introduction to logic

Question

The prisoner CONTRACTED hepatitis in jail. This is a consequence of the following facts: The INCUBATION period is from 10 to 50 days yet the prisoner has been in jail for over FOUR months. [Now,] if the incubation period is from 10 to 50 days, then providing the man has been in jail over four months then he must have contracted hepatitis in jail.

Answer 1

Hi, I'm happy to help. I'm not entirely sure what's being asked, but I think you might be wondering if the argument is sound. Let's start by standardizing the argument:

P1. The incubation period for hepatitis is from 10 to 50 days.

P2. The prisoner has been in jail for over four months (i.e., more than 120 days).

P3. If the incubation period for hepatitis is from 10 to 50 days and if the prisoner has been in jail over four months, then the prisoner must have contracted hepatitis in jail.

C1, Therefore, the prisoner must have contracted hepatitis in jail.

This is a deductive argument, so it is sound if two conditions are met: (i) the form is proper (meaning the truth of the conclusion would necessarily follow from the truth of the premises), and (ii) the premises are true. On the first count, the argument does have a valid form as P3 gives us a conditional which, together with P1 and P2, guarantees the conclusion by way of modus ponens.

The harder part is determining if the premises are true. P2 (“The prisoner has been in jail for over four months”) we can treat as true ex hypothesi, which means true for the sake of the problem. Logic exercises often give us certain statements as fixed facts, and we assume them to be true even if in real life we’d want to double-check.

P1 (“The incubation period for hepatitis is from 10 to 50 days”) is not as straightforward as it might look, because it depends on what exactly is meant by incubation. If “incubation” is being used in the standard sense to mean the time between infection and the onset of symptoms, then this premise might actually be false. It is possible that hepatitis could remain asymptomatic for much longer than 50 days. If, on the other hand, “incubation” is being used in a narrower or more technical sense, then perhaps the 10–50 day window is correct, but I am not sure. Either way, this is a question that would require consultation with medical experts or credible research to confirm.

It’s hard to evaluate P3 because it turns on what P1 means by “incubation.” If the incubation claim in P1 is false, then the antecedent of P3 is false and a conditional with a false antecedent is automatically true (vacuously). That would make P3 true but still leave the argument unsound because P1 is false. If, on the other hand, the incubation claim is true under whatever its intended definition, then we should evaluate P3 the way that conditionals are evaluated: look for a counterexample where the antecedent is true (10–50 day incubation and >4 months in jail) but the consequent is false (the prisoner did not contract hepatitis in jail). Whether such a case is medically possible depends on what “incubation” means and on facts about asymptomatic or delayed presentation.

So, although this argument is valid, I do not think it is sound because I am inclined to think P1 is false. From what I can tell, "incubation" standardly means the time between initial infection and the onset of symptoms, and it seems from my research hepatitis can be asymptomatic for much longer than 50 days. If P1 is false, the argument cannot be sound regardless of the truth values of the other premises.

Hope this helps!