The first form we will study is a conditional in which the consequent is true only with a certain probability when the antecedent is true.14 Another way of saying this is that the consequent is true for only a proportion of the time when the antecedent is true. This means that part of the time the antecedent will be true and the consequent will be untrue. (This concept will be studied again as the topic of conditional probability in the unit on statistical reasoning.)
14 Logic Note. Remember that the antecedent of the conditional is the “if” part of the sentence and the
The following example illustrates this type of conditional:
If an agent applies for every vacancy in Sector X, he or she will have a 30% chance of being promoted in Sector X this year.
Agent Jones will apply for every vacancy in Sector X this year.
Therefore, Agent Jones will have a 30% chance of being promoted in Sector X
this year.
Notice how different this formula is from the conditional used in deduction. The formula yields only a probabilistic conclusion. Agent Jones may or may not be promoted in Sector X if she applies for every vacancy.
Unlike the deductive conditional, this form does not permit a definite conclusion to be drawn if the consequent is denied.15 In other words, from the information that an agent was not promoted, we cannot draw a conclusion about the probability that the agent applied for every vacancy.
Like the deductive conditional, this form does not permit a conclusion if the antecedent is negated (the inverse) or the consequent is affirmed (the converse). In other words, from information that an agent did not apply for all vacancies in Sector X, we would not be able to estimate the likelihood that the agent was not promoted in Sector X (the inverse). From information that the agent was promoted in Sector X, we would not be able to estimate the probability that the agent applied for all vacancies in Sector X (the converse).
It is interesting to note here that the probability that an event will not occur is equal to 1 minus the probability that it will occur. Therefore, we can conclude that if a person does apply for every vacancy, the person has a 70% chance of not being promoted. We can say that these two probabilities (.3 and .7) are the complements of each other, because the two probabilities add up to 1.
The Biconditional With Probabilities
We can define a biconditional premise that has two conditional probabilities associated with it. It would retain the symmetry associated with the deductive biconditional. Such a biconditional would look like this:
15 Logic Note. Another way of saying this is that the contrapositive is not a valid conclusion from this type of
If a person has a certain type of infection, diagnostic test A has a 90% chance of giving a positive diagnosis.
If a person (whose infection status is unknown) receives a positive diagnosis on test A, there is a 90% chance that the person has this type of infection.16
Individual B is known to have this type of infection.
Therefore, individual B has a 90% chance of receiving a positive diagnosis on test A.
The valid conclusion shown above is based on affirming the first part of this biconditional. Another valid conclusion would be:
Individual C (whose infection status is unknown) has received a positive diagnosis on test A.
Therefore, individual C has a 90% chance of having this type of infection.
Like the inductive conditional, this form still would not give a definite conclusion if either the antecedent or the consequent were negated. In other words, from the information that a person did not have this type of infection, we would not be able to draw any conclusion about the probability that the person would have a positive diagnosis on test A. From information that a person received a negative diagnosis on test A, we would not be able to draw a conclusion about the probability that the person had the infection. Therefore, this biconditional has some limitations on the types of conclusions that can be drawn, in contrast to the deductive biconditional.
Self-Test: Section III.C.1 (answers are given on page 127)
1. The following paragraph contains a conditional statement with a probability. Given the information in the paragraph, decide which of conclusions a through d are correct inferences.
Paragraph
In a certain sector it was found that if a vehicle transporting illegal aliens was apprehended within the sector by any law enforcement authority, there was a 20% chance that it was stopped for speeding. Vehicle A was apprehended by a law enforcement officer and was found to be transporting illegal aliens. Vehicle B was also transporting illegal aliens, but it was not apprehended.
16 Logic Note. For purposes of simplicity, the same probability (.90) is used for the two parts of the
Conclusions
a. There is a 20% chance that vehicle A was stopped for speeding. (Correct / incorrect)
b. There is an 80% chance that vehicle B was not stopped for speeding. (Correct / incorrect)
c. There is an 80% chance that vehicle A was not stopped for speeding. (Correct / incorrect)
d. Of all vehicles stopped for speeding in this sector, 20% were transporting illegal aliens. (Correct / incorrect)
2. The following paragraph contains a biconditional with a probability. Given the information in the paragraph, decide which of conclusions a through d are correct inferences.
Paragraph
A librarian in a law enforcement agency, after studying circulation records and surveying users, was able to draw two conclusions about library use. First, if a book was checked out of the library for use on a specific work project, there was a 70% chance that it would be returned on time. Also, of all the books that were returned on time, 70% had been signed out for use on a specific work project.
Conclusions
a. If a book was signed out for some reason other than a specific work project, there was a 70% chance that it would not be returned on time. (Correct / incorrect)
b. Of 1,000 books returned on time in a one-month period, probably about 700 of them were signed out for use on a specific work project. (Correct / incorrect)
c. In one month, 500 books were signed out for specific work projects, and thus it can be expected that approximately 350 of them will be returned on time.
(Correct / incorrect)
d. If a book was not returned on time, it was more likely than not signed out for a reason other than a specific work project. (Correct / incorrect)
Section III.C.2: Connectives With an Added Premise Containing a Probability