College Mathematics 10+5 pts
"Consider the following argument:
a. George and Mary are not both innocent.
b. If George is not lying, Mary must be innocent.
c. Therefore, if George is innocent, then he is lying.
Let ???? be the proposition "George is innocent", m be the proposition "Mary is innocent", and let ???? be the proposition "George is lying"."
1. Write a propositional formula F involving variables g, m, l such that the above argument is valid if and only if F is valid.
2. Is the above argument valid? If so, prove its validity by proving the validity of F. If not, give an interpretation under which F evaluates to false.
Answer
1. [tex]F = ((g\cap \sim m)\cup(\sim g\cap m)\cap(\sim l\to m))\to(g\to l) [/tex]
2. It is valid. See explanation for proof.
Step-by-step explanation
1. Statement a is an exclusive OR of [tex]g[/tex] and [tex]m[/tex]. This is because only one of them is innocent while the other is not. That is [tex]g[/tex] is true and [tex]m[/tex] is false or [tex]g[/tex] is false and [tex]m[/tex] is true.
Statement b is an implication that [tex]\sim l \to m[/tex].
Statement c is another implication that [tex]g\to l[/tex].
The presence of the word "therefore" in statement c means it is an implication from statement a AND statement b. So we have
[tex]F = ((g\cap \sim m)\cup(\sim g\cap m)\cap(\sim l\to m))\to(g\to l) [/tex]
2.
The validity will be proved using a truth table. [tex]F[/tex] is valid if the last column contains only true values i.e. truth values of T.
From the table (in the attached image), [tex]c[/tex] represents the statement a, [tex]d[/tex] represents statement b and [tex]f[/tex] represents statement c. The last column, which represents [tex]F[/tex], is valid as all its entries are T.
