§4 Common Logical Patterns

Recognizing standard patterns of reasoning allows us to identify valid and invalid logic instantly. These forms act as templates; once you identify the pattern, you know the logical relationship between the premises and the conclusion without having to re-analyze the content every time.

4.1 Valid Argument Forms

In these patterns, if the premises are true, the conclusion is guaranteed to be true.

Modus Ponens (Affirming the Antecedent)

This is the most straightforward logical form. It starts with a conditional (“If-Then”) statement and confirms that the first part is true.

• Form: If $P$, then $Q$. $P$. Therefore, $Q$.

• Example: If it is raining, the ground is wet. It is raining. Therefore, the ground is wet.

Modus Tollens (Denying the Consequent)

This form works in reverse. It starts with a conditional and confirms that the result (the second part) did not happen.

• Form: If $P$, then $Q$. Not $Q$. Therefore, not $P$.

• Example: If it is raining, the ground is wet. The ground is not wet. Therefore, it is not raining.

Disjunctive Syllogism

This form deals with “Either-Or” statements. If you are presented with two options and one is eliminated, the other must be true.

• Form: Either $P$ or $Q$. Not $P$. Therefore, $Q$.

• Example: Either we go to the movies or the park. We aren’t going to the movies. Therefore, we are going to the park.

Hypothetical Syllogism

This form chains two conditional statements together to create a new one.

• Form: If $P$, then $Q$. If $Q$, then $R$. Therefore, if $P$, then $R$.

• Example: If I study, I will pass. If I pass, I will graduate. Therefore, if I study, I will graduate.

4.2 Formal Fallacies (Invalid Patterns)

These patterns look similar to valid ones but contain a logical “break” that makes them invalid. In these cases, even if the premises are true, the conclusion could still be false.

Affirming the Consequent

This mistake assumes that because the result happened, the specific cause you identified must have been the reason.

• Invalid Form: If $P$, then $Q$. $Q$. Therefore, $P$.

• Example: “If it rains, the ground is wet. The ground is wet, so it must be raining.”

• The Flaw: This ignores other possible causes for $Q$. The ground could be wet because a sprinkler was on or someone spilled a bucket of water.

Denying the Antecedent

This mistake assumes that if the cause is absent, the result cannot happen.

• Invalid Form: If $P$, then $Q$. Not $P$. Therefore, not $Q$.

• Example: “If it rains, the ground is wet. It’s not raining, so the ground isn’t wet.”

• The Flaw: Again, this ignores alternative causes. Just because it isn’t raining doesn’t mean the ground isn’t wet from something else.