(related to Lemma: Hypothetical Syllogism)
We want to prove that the hypothetical syllogism is a valid logical argument. * The hypothetical syllogism can be formulated in propositional logic as $((p\Rightarrow q)\wedge (q\Rightarrow r))\Rightarrow (p\Rightarrow r).$ * Using the definition of implication "$\Rightarrow$", we can construct the following truth table:
$[[p]]_I$ |
$[[q]]_I$|
$[[r]]_I$|
$[[p\Rightarrow q]]_I$|
$[[q\Rightarrow r]]_I$|
$[[p\Rightarrow r]]_I$:------------- |:------------- |:------------- |:------------- |:------------- |:-------------
$0$|
$0$|
$0$|
$1$|
$1$|
$1$$0$
|
$0$|
$1$|
$1$|
$1$|
$1$$0$
|
$1$|
$0$|
$1$|
$0$|
$1$$0$
|
$1$|
$1$|
$1$|
$1$|
$1$$1$
|
$0$|
$0$|
$0$|
$1$|
$0$$1$
|
$0$|
$1$|
$0$|
$1$|
$1$$1$
|
$1$|
$0$|
$1$|
$0$|
$0$$1$
|
$1$|
$1$|
$1$|
$1$|
$1$`
This result can also be achieved by calculating the truth table of the whole expression of the hypothetical syllogism, which confirmes that it is a tautology. Please click evaluate to verify this: