P( whole argument is wrong ) = P( first subargument is wrong ) * P( second subargument is wrong | first subargument is wrong )
P( whole argument is wrong ) is not P( first subargument is wrong AND second subargument is wrong), so the above conditional probability decomposition is incorrect.
Let’s say I am arguing that the sky is blue because the aliens have landed and that’s the color they painted it. The probability of the aliens having landed is very low, and the probability of the aliens painting the sky blue given that they’ve landed is also fairly low, but the probability of the sky being blue is quite high.
In other words, incorrect arguments in favor of a proposition don’t make the proposition less likely.
I don’t believe he’s speaking of two subarguments which together imply the main argument, but two subarguments each of which independently implies the main argument. Thus, they would both have to be false
The logical operator between the two subarguments was ambiguous—I assumed the total argument would be something like a lemma and a theorem that depends on the lemma, not a disjunction of propositions.
If the arguments are chained together, then this is true, but the original poster was talking about independent lines of reasoning leading to the same conclusion. For arguments which are truly independent, then his formulation is correct.
P( whole argument is wrong ) is not P( first subargument is wrong AND second subargument is wrong), so the above conditional probability decomposition is incorrect.
Could you expand? I don’t follow you.
Let’s say I am arguing that the sky is blue because the aliens have landed and that’s the color they painted it. The probability of the aliens having landed is very low, and the probability of the aliens painting the sky blue given that they’ve landed is also fairly low, but the probability of the sky being blue is quite high.
In other words, incorrect arguments in favor of a proposition don’t make the proposition less likely.
For the argument to be wrong, only one of its subarguments has to be wrong. So the correct equation is
P(whole argument is wrong) = 1 - P(first subargument is right) * P(second subargument is right | first argument is right)
I don’t believe he’s speaking of two subarguments which together imply the main argument, but two subarguments each of which independently implies the main argument. Thus, they would both have to be false
That’s right. Unless one has negative relevance. Happens all to often.
The logical operator between the two subarguments was ambiguous—I assumed the total argument would be something like a lemma and a theorem that depends on the lemma, not a disjunction of propositions.
If the arguments are chained together, then this is true, but the original poster was talking about independent lines of reasoning leading to the same conclusion. For arguments which are truly independent, then his formulation is correct.
Alrighty, I gotcha.
Not quite. You can combine the arguments together in all sorts of ways.