Hypothetical Syllogism/Examples/Ancient Chinese Proverb

From ProofWiki
Jump to navigation Jump to search

Example of Use of Hypothetical Syllogism

If there is light in the soul,
then there will be beauty in the person.
If there is beauty in the person,
then there will be harmony in the house.
If there is harmony in the house,
then there will be order in the nation.
If there is order in the nation,
then there will be peace in the world.


The conclusion is:

If there is light in the soul, then there will be peace in the world.


Proof

Let:

$p$ denote the statement:
There is light in the soul.
$q$ denote the statement:
There is beauty in the person.
$r$ denote the statement:
There is harmony in the house.
$s$ denote the statement:
There is order in the nation.
$t$ denote the statement:
There is peace in the world.


The proverb can be expressed in symbols as follows:

\(\text {(1)}: \quad\) \(\ds p\) \(\implies\) \(\ds q\)
\(\text {(2)}: \quad\) \(\ds q\) \(\implies\) \(\ds r\)
\(\text {(3)}: \quad\) \(\ds r\) \(\implies\) \(\ds s\)
\(\text {(4)}: \quad\) \(\ds s\) \(\implies\) \(\ds t\)


Hence we have:

\(\ds p\) \(\implies\) \(\ds q\) from $(1)$
\(\, \ds \land \, \) \(\ds q\) \(\implies\) \(\ds r\) from $(2)$
\(\text {(5)}: \quad\) \(\ds \vdash \ \ \) \(\ds p\) \(\implies\) \(\ds r\) Hypothetical Syllogism


\(\ds p\) \(\implies\) \(\ds r\) from $(5)$
\(\, \ds \land \, \) \(\ds r\) \(\implies\) \(\ds s\) from $(3)$
\(\text {(6)}: \quad\) \(\ds \vdash \ \ \) \(\ds p\) \(\implies\) \(\ds s\) Hypothetical Syllogism


\(\ds p\) \(\implies\) \(\ds s\) from $(6)$
\(\, \ds \land \, \) \(\ds s\) \(\implies\) \(\ds t\) from $(4)$
\(\ds \vdash \ \ \) \(\ds p\) \(\implies\) \(\ds t\) Hypothetical Syllogism

$\blacksquare$


Sources