argument relating three sets permits a valid conclusion. The first three rules are very easy to apply. The fourth and fifth rules require additional explanation, because they introduce the new concept of a distributed term.
1. If both premises are negative, there is no conclusion. (If both premises are negative, the three sets are completely separate and no further conclusion can be drawn.)
2. If one premise is negative, the conclusion is negative. (If one premise is negative, the essence of the syllogism is to disconnect one set from another through its disconnection from the middle term.)
3. If neither premise is negative, the conclusion is affirmative. (If both premises are affirmative, there is no reason to conclude anything negative.)
4. The middle term must be distributed at least once in order for there to be a valid conclusion. (See discussion immediately below.)
5. No term may be distributed in the conclusion if it is not distributed in the premises. (See discussion immediately below.)
When we say a term is distributed in a premise, we mean that the premise says something about the entire set represented by the term.
• In the statement All [members of set A] are [members of set B], set A is distributed because the statement says that the entire set A is included in set B. That is to say, the statement refers to set A in its entirety.
• In the statement No [members of set A] are [members of set B], set A and set B are both distributed because all of set A is excluded from all of set B.
• In the statement Some [members of set A] are [members of set B], neither term is distributed because we only know about part of set A and part of set B.
• In the statement Some [members of set A] are not [members of set B], the set B is distributed because the statement says that there are some members of set A that are excluded from all of set B. (This explanation is a little hard to understand, but it becomes clear when pictured in a diagram. There, the portion of set A is drawn outside the entire set B; hence, it is clearer that we are referring to the whole of set B.)
From the five basic rules, two more very useful rules can be derived:
6. If both premises refer to only part of a set, there is no conclusion. (It is impossible to relate the parts of the sets that are referred to by the vague quantifier “some.”)
7. If one premise refers to only part of a set, the conclusion refers to only part of a set. (If one premise refers to part of a set, the conclusion cannot generalize legitimately beyond that.)
To gain further appreciation of these rules, we can apply them to some of the syllogisms that have already been presented in Section II.B.1. The syllogism in Example 1 has the following form (with M representing the term that is common to the two premises):
Premise 1: All [members of set M] are [members of set B] Premise 2: All [members of set A] are [members of set M] Conclusion: All [members of set A] are [members of set B]
This syllogism has two affirmative premises, so the conclusion must be affirmative (rule 3). The middle term M is distributed in the first premise (rule 4). Set A is distributed in the conclusion and in the second premise (rule 5).
Now let us apply the rules to Example 3 in Section II.B.1, which does not have a valid conclusion. This syllogism has the following form:
Premise 1: All [members of set B] are [members of set M] Premise 2: All [members of set A] are [members of set M]
Invalid Conclusions: All [members of set A] are [members of set B] Some [members of set A] are [members of set B]
Since both premises are affirmative, the syllogism would have to have an affirmative conclusion (rule 3). All [members of set A] are [members of set B] and Some [members of set A] are [members of set B] are therefore possible conclusions. However, the middle term M is not distributed in either premise (violation of rule 4). Therefore, the syllogism does not have a valid conclusion.
The Self-Test will give you practice in applying these rules. Self-Test: Section II.B.2 (answers are given on page 118)
In exercises 1 through 4, indicate which syllogisms are valid. For valid syllogisms, decide which syllogistic rules apply. For invalid syllogisms, decide which syllogistic rules are violated.
1. Premise 1: All ICE Agents are Department of Homeland Security employees. Premise 2: All ICE Agents are Federal Government employees.
Conclusion: All Federal Government employees are Department of Homeland Security employees.
(valid / invalid)
If valid, which rules apply? If invalid, which rules are violated? 2. Premise 1: All drug smugglers are violators of anti-drug laws.
Premise 2: No Station 2 detainees are violators of anti-drug laws. Conclusion: No Station 2 detainees are drug smugglers.
(valid / invalid)
If valid, which rules apply? If invalid, which rules are violated? 3. Premise 1: Some contraband seized was seized by DHS.
Premise 2: All contraband seized becomes property of the U.S. Government. Conclusion: Some property of the U.S. Government is contraband seized by the DHS.
(valid / invalid)
If valid, which rules apply? If invalid, which rules are violated?
4. Premise 1: All illegal immigrants are under the jurisdiction of the DHS. Premise 2: Some deported persons are illegal immigrants.
Conclusion: Some deported persons are not under the jurisdiction of the DHS. (valid / invalid)
If valid, which rules apply? If invalid, which rules are violated?
Section II.B.3. Illogical Biases in Relating Three Sets
It has been shown in numerous studies that people make many mistakes when reasoning with three sets. We could say that people have many illogical biases in this type of reasoning. The purpose of this section is to show you the most prevalent of these biases so that you can avoid them yourself.
One of the common mistakes of reasoning is to think that a valid conclusion can be drawn from premises that do not warrant a valid conclusion. For example, in the following two cases no conclusion can be drawn, although people tend to think otherwise.
Premise 1: No biased persons make fair decisions. Premise 2: No adjudicators are biased.
Invalid Conclusion: No adjudicators make fair decisions. Premise 1: Some detainees were deported.
Premise 2: Some deported persons came from overseas.
Invalid Conclusion: Some persons who came from overseas were detainees.
If you read the Advanced Topic on syllogistic rules in the previous section, you will see that, in the first example, no conclusion can be drawn because both premises are negative and, in the second example, no conclusion can be drawn because both premises refer only to part of a set. If you construct diagrams for these two forms, you will appreciate graphically the
invalidity of these forms. In the first form, the sets are totally disconnected from one another, thus allowing no conclusion. In the second form, the parts of the set denoted by the vague quantifier “some” are undefined and hence cannot be related to one another.
Another common error is to draw a conclusion about an entire set when a conclusion is warranted about only part of a set. The next two examples show this type of error.
Premise 1: All primary inspections are conducted by one inspector. Premise 2: All primary inspections are brief.
Invalid Conclusion: All brief inspections are conducted by one inspector. Valid Conclusion: Some brief inspections are conducted by one inspector.
The first conclusion is invalid because Premise 2 does not give us information about all brief inspections. (As you will recall from your study of the converse in Section II.A.3, you cannot conclude that “All brief inspections are primary inspections” from the statement that “All primary inspections are brief.”) We can conclude from that premise only that “some brief inspections are primary inspections.” Therefore, we can only conclude that “some brief inspections are conducted by one inspector.”
Premise 1: No Texans live in Florida. Premise 2: All Floridians live in the U.S.A.
Invalid Conclusion: No persons who live in the U.S.A. are Texan. Valid Conclusion: Some persons who live in the U.S.A. are not Texan.
The invalid conclusion to this syllogism is based on the same type of reasoning error that characterized the previous invalid syllogism. From the statement that “All Floridians live in the U.S.A.” (Premise 2), you can only conclude that “Some people who live in the U.S.A. are Floridians.” Therefore, you can only conclude that “Some people who live in the U.S.A. (the Floridians) are not Texans.”
What is the significance of these illogical biases?
These illogical biases serve as a caution to you (and to all of us) to be very careful when drawing conclusions about the relationships among sets. An incorrect conclusion may “feel” right at first and you may be swayed into faulty conclusions that will affect the integrity of your decision making on the job. The antidote to illogical biases is good reasoning: your goal is to become a top reasoner by fine-tuning your skills through the use of this manual.
Self-Test: Section II.B.3 (answers are given on page 119)
For each syllogism below, indicate if the conclusion is valid (true). If the conclusion is not valid (that is, if it is false or if there is insufficient information), determine if there is a different, valid conclusion.
1. Premise 1: No Special Agents failed to qualify with their firearm last week. Premise 2. All employees who failed to qualify with their firearm last week must qualify this week.
Conclusion: No employees who have to qualify with their firearm this week are Special Agents.
(valid / invalid)
If applicable, valid conclusion:
2. Premise 1: All requests for computer assistance must be made to the Help Desk. Premise 2: All Help Desk requests are entered into the service system.
Conclusion: All entries into the service system are requests for computer assistance. (valid / invalid)
If applicable, valid conclusion:
3. Premise 1: All furniture in the ABC building is the property of DHS. Premise 2: Some of this department’s furniture is in the ABC building. Conclusion: Some of this department’s furniture is the property of DHS. (valid / invalid)
If applicable, valid conclusion:
4. Premise 1: Some space on this floor belongs to the Logistics office. Premise 2: Some space of the Logistics office is assigned to Jim. Conclusion: Some space assigned to Jim is on this floor.
(valid / invalid)
If applicable, valid conclusion: