8.3 Disjunctive Syllogisms - My Notes on "The Reasonable Person" By Mark Grannis

Disjunctive Proposition

"A compound proposition in which the two categorical propositions are joined in an 'either-or' relationship." 

Disjunctive Syllogism

"A compound syllogism in which a disjunctive major premise and a categorial minor premise affirming or denying one of the alternates lead to a categorical conclusion." 

Example:

Either P or Q;                 Either Smith is falsely accused or he broke the law;

~P;                                  Smith is not falsely accused; 

Therefore Q.                   Smith broke the law. 

This works because we are denying one of the alternates from the major premise, and therefore leaving the other. We could not make this work if we affirmed one of them because there could be more than two possibilities. This leads to:

Inclusive Disjunction - "A disjunctive proposition in which more than one alternant can be true." 

This is where the two alternates are NOT mutually exclusive, and could both be true. 

Exclusive Disjunction - "A disjunctive proposition in which, necessarily, only one alternant can be true." 

This is where of the two alternates ONLY one can be true. 

RULE: "...we presume all disjunctions to be inclusive unless we can know for certain that they are exclusive." 

How to Tell Exclusive Disjunction

1 - The alternants are contradictories on the square of opposition. 

2 - "it might be that the two alternants are mutually exclusive for some other reason known to us." 

3 - The disjunctive proposition includes "but not both" clearly setting the two in opposition. 

Tollendo Ponens

These are two valid moods of the disjunctive syllogism because as mentioned above by denying one of the alternants we can affirm the other without making a universal claim on every possibility. 

Either P or Q;                 Either P or Q; 

~P;                                 ~Q; 

Therefore Q.                  Therefore Q.

Ponendo Tollens 

These two moods affirm one of the alternants and deny the other in the conclusion but this is only valid if the alternants are exclusive and the only two possibilities. 

Either P or Q;                 Either P or Q; 

P;                                    Q; 

Therefore ~Q.                 Therefore ~P.