If the consequents are mutually exhaustive, then is the antecedent false

logicpropositional-calculussolution-verification

This is a follow-up inquiry to If the antecedents are mutually exclusive, then is the consequent true?.

Suppose I know that the implications
$$P \implies C_1\\
P \implies C_2\\
P \implies C_3$$ are true for some premise $P$ and some conditions $C_1, C_2$, and $C_3.$

If $C_1, C_2$, and $C_3$ are mutually exhaustive conditions, does it follow that $P$ is false?

MY ATTEMPT

I noticed that the three implications are logically equivalent to
$$\left(P \implies C_1\right) \iff \left(\lnot P \lor C_1\right)\\
\left(P \implies C_2\right) \iff \left(\lnot P \lor C_2\right)\\
\left(P \implies C_3\right) \iff \left(\lnot P \lor C_3\right)$$
by Material Implication.

Hence,
$$\left(\lnot P \lor C_1\right) \land \left(\lnot P \lor C_2\right) \land \left(\lnot P \lor C_3\right)$$
must be true.

Using Distribution of Disjunction over Conjunction, we obtain
$$\Big(\left(\lnot P \lor C_1\right) \land \left(\lnot P \lor C_2\right) \land \left(\lnot P \lor C_3\right)\Big) \iff \Big(\lnot P \lor \left(C_1 \land C_2 \land C_3\right)\Big).$$

But the conditions $C_1, C_2$, and $C_3$ are mutually exhaustive. This means that
$$C_1 \land C_2 \land C_3 \tag{*}$$
must be false.

We finally get that $\lnot P$ must be true, or in other words, $P$ must be false.

Is the proof argument in the section marked with a $(*)$ sound? I have some doubts since I am not sure if I should have used Distribution of Disjunction over Disjunction instead.

Best Answer

$$P \implies C_1\\ P \implies C_2\\ P \implies C_3$$ If $C_1, C_2$, and $C_3$ are mutually exhaustive conditions, does it follow that $P$ is false?

No: put the tautology $(X\lor\lnot X)$ as each of $P,C_1,C_2,C_3.$

But the conditions $C_1, C_2$, and $C_3$ are mutually exhaustive. This means that $$C_1 \land C_2 \land C_3 \tag{*}$$ must be false.

No, the three conditions being (collectively/mutually) exhaustive just means that at least one of them is satisfied; this is corroborated by goblinGONE in your linked page.

On the other hand, sentence (*) being false means that at least one the three conditions is not satisfied, that is, their truth sets are collectionwise-disjoint. (Note that, by Definition #2 here, this does not mean that the three conditions are mutually exclusive either.)

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