Riddle on implication

Question

There was this question I don't understand in quizz.

Prosecutor says: If he is guilty, he must have had accomplice.

The question was: which of the following proves that he was guilty.

And the correct answer was that:"the prosecutor made false claim". Other options were:

• prosecutor was telling truth
• if he is guilty he had no accomplice

Clearly that fact that prosecutor was wrong can't imply that he was guilty otherwise I could go say similar wrong statement, and none would arrest anyone right?

IMHO this is related to implication because if prosecutor was lying implication above is FALSE hence, the premise he is guilty must be true, and the other about accomplice false, but still this doesn't prove person was guilty right in practice? what am I missing about implication?

Answer 1

Perhaps this is what the quiz means. Let $G$ mean "The defendant is guilty" and $A$ mean "The defendant had an accomplice" We can symbolize "If the defendant is guilty, he must have had an accomplice" as

$$G \implies A$$

Now, by the rule of conditional exchange, the above sentence is equivalent to

$$\neg G \lor A$$

What happens if we negate this? Well, by DeMorgan's Law, we get

$$G \land \neg A$$

Therefore, if $G \implies A$ is false, $G \land \neg A$ is true

Thus, we know that the defendant is guilty and did not have an accomplice because the prosecutor's statement is false. In general, the only possible way that a conditional is false is if the antecedent (in this case $G$) is true and the consequent (in this case $A$) is false.

Note: $\lor =$ "or", $\land =$ "and", $\neg =$ not

Answer 2

I have a different take on this. IMHO mathematical logic isn't that far off from regular speech in this instance.

In what way can $A \implies B$ be false? It can only be false iff the reality is a counter-example, i.e. $A$ is true but $B$ is false. Therefore "$A \implies B$ is false" does indeed imply $A$ is true.

However, what's happening here:

• The prosecutor can say $A \implies B$.
• If the prosecutor lied then $A$ is true (the defendant is guilty).
• But the prosecutor has not proven (beyond a reasonable doubt) that he is lying!
• Therefore the jury should not convict just based on the prosecutor's statement -- after all it might be a true claim :) in which case it does not establish guilt or innocence at all.

Answer 3

If the prosecutor was lying then the statement: "If he is guilty then he had accomplices" is false. The ONLY way $P \implies Q$ can be false is if $P$ is true and $Q$ is false. So if the prosecutor is false, He is guilty and he didn't have accomplices.

but still this doesn't prove person was guilty right in practice?

well, kind of. The thing is we have to prove the prosecutor was lying. In practice we don't have robotic lying prosecutors who be some law of magic are only capable of uttering statements that logically false.

How would we know the prosecutor is lying. If we hooked him to a lie detector test and he failed well, is that convincing that he failed because he is lying? if a polygraph detects lies by emotional factors are the emotional factors based on systematic logic?

Reading the comments you seem to think that for any $A$ we can find a $B$ where $A \implies B$ is false and that this is paradoxical as that "forces" $A$ to be true. But if $A$ is false we can never find a false $A\implies B$.

Or maybe you seem to think we can turn a light switch and make $A\implies B$ be false. And that doing so will make $A$ be true. .... Well, what of it? If these magical truth light switches exist and we can mat $A\implies B$ be false, the very same light switch will also make $A$ be true. What's strange about that?

And if these light switches don't exist the only way we can determine if $A\implies B$ is true or false is checking if $A$ and $B$ are true and false.

If $A$ is true and $B$ is true: Then $A\implies B$ is true.

If $A$ is true and $B$ if false; Then $A\implies B$ is false.

If $A$ is false and $B$ is true; then $A\implies B$ is true.

If $A$ is false and $B$ is true; then $A\implies B$ is true.

.....

Maybe you are worried about cause and effect? It seeems wierd that "prosecutor is lying" can cause real world events to be true or false. But that's backwards. It's the real world events that cause the prosecutors statement to be true of false. All we are doing is looking at the results and determining what happens.

Imagine this.... suppose we lived in a universe where the ONLY way I could have rocks in my driveway is if Ragnorak occurred, a black cat ate the Thunder Good Thor and spat out his bones on my driveway and they turned to rocks. It's a law of physics in this universe that the is the ONLY way I can have rocks in my driveway.

So one night after a sound sleep on a quiet night I go out to fetch the paper and I find rocks on my driveway... Hmm, I guess that means Ragnorak occurred, a black cat ate the Thunder Good Thor and spat out his bones on my driveway.

That's strange, I didn't hear anything but that's the ONLY explanation so it must be true.

Now imagine a skeptic walks along and says: That doesn't make sense, you can make Ragnorak occurred a black cat eat the Thunder Good Thor and spit out his bones on your driveway, simply by throwing rocks in your driveway.

But the thing is .... I can't simply throw rocks in my driveway. Rocks can not appear in my drivewar. Rocks in my driveway didn't cause Ragnorak to occur, and a black cat to eat the Thunder Good Thor and to spit out his bones on my driveway. Ragnorak occurring and a black cat eating the Thunder Good Thor and spittin out his bones on my driveway caused the rocks.