8.2 Conditional Syllogisms - My Notes from "The Reasonable Person" by Mark Grannis

There are two types of conditional syllogisms: the "pure conditional syllogism" and the "mixed conditional syllogism". 

The Mixed Conditional Syllogism 

"A compound syllogism in which a conditional major premise and a categorical minor premise lead to a categorical conclusion." 

Major Premise - Conditional proposition

Minor Premise - Categorical proposition 

Conclusion - Categorical proposition 

Example

If P, then Q;         If Felix purrs, then Felix is a cat;

P;                         Felix purrs; 

Therefore: Q        Felix is a cat.

*Remember "terms" here aren't just words, but categorical propositions. 

*Also, P is considered the "antecedent" and Q the "consequent". 

Modus Ponens 

The above example "If P, then Q; P; Therefore Q" is a classic and valid syllogism known as a "modus ponens". 

It is also known as "arguing forward" or the "constructive mood". 

Modus Tollens 

This is the other valid mood of the mixed conditional syllogism. 

If P, then Q;         If Felix purrs, then Felix is a cat;

~ Q;                     Felix is not a cat; 

Therefore: ~P      Felix does not purr.

It is also known as "arguing backwards" or the "destructive mood".

These are the only two valid moods for a mixed conditional syllogism. There are two more that are invalid, the "fallacy of affirming the consequent" and the "fallacy of denying the antecedent." 

Fallacy of Affirming the Consequent 

If P, then Q;         If there is lighting, then the beach is closed; 

Q;                        The beach is closed; 

Therefore, P.       There is lighting.

Here, affirming that consequent (that the beach is closed) does not prove the antecedent (that there is lighting). There could be other reasons why the beach is closed. We cannot therefore affirm the consequent to prove the antecedent (that's going from universal to particular), but we can do the opposite, affirm the antecedent to prove the consequent (going from particular to universal). 

Fallacy of Denying the Antecedent 

If P, then Q;         If there is lightning, then the beach is closed;

~P;                      There is no lightning; 

Therefore ~Q.     The beach is not closed. 

Again, the beach could be closed for other reasons.

Pure Conditional Syllogism 

"A compound syllogism in which both premises and the conclusion are conditional propositions." 

In other words, both premises contain conditional propositions. 

For Example

If Q, then R;                        If she is made of wood, then she is a witch; 

If P, then Q;                         If she weighs the same as a duck, then she is made of wood; 

Therefore, If P, then R.        Therefore, If she weighs the same as a duck, then she is a witch.

*Notice how this mirrors the major term, minor term, and middle term in a categorical proposition. 

*Just as the mixed conditional syllogism, there are two valid and invalid moods of the pure conditional syllogism.