Some dogs are mammals. Some cats are mammals. Therefore some dogs are cats.
Usually, an invalid syllogism couched in familiar terms feels wrong, even if the conclusion is true. Further brief thought then generates a variant that proves its invalidity. Using the ‘animal tree’ to test syllogisms can generally avoid the juxtaposition of invalid logic and true conclusion: simply confine each statement to adjacent levels in the animal tree, rather than creating statements like ‘some dogs are animals’ that skip a level.
Hypothetical Syllogisms
Like categorical syllogisms, hypothetical syllogisms consist of two premises and a conclusion. Unlike categorical syllogisms, one or both of the premises in a hypothetical syllogism is a conditional statement: ‘if A, then B’.
We can express a conditional, or if/then, statement symbolically as A⇒B. The statement A⇒B can be read as ‘A implies B’ or as ‘if A, then B’; the two are logically equivalent. Both statements state that A is a necessary and sufficient condition for B.
If both premises in a hypothetical syllogism are if/then statements, then only three forms of syllogism are possible:
| Valid | Invalid | Invalid |
|---|---|---|
| S⇒M. | S⇒M. | M⇒S. |
| M⇒P. | P⇒M. | M⇒P. |
| ∴ S⇒P. | ∴ S⇒P. | ∴ S⇒P. |
Another type of hypothetical syllogism has one if/then statement, a statement that one of the two conditions is present or absent, and a conclusion about whether the other condition is present or absent. Symbolically, we can indicate presence (or truth) by S or P, and absence by -S or -P. If only one premise is an if/then statement, two valid and two invalid forms of syllogism are possible:
| Valid | Invalid | Invalid | Valid |
|---|---|---|---|
| S⇒P | S⇒P | S⇒P | S⇒P |
| S | -S | P | -P |
| ∴P | ∴ -P | ∴S | ∴ -S |
As with categorical syllogisms, hypothetical syllogisms are readily testable through substitution. The substitution that I use treats if/then as a mnemonic for ‘if the hen’:
A: if the hen lays an egg; B: we cook omelettes; C: we eat omelettes.
This substitution readily distinguishes invalid from valid hypothetical syllogisms:
Valid: A⇒B. If the hen lays an egg, then we cook omelettes.
B⇒C. If we cook omelettes, then we eat omelettes.
∴ A⇒C. Therefore, if the hen lays an egg, we eat omelettes.
