Help understanding a logical reasoning question

Question

I'm having trouble with a logic question and was hoping someone could help me out. The statement goes like this:

“If all the accused were innocent, some would be acquitted. No one was acquitted therefore all must be guilty.”

I have identified the following propositional variables:

Based on the statement, I think we can infer that if all the accused were innocent (A is true), then some would be acquitted (B is true). Additionally, the statement tells us that no one was acquitted (C is true), which means that B must be false. Finally, the statement concludes that all must be guilty (D is true).

I've tried creating a truth table to confirm my reasoning, but I'm not sure how to incorporate the C variable. Here's what I've got so far:

Can anyone help me figure out how to incorporate the C variable in my truth table? And am I on the right track with my reasoning?

Thanks in advance for any help!

Answer 1

“If all the accused were innocent, some would be acquitted. No one was acquitted therefore all must be guilty.”

For consistency with $C,$ you should change the 'are' in $B$ to 'was'. Or just rewrite $B$ as $\lnot C.$ The argument, clearer, is:

• If all the accused were innocent, some would be acquitted; no one was acquitted. Therefore, all must be guilty.

Using propositional logic, it means \begin{align}{}&A→B\\\text{and}\quad{}& C;\\\therefore\quad\quad {}&D.\tag1\end{align} If we allow quantifiers, then (here $x$ stands for accused persons): \begin{align}{}&\forall x\;I(x)\to \exists x\;Q(x)\\\text{and}\quad{}& \lnot\exists x\;Q(x);\\ \therefore\quad\quad{}&\forall x\;\lnot I(x).\tag2\end{align}

You have correctly identified that $$C\equiv\lnot B,$$ but comparing columns 3 and 5 reveals that you are mistakenly believing that $$C\equiv D$$ (being acquited and being innocent are not synonyms!). Also, column 5 indicates that you are misreading column 6 as $$(A\to B)\land(\lnot B\to D)$$ rather than $$((A\to B)\land\lnot B))\to D.$$ Correcting your truth table will show that argument $(1)$ is not a tautology (its column is not all T).

Truth tables aren't able to handle the quantifiers in argument $(2),$ which is in fact invalid, since Gary but not Ivan being guilty makes its premises true but its conclusion false.