Why these two ways of constructing an argument produce different results?

Question

I have two arguments which I want to combine. Depending on the way I do it I get different results.

Argument #1

P1) If a person is A, then it's likely that that person is also B.

P2) This person is A.

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C) This person is likely to be B.

Here, "likely" is interpreted as having a probability greater than 0.5. Using conditional probability we can write P1 as

Pr(B|A) > 0.5

This argument is deductively valid. If our premises are true we can confidently say that the person is more likely to be B than not.

Argument #2

P1) If a person is C, then it's likely that that person is also B.

P2) This person is C.

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C) This person is likely to be B.

Just like with the first argument we can write P1 as

Pr(B|C) > 0.5

And again, taking this argument in isolation and assuming the premises are true, we can say that the person in question is more likely to be B than not.

Finally, let's assume that we have both arguments and their second premises refer to the same person. Is it likely that that person is B? If I use arguments' conclusions as premises for my third argument then the inference is trivial.

Argument #3

P1) This person is likely to be B. (From argument #1.)

P2) This person is likely to be B. (From argument #2.)

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C) This person is likely to be B.

I expect to come to the same conclusion if I combine premises from arguments #1 and #2.

Argument #4

P1) If a person is A, then it's likely that that person is also B.

P2) This person is A.

P3) If a person is C, then it's likely that that person is also B.

P4) This person is C.

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C) This person is likely to be B.

However, if I try to prove this using conditional probabilities I can't get a definitive answer. What we need to prove now is that Pr(B|A,C) > 0.5. However, as was shown in Information paradox: the more we know, the less confident we are, premises

P1) Pr(B|A) > 0.5 and

P3) Pr(B|C) > 0.5

do not guarantee that

Pr(B|A,C) > 0.5

There must be a mistake in the way argument #3 or argument #4 is constructed but I don't see it. Or my assumption that the two ways of argument construction are equivalent may be wrong. So why the conclusions are different?

Answer 1

The apparent difficulties in this argument arise from the way the words and symbols that you are using are working together to create an imprecise discussion regarding the probabilities of picking an individual from a group versus the probabilities of looking at the traits of a certain individual. We get in trouble when we don't think precisely about the difference between these two statements: 1) I am going to pick an individual from a group and the probability they have various traits is X. 2) I have an individual in front of me, the probabilities this person has these trait is X. The symbols that you are using do not sufficient context to make this distinction on their own. The picture below is a Ven diagram depicting a situation where the three conditional statements can all be true. (Those three are underlined.)

I suspect there is a convention using subscripts that could solve the problem. But the following tries to make clear the distinction you would have to capture in the symbols for the symbolic argument not to give you a problem.

Now lets translate your conditional probabilities into words that are less problematic:

The probability that I pick any individual from this set and they have trait B, given they came from set A, is greater than 50%. (51%) The probability that I pick any individual from this set (NOT NECESSARILY THE SAME INDIVIDUAL!) and they have trait B, given they came from set C, is greater than 50%. (52%) The probability that I pick any individual from this set (NOT NECESSARILY THE SAME INDIVIDUAL!) and they have trait B, given they came from a set of individuals that are both A&C, is less than 50%. (20%)

Were we to tell the story of one individual the statements could look like this:

The probability that I pick an individual from this set and they have trait B, given they came from set A, is greater than 50% (51%). The probability that THAT THAT SAME INDIVIDUAL has trait B, given I add the information that they have trait C is less than 50%. (20%). The probability that THAT THAT SAME INDIVIDUAL has trait B, given I redundantly tell you that they have trait A&C is less than 50%. (20%).

OR to make a similar contrast

I have an individual in front of me. I have picked them from this group of 345 people. (see diagram) When I tell you they have trait A, what is the probability that they have trait B? Answer: 51% Forget I told you that they have trait A. When I tell you they have trait C, what is the probability that they have trait B? Answer: 52% Forget I told you that they have trait A or C. When I tell you they have trait A and trait C, what is the probability that they have trait B? Answer: 20%

versus

I have picked them from this group of 345 people. (see diagram) When I tell you they have trait A, what is the probability that they have trait B? Answer: 51% Now don't forget what you already know. When I now tell you they have trait C, what is the probability that they have trait B? Answer: 20% Now don't forget what you already know. When I tell you they have trait A and trait C (WHICH I ALREADY TOLD YOU), what is the probability that they have trait B? Answer: 20%

Answer 2

You're missing information on whether or not A AND C is representative of A and representative of B. I.e. it could be that A AND C represents 49% of A, and 49% of C, and that NONE of them are B.

Let's try an example:

A are Probably B, C are probably B, Therefore (A and C) are probably B

E.g. A = American Females, B = 'White', C = American Politicians

An American Female is probably White, An American Politician is probably White, Therefore A Female American Politician is probably White

Is obviously not necessarily true. Unless you add something like "American female politicians are probably white"

Answer 3

I'll throw my hat on the ring and say that argument 4 sends problematic, because your are (unnecesarily?) reversing causal relationship between attributes.

Does that make any sense to you?

Answer 4

I think the problem is with Argument #3. The reason why that argument seems to always confirm that the person is likely to be B is in the way we interpret it.

Firstly, we need to recognise that the premises are descriptions of two different findings. Although they look exactly the same they do not necessarily represent the result of the same assessment or measurement.

Secondly, the term "likely" is vague. It indicates that a probability of an event is more than 0.5 which gives us a range of possible values. The actual values hidden by the quantifier "likely" may be different in each premise.

Given these two points, let's imagine a situation in which both premises are true and we know the actual probabilities are different in each case. If we had those probabilities we could refine Argument #3 like this

P1) Probability of this person being B is x.

P2) Probability of this person being B is y.

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C) This person is likely to be B.

where x>0.5, y>0.5 and x is not equal y.

In such case, the argument would be valid because the conjunction of the premises is always false, but it would not be sound. This situation is possible because we lose information about the person by not mentioning conditions A and C in Argument #3.

In summary, the apparent soundness of Argument #3 is caused by imprecise description of prior findings where specific values are replaced by vague quantifiers.