Despite the form, statement (b) is not actually a logical conjunction. It is a statement about the collective of both parents.
This becomes clearer if we strengthen the statement slightly to “Alvin’s mother and father are responsible for all of his genome”. It’s much more clear now that it is not a logical conjunction. If it were, it would mean “Alvin’s mother is responsible for all of his genome and Alvin’s father is responsible for all of his genome”, both of which are false.
This is probably meant as a reply to this comment.
In Eliezer Yudkowsky’s essay Burdensome Details, I get stuck by the first paragraph on the Conjunction Fallacy:
The conjunction fallacy is when humans assign a higher probability to a proposition of the form “A and B” than to one of the propositions “A” or “B” in isolation, even though it is a theorem that conjunctions are never likelier than their conjuncts.
According to Wikipedia’s entry on Conjunction Fallacy:
″… the probability of two events occurring together (in “conjunction”) is always less than or equal to the probability of either one occurring alone...”
I can see now how Yudkowskys statement the ”...conjunctions are never likelier than their conjuncts.” can also be stated as [the probability of two events occuring together] is always less than or equal to the probability of either one occurring alone...”
What I don’t get is how, in a case where the conjunction is truer than either conjunct alone, this can hold true. Maybe I am approaching this wrong, but here is an example I’m having trouble with:
Alvin is a human being, average in every way. He was conceived and birthed naturally by his parents. Which of the following are true?
a) Alvin’s mother is responsible for his genome
b) Alvin’s mother and father are responsible for his genome.
Since each of his parents contribute 23 chromosomes to his genome, for a total of 46, isn’t this a case where the conjunction is 100% true while the lone statement of a) is only 50% true?
I’ve never taken a formal logic class and stopped at pre-calc, so I know I’m missing something somewhere. Is it that the conjuncts are 2 similar to be considered as separate statements? I suppose common knowledge would suggest that by stating “Alvins mother contributed half his genes” automatically infers that his father contributed the other half, so that the statement ‘b’ is really just restating ‘a’?
And after reading up a little I also found this article from 2015 stating you actually get more genes from your mother when you consider the sets of both nuclear dna and mitochondrial dna. Although the Wikipedia entry for Human Genome states that these 2 sets are “usually treated separately” (in discussions of who is responsible for your dna, I guess?)
So after trusting what I learned in bio 101 (or where ever) that both my parents are responsible for my genome, now I’m confused about whether it’s actually true, AND how to think about the issue from a perspective trying to take in the conjunction fallacy.
Could someone help clear this up for me? Thanks in advance.
The short answer is that both statements are true, so the “or equal to” part is satisfied and the inequality holds. The values involved are probabilities over binary truth/falsity, not degrees of truth.
That clears up the issue for me. Thank you!
Maybe read up on the concepts of outcome, sample space, event, probability space, and see what the probability of intersection of events means in terms of all that. It’s this stuff that’s being implicitly used, usually it should be clear how to formulate the informal discussion in these terms. In particular, truth of whether an outcome belongs to an event is not fuzzy, it either does or doesn’t, as events are defined to be certain sets of outcomes.
(Also, the reasons behind omission of the “or equal to”s you might’ve noticed are discussed in 0 And 1 Are Not Probabilities, though when one of the events includes the other this doesn’t apply in any straightforward sense.)
Thanks for the suggestions, and now that I understand the idea that the probability values correspond to a binary interpretation of the events, it makes these areas easier to navigate for me in discussion.
In particular, truth of whether an outcome belongs to an event is not fuzzy, it either does or doesn’t, as events are defined to be certain sets of outcomes.
This definitely stands as a hard to argue against idea, and it makes sense when seen from the viewpoint of rational humans interpreting data from systems based on binary calculations and logic.
Do you think it’s possible that there is a better way than Binary logic to compute and reason though?
Not being familiar with the literature, I wonder if it’s possible that because we have relied on binary logic to compute and reason for so long, it’s created a false dichotomy in our understanding of reality. Is there another way to reason that works better, based on quantum computing rational?
How the next decade will add to the discussion of reality in terms of the advances in Quantum computing seems to be debatable. Translating probability values into either true or false logic is a step in binary computing that I believe quantum computing skips, and so the data returned takes in to account I think, the cases in which ”...one of the events includes the other...” in a more or less straightforward way.
At this point though, (I could be wrong) I believe there is still more of a front end system that runs binary to interpret the Quantum calculations of a quantum computer, because when the data returned isn’t binary, we’re still trying to figure out what it’s good for.
In relation to events and how long or little they last, this whole area of Quantum clocks is interesting to me. We can measure time more accurately because of them, but it seems like so much of the science in common use still relies on the second as the base measurement. Maybe the second is the bottom limit of what humans can somewhat accurately perceive without aid of a tool like a watch, which makes a case for basing measurements of time using more accurate methods off of the second.
Is it possible we could create wet ware with augmented vision which would allow us to ‘perceive’ smaller and smaller units of time, or would we just be better off trying to figure out how to slow down time? Sometimes rationally speaking, in the light of all these scientific advances, it gets a little harder to appreciate humans when you consider our limited abilities. I think it’s our ability to conceptualize these phenomenon though that is our ‘saving grace.’
At this level of technical discussion it’s hopeless to attempt to understand anything. Maybe try going for depth first, learning some things at least to a level where passing hypothetical exams on those topics would be likely, to get a sense of what a usable level of technical understanding is. Taking a wild guess, perhaps something like Sipser’s “Introduction to the Theory of Computation” would be interesting?
I just downloaded the 2nd edition. Thank you for the suggestion.