As one gets rusty in the common basics due to the day-by-day programming I try to refresh my fundmantal knowledge from time to time. This weekend it was maths, or to be more precisely, propositional calculus (this is what programmers call boolean algebra).
There is one thing over which I stumbled when scanning literature, and I actually couldn’t remember it from my time at college. So I discussed the issue with my sparrings partner (a non-MINT) and we came up with a philosophical approach, which I’d like to share today.
The issue is: Why is the result of A ⇒ B always true in case A is false? The question rised because obvisouly if A is false no assumption could be made about B due to the directionality of the implication, hence, if A is not true, nobody knows what B is.
And indeed, that is exactly the approach to explaining it from a philosphical aspect. I actually shouldn’t dare to admit that the solution came up to my mind when I was about to clean the toilet. My wife passed by, and she was wearing a T-shirt with “Shroedinger’s Cat” on it. She’s rather addicted to philosophy. In that moment I instantly had the solution in my mind, and it was clear what the mathematicians mean with “Ex falso quodlibet“. So to not tantalise you more, here is the answer.
Let’s assume the proposition “If she had no sex she cannot be pregnant”. This implies the premise that for getting pregnant, it is necessary to have sex (ok, let’s ignore Virgin Mary and assume that in-vitro fertilisation is some way of sex). Hence, the following philosophical assumptions are:
- After she had sex, she got pregnant. Clearly this is true due to the nature of biology.
- After she had sex, she did not get pregnant. To the dissappointment of many couples, this also is true.
- After she did not have sex, she got pregnant. With the exception of Virgin Mary, mankind agreed on this to be false.
- After she did not have sex, she did not get pregnant. All educated teenagers should know that this is true.
In fact, if the had sex, we cannot assume that she is pregnant or not. It is just as likely. But we know for sure, that if the is pregnant, she definitively had sex. This is just like Shroedinger explained (we cannot say whether she is pregnant or not until she had sex, due to the nature of the implication operator).
Hence, the following truth table results from this purely philosophical insight into biology:
She had no sex | She is not pregnant || She no had sex ⇒ She is not pregnant
TRUE | TRUE || TRUE
TRUE | FALSE || FALSE
FALSE | TRUE || TRUE
FALSE | FALSE || TRUE
As we see from the truth table, if she did have sex (i. e. the premise is not fulfilled), it could be both, that she got pregnant or she did not, but in both cases, the implication result is true. Hence, for the outcome of the implication, it does not play any role whether she is pregnant or not, until she did not have sex. So A ⇒ B is the same as ¬A∨B: The implication is true in case she did have sex or in case she is not pregnant.
The cases that she had no sex but actually is pregnant is the sole one where the proposition would render false, which is proven by applying DeMorgan’s rule, resulting in A∧¬B.