Section 1.8 of chapter 1 of the COMP2310 courseware discusses a concept called Applications. This describes using what was learned about formal proofs in propositional logic to test the validity of certain real-world arguments.

In this blog, I will go over an example to see how we can use propositional logic to prove if certain arguments make sense. Each proof follows a given structure:

1. Denote each proposition with a variable (usually upper-case but doesn’t have to be), such as $A$, $B$, $C$, etc.
2. Translate each statement into a wff.
3. Prove the conclusion with the given premises.

Some general tips when solving:

1. Try to only denote the positive expressions as variables (i.e. if the statement reads “John does not walk”, denote J as “John walks” and convert the expression to $\sim J$).
2. Once the statements are properly translated into wffs, do your best to try and understand conceptually why the logic works out.

Example

There is a poker game going on. If Nick wins, then Matt will throw a party. If Spencer wins, then Tom will throw a party. It is the case that either Nick or Spencer will win. If Nick wins, then Tom will not throw a party, and if Spencer wins, then Matt will not throw a party. Therefore, Matt will throw a party if and only if Tom does not throw a party.

Solution

Let N denote “Nick wins”,
$\quad$ S denote “Spencer wins”,
$\quad$ M denote “Matt will throw a party”,
$\quad$ T denote “Tom will throw a party”.

The sentence “There is a poker game going on” does not need to be translated at all, as there is nothing useful about it.
The sentence “If Nick wins, then Matt will throw a party” can be translated to $N\Rightarrow M$
The sentence “If Spencer wins, then Tom will throw a party” can be translated to $S\Rightarrow T$
The sentence “It is the case that either Nick or Spencer will win” can be translated to $(N\lor S)\land\sim(N\land S)$
The sentence “If Nick wins, then Tom will not throw a party, and if Spencer wins, then matt will not throw a party ” can be translated to $(N\Rightarrow\sim T)\land (S\Rightarrow\sim M)$
The conclusion “Matt will throw a party if and only if Tom does not throw a party” can be translated to $M\Leftrightarrow\sim T$.

We thus have:
$\qquad P_1:\quad N\Rightarrow M$
$\qquad P_2:\quad S\Rightarrow T$
$\qquad P_3:\quad (N\lor S)\land\sim(N\land S)$
$\qquad P_4:\quad (N\Rightarrow\sim T)\land (S\Rightarrow\sim M)$
$\qquad C:\quad M\Leftrightarrow\sim T$

and we are to prove that $P_1,P_2,P_3,P_4\vdash C$.

1. $N\Rightarrow M\quad$ from $\Gamma$
2. $S\Rightarrow T\quad$ from $\Gamma$
3. $(N\lor S)\land\sim(N\land S)\quad$ from $\Gamma$
4. $(N\Rightarrow\sim T)\land(S\Rightarrow\sim M)\quad$ from $\Gamma$
5. $(S\Rightarrow\sim M)\land(N\Rightarrow\sim T)\quad$ 4, E9
6. $(S\Rightarrow\sim M)\quad$ 5, I2
7. $(\sim\sim M\Rightarrow\sim S)\quad$ 6, E19
8. $(M\Rightarrow\sim S)\quad$ 7, E15
9. $N\lor S\quad$ 3, I2
10. $S\lor N\quad$ 9, E10
11. $\sim\sim S\lor N\quad$ 10, E15
12. $\sim S\Rightarrow N\quad$ 11, E18
13. $M\Rightarrow N\quad$ 8, 12, I5
14. $N\Rightarrow\sim T\quad$ 4, I2
15. $M\Rightarrow\sim T\quad$ 13, 14, I5
16. $\sim T\Rightarrow\sim S\quad$ 2, E19
17. $\sim T\Rightarrow N\quad$ 16, 12, I5
18. $\sim T\Rightarrow M\quad$ 17, 1, I5
19. $(M\Rightarrow\sim T)\land(\sim T\Rightarrow M)\quad$ 15, 18, I6
20. $M\Leftrightarrow\sim T\quad$ 19, E20

Hence, $P_1,P_2,P_3,P_4\vdash C\qquad\blacksquare$