have a valid form of both moods if the conjuncts are mutually exclusive. Then either affirming or denying one tells you something about the other. There are three indicators of a formally perfect conjunctive syllogism. 

First Indicator - The two conjuncts are contradictories on the square of opposition. (Here we have a formal opposition) 

Second Indicator - "the two conjuncts are mutually exclusive for some other reason know to us." (Here we have a material opposition) 

Third Indicator - It explicitly tells you in the major premise.