Hypothetical Syllogism/Examples/Ancient Chinese Proverb
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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
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...: Ponderable $1.1.2$