Introduction to Conjunctions and Disjunctions

("And" and "Or")

The simplest deductions are those involving two or more things connected by "and" (a conjunction) or "or" (a disjunction), and are governed by the following rules:

"And" (and "but"): Affirm all, negate one.

  • To affirm an "and" claim, all parts of the "and" must be affirmed as true.
  • To negate an "and" claim, at least one part of the "and" must be negated as false.

"Or": Affirm one, negate all.

  • To affirm an "or" claim, at least one part of the "or" must be affirmed as true.
  • To negate an "or" claim, all parts of the "or" must be negated as false.


Example 1. Consider the claim, "Pat and Juan have arrived." If that claim is affirmed as true, we can conclude that both Pat and Juan have arrived, because all parts must be affirmed. If, however, the claim is false, then we can conclude that at least one of the terms must be negated: either Pat has not arrived, or Juan has not arrived, or neither has arrived.

Example 2. Consider the claim, "Farida or Marcia has won the race." If true, then (because at least one part of the "or" must be affirmed) one of the following must be true: Farida has won, Marcia has won, or they both have won. If false, then neither Farida nor Marcia have won.

These rules always apply, even when the deductions are complicated by more elements ("Farida, Marcia, Pat, and Juan"), the use of negatives ("Farida and not Marcia"), or some combination of these. So, to affirm the claim "Farida and Pat but not Marcia or Juan have finished," we would employ the following steps:

  1. To negate the "or" ("not Marcia or Juan"), we would negate both parts, concluding that Marcia has not finished and Juan has not finished.
  2. To affirm the "and" ("Farida and Pat"), we would affirm both parts, concluding that Farida has finished and Pat has finished.
  3. To affirm the "but" (which operates logically as an "and"), we would affirm all parts, concluding that Farida has finished, Pat has finished, Marcia has not finished, and Juan has not finished. All these must be true in order for the claim "Farida and Pat but not Marcia or Juan have finished" to be true.


Inclusive and Exclusive "Or"

Sometimes, "or" is used in an exclusive sense. For example, you might read on a menu, "Soup or salad comes with the dinner." This means, "soup or salad, but not both," because the menu is describing what is included with the price of the meal. However, if there is no contextual reason to think otherwise, assume every "or" is inclusive--that is, "A or B or both." The difference between the inclusive and exclusive "or," then, has to do with cases in which "both" are true. Since "Soup or salad comes with the dinner" is exclusive, it places "soup and salad" outside the range of things that are included in the price of the meal.

Now suppose an advisor tells you that you can take English 7 or History 60 to satisfy a critical thinking requirement. Though, in this case, it's clear that you don't need to take both, that "or" is still inclusive, because if you did take both, you would still be satisfying the requirement: English 7 or History 60 or both satisfy the requirement.

As a result, the use of "and/or" is unnecessarily confusing and should be avoided, since "or" by itself, in the absence of any exclusionary language or context, means the same thing. Using "A or B or both" makes the possibilities clearer but, in most cases, a simple "or" will suffice.



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1. John claims, "I returned the book to the library on Monday night or Tuesday morning." If he returned the book to the library on Monday night, then his claim is true.

 Valid

 Invalid

  Unknown

2. John claims, "I returned the book to the library on Monday night or Tuesday morning." If the claim is true, and if he did not return the book to the library on Monday night, then he must have returned it Tuesday morning.

  Valid

  Invalid

  Unknown

3. John claims, "I returned the book to the library on Monday night or Tuesday morning." If the claim is true, and if he did return the book to the library on Monday night, then he must not have returned it Tuesday morning.

  Valid

  Invalid

  Unknown

4. John claims, "I returned the book to the library on Monday night or Tuesday morning." Even if the claim is false, he may have returned the book on Monday night.

  Valid

 Invalid

  Unknown

5. John claims, "I returned the book to the library on Monday night or Tuesday morning." If John returned the book on Tuesday afternoon, then his claim is false.

  Valid

  Invalid

  Unknown

6. John claims, "I returned the book to the library on Monday night or Tuesday morning." Even if he returned the book on Tuesday morning, the claim may still be false.

  Valid

  Invalid

  Unknown

7. Juanita says, "I have midterms on Monday and Tuesday." Juanita might be wrong, even if she does have a midterm on Tuesday.

  True

  False

  Unknown

8. Juanita says, "I have midterms on Monday and Tuesday." If Juanita's claim is true, does she have a midterm on Monday?

  Yes

  No

  Unknown

9. Your boss guarantees you a raise if you'll relocate to another city, or a promotion but no raise if you stay where you are. In this example, is the "or" inclusive or exclusive?

  Inclusive

  Exclusive

  Unknown

10. Your boss guarantees you a raise if you'll relocate to another city, or a promotion but no raise if you stay where you are. Assuming that the guarantee is true, if you don't get a raise, are you sure of getting a promotion?

  Yes

  No

  Unknown

1. John claims, "I returned the book to the library on Monday night or Tuesday morning." If he returned the book to the library on Monday night, then his claim is true.

You answered:

  Valid

Correct!

To affirm an "or" claim, at least one element needs to be affirmed. In this case, one or both of the elements, "Monday night" and "Tuesday morning," must be affirmed for the claim to be true. Since "Monday night" has been affirmed, it is valid to conclude that the claim is true.

1. John claims, "I returned the book to the library on Monday night or Tuesday morning." If he returned the book to the library on Monday night, then his claim is true.

You answered:

  Invalid

To affirm an "or" claim, at least one element needs to be affirmed. In this case, one or both of the elements, "Monday night" and "Tuesday morning," must be affirmed for the claim to be true. Since "Monday night" has been affirmed, it is valid to conclude that the claim is true.

1. John claims, "I returned the book to the library on Monday night or Tuesday morning." If he returned the book to the library on Monday night, then his claim is true.

You answered:

  Unknown

To affirm the whole "or" claim, at least one element needs to be affirmed. In this case, one or both of the elements, "Monday night" and "Tuesday morning," must be affirmed for the claim to be true. Since "Monday night" has been affirmed, it is valid to conclude that the claim is true.

2.John claims, "I returned the book to the library on Monday night or Tuesday morning." If the claim is true, and if he did not return the book to the library on Monday night, then he must have returned it Tuesday morning.

You answered:

  Invalid

To affirm an "or" claim, at least one element needs to be affirmed. In this case, since the claim has been affirmed as true, we can conclude that one or both of the elements, "Monday night" and "Tuesday morning," are true. Since we also know that "Monday night" is false--that is, John did not return the book on Monday night--we must conclude he did return the book to the library on Tuesday morning.

2.John claims, "I returned the book to the library on Monday night or Tuesday morning." If the claim is true, and if he did not return the book to the library on Monday night, then he must have returned it Tuesday morning.

You answered:

  Valid

Correct!!

To affirm an "or" claim, at least one element needs to be affirmed. In this case, since the claim has been affirmed as true, we can conclude that one or both of the elements, "Monday night" and "Tuesday morning," are true. Since we also know that "Monday night" is false--that is, John did not return the book on Monday night--we must conclude he did return the book to the library on Tuesday morning.

2.John claims, "I returned the book to the library on Monday night or Tuesday morning." If the claim is true, and if he did not return the book to the library on Monday night, then he must have returned it Tuesday morning.

You answered:

  Unknown

To affirm an "or" claim, at least one element needs to be affirmed. In this case, since the claim has been affirmed as true, we can conclude that one or both of the elements, "Monday night" and "Tuesday morning," are true. Since we also know that "Monday night" is false--that is, John did not return the book on Monday night--we must conclude he did return the book to the library on Tuesday morning.

3.John claims, "I returned the book to the library on Monday night or Tuesday morning." If the claim is true, and if he did return the book to the library on Monday night, then he must not have returned it Tuesday morning.

You answered:

  Valid

To affirm an "or" claim, one or more elements need to be affirmed. In this case, since the claim has been affirmed as true, we can conclude that one or both of the elements, "Monday night" and "Tuesday morning," are true. Knowing that one is true, "Monday night," does not rule out the other, "Tuesday morning," being true as well. If you based your conclusion on the assumption that John could not have returned the book twice, you may be depending too much on your "common sense"--that is, on your initial reaction rather than a more careful analysis. Certainly John may have returned the same book to the library on both Monday night and Tuesday morning if, for example, he borrowed that book both on Monday morning and again on Monday night.

3.John claims, "I returned the book to the library on Monday night or Tuesday morning." If the claim is true, and if he did return the book to the library on Monday night, then he must not have returned it Tuesday morning.

You answered:

  Invalid

Correct!!

To affirm an "or" claim, one or more elements need to be affirmed. In this case, since the claim has been affirmed as true, we can conclude that one or both of the elements, "Monday night" and "Tuesday morning," are true. Knowing that one is true, "Monday night," does not rule out the other, "Tuesday morning," being true as well; so the conclusion that he could not have returned the book on Tuesday morning is invalid.

3. John claims, "I returned the book to the library on Monday night or Tuesday morning." If the claim is true, and if he did return the book to the library on Monday night, then he must not have returned it Tuesday morning.

You answered:

  Unknown

To affirm an "or" claim, one or more elements need to be affirmed. In this case, since the claim has been affirmed as true, we can conclude that one or both of the elements, "Monday night" and "Tuesday morning," are true. Knowing that one is true, "Monday night," does not rule out the other, "Tuesday morning," being true as well. Though it is unknown whether John returned the book on Tuesday morning, it is invalid to conclude he did not.

If you allowed yourself to be confused by the idea that John may have returned the book twice, you may be depending too much on your "common sense"--that is, on your initial reaction rather than a more careful analysis. Certainly John may have returned the same book to the library on both Monday night and Tuesday morning if, for example, he borrowed that book both on Monday morning and again on Monday night.

4. John claims, "I returned the book to the library on Monday night or Tuesday morning." Even if the claim is false, he may have returned the book on Monday night.

You answered:

  Valid

To negate an "or" claim, all elements must be negated. In this case, therefore, neither of the elements, "Monday night" or "Tuesday morning," can be true, so John can not have returned the book on Monday night, regardless of what he did at any other time.

4. John claims, "I returned the book to the library on Monday night or Tuesday morning." Even if the claim is false, he may have returned the book on Monday night.

You answered:

 Invalid

Correct!!

To negate an "or" claim, all elements must be negated. In this case, therefore, neither of the elements, "Monday night" or "Tuesday morning," can be true, so John can not have returned the book on Monday night, regardless of what he did at any other time.

4. John claims, "I returned the book to the library on Monday night or Tuesday morning." Even if the claim is false, he may have returned the book on Monday night.

You answered:

  Unknown

To negate an "or" claim, all elements must be negated. In this case, therefore, neither of the elements, "Monday night" or "Tuesday morning," can be true, so John can not have returned the book on Monday night, regardless of what he did at any other time.

5. John claims, "I returned the book to the library on Monday night or Tuesday morning." If John returned the book on Tuesday afternoon, then his claim is false.

You answered:

  Valid

To determine if an "or" claim is true or not, we must be able either to affirm one of the elements, or negate all of them. In this case, "Tuesday afternoon" only negates "Monday night" and "Tuesday morning" if you assume John can return the book only once--something we are not told. Because John may have returned the book both Monday night and Tuesday afternoon, for example, affirming that he did return it Tuesday afternoon does not rule out the possibility that he also returned it earlier.

5. John claims, "I returned the book to the library on Monday night or Tuesday morning." If John returned the book on Tuesday afternoon, then his claim is false.

You answered:

  Invalid

To determine if an "or" claim is true or not, we must be able either to affirm one of the elements, or negate all of them. In this case, "Tuesday afternoon" only negates "Monday night" and "Tuesday morning" if you assume John can return the book only once--something we are not told. Because John may have returned the book both Monday night and Tuesday afternoon, for example, affirming that he did return it Tuesday afternoon does not rule out the possibility that he also returned it earlier.

5. John claims, "I returned the book to the library on Monday night or Tuesday morning." If John returned the book on Tuesday afternoon, then his claim is false.

  Unknown

Correct!!

To determine if an "or" claim is true or not, we must be able either to affirm one of the elements, or negate all of them. In this case, "Tuesday afternoon" only negates "Monday night" and "Tuesday morning" if you assume John can return the book only once--something we are not told. Because John may have returned the book both Monday night and Tuesday afternoon, for example, affirming that he did return it Tuesday afternoon does not rule out the possibility that he also returned it earlier.

6. John claims, "I returned the book to the library on Monday night or Tuesday morning." Even if he returned the book on Tuesday morning, the claim may still be false.

You answered:

 Valid

An "or" claim is true if at least one of its elements is true. In this case, the claim is true if either "Monday night" or Tuesday morning" is affirmed, and we are told John did in fact return his book on Tuesday morning." The conclusion that the claim may be false is therefore invalid.

6. John claims, "I returned the book to the library on Monday night or Tuesday morning." Even if he returned the book on Tuesday morning, the claim may still be false.

You answered:

 Invalid

Correct!!

An "or" claim can only be false if all its elements are negated. In this case, for the claim to be false, both "Monday night" and Tuesday morning" must be negated. If "Tuesday morning" is affirmed, as it is above, then the claim is true.

6. John claims, "I returned the book to the library on Monday night or Tuesday morning." Even if he returned the book on Tuesday morning, the claim may still be false.

You answered:

 Unknown

An "or" claim is true if at least one of its elements is true. In this case, the claim is true if either "Monday night" or Tuesday morning" is affirmed, and we are told John did in fact return his book on Tuesday morning." The conclusion that the claim may be false is therefore invalid.

7. Juanita says, "I have midterms on Monday and Tuesday." Juanita might be wrong, even if she does have a midterm on Tuesday.

You answered:

 True

Correct!!

To affirm an "and" claim, all elements must be affirmed. So, to ensure that Juanita is right, midterms on both Monday and Tuesday must be affirmed. In this example, no midterm on Monday would mean that Juanita does not have "midterms on Monday and Tuesday," and thus her claim would be false. (If Juanita had said, "I have midterms on Monday or Tuesday," however, the claim would be true with just Tuesday affirmed.

7. Juanita says, "I have midterms on Monday and Tuesday." Juanita might be wrong, even if she does have a midterm on Tuesday.

You answered:

 False

To affirm an "and" claim, all elements must be affirmed. So, to ensure that Juanita is right, midterms on both Monday and Tuesday must be affirmed. In this example, no midterm on Monday would mean that Juanita does not have "midterms on Monday and Tuesday," and thus her claim would be false. (If Juanita had said, "I have midterms on Monday or Tuesday," however, the claim would be true with just Tuesday affirmed.

7. Juanita says, "I have midterms on Monday and Tuesday." Juanita might be wrong, even if she does have a midterm on Tuesday.

You answered:

 Unknown

To affirm an "and" claim, all elements must be affirmed. So, to ensure that Juanita is right, midterms on both Monday and Tuesday must be affirmed. In this example, no midterm on Monday would mean that Juanita does not have "midterms on Monday and Tuesday," and thus her claim would be false. (If Juanita had said, "I have midterms on Monday or Tuesday," however, the claim would be true with just Tuesday affirmed.)

8. Juanita says, "I have midterms on Monday and Tuesday." If Juanita's claim is true, does she have a midterm on Monday?

You answered:

 Yes

Correct!!

For an "and" claim to be true, all elements must be true. Therefore, Juanita must have at least one midterm on Monday, and at least one midterm on Tuesday.

8. Juanita says, "I have midterms on Monday and Tuesday." If Juanita's claim is true, does she have a midterm on Monday?

You answered:

 No

For an "and" claim to be true, all elements must be true. Therefore, Juanita must have at least one midterm on Monday, and at least one midterm on Tuesday.

8. Juanita says, "I have midterms on Monday and Tuesday." If Juanita's claim is true, does she have a midterm on Monday?

You answered:

 Unknown

For an "and" claim to be true, all elements must be true. Therefore, Juanita must have at least one midterm on Monday, and at least one midterm on Tuesday.

9. Your boss guarantees you a raise if you'll relocate to another city, or a promotion but no raise if you stay where you are. In this example, is the "or" inclusive or exclusive?

You answered:

 Exclusive

Correct!!

The context makes it clear that you cannot do both: relocate and stay put, receive a raise and not receive a raise.

  • Exclusive: "A or B, but not both."
  • Inclusive: "A or B, or both."
Remember to assume an "or" is inclusive, unless there are specific indications otherwise. The fact that this "or" is exclusive has logical consequences, as you will see in the next exercise.

9. Your boss guarantees you a raise if you'll relocate to another city, or a promotion but no raise if you stay where you are. In this example, is the "or" inclusive or exclusive?

You answered:

 Inclusive

The context makes it clear that you cannot do both: relocate and stay put, receive a raise and not receive a raise.

  • Exclusive: "A or B, but not both."
  • Inclusive: "A or B, or both."
Remember to assume an "or" is inclusive, unless there are specific indications otherwise. The fact that this "or" is exclusive has logical consequences, as you will see in the next exercise.

9. Your boss guarantees you a raise if you'll relocate to another city, or a promotion but no raise if you stay where you are. In this example, is the "or" inclusive or exclusive?

You answered:

 Unknown

The context makes it clear that you cannot do both: relocate and stay put, receive a raise and not receive a raise.

  • Exclusive: "A or B, but not both."
  • Inclusive: "A or B, or both."
Remember to assume an "or" is inclusive, unless there are specific indications otherwise. The fact that this "or" is exclusive has logical consequences, as you will see in the next exercise.

10. Your boss guarantees you a raise if you'll relocate to another city, or a promotion but no raise if you stay where you are. Assuming that the guarantee is true, if you don't get a raise, are you sure of getting a promotion?

You answered:

 Yes

Correct!!

The two choices, "a raise" and "a promotion but no raise" are connected by an exclusive "or." That means you can have one or the other, but not both. (You boss's "guarantee" takes care of the other possibility, "neither.") Thus, if you do not get "a raise," you must get "a promotion but no raise." In that last phrase, the "but" is a conjunction, like "and," so for the phrase to be true, both elements must be true: "promotion" and "no raise."

10. Your boss guarantees you a raise if you'll relocate to another city, or a promotion but no raise if you stay where you are. Assuming that the guarantee is true, if you don't get a raise, are you sure of getting a promotion?

You answered:

 No

The two choices, "a raise" and "a promotion but no raise" are connected by an exclusive "or." That means you can have one or the other, but not both. (You boss's "guarantee" takes care of the other possibility, "neither.") Thus, if you do not get "a raise," you must get "a promotion but no raise." In that last phrase, the "but" is a conjunction, like "and," so for the phrase to be true, both elements must be true: "promotion" and "no raise."

10. Your boss guarantees you a raise if you'll relocate to another city, or a promotion but no raise if you stay where you are. Assuming that the guarantee is true, if you don't get a raise, are you sure of getting a promotion?

You answered:

 Unknown

The two choices, "a raise" and "a promotion but no raise" are connected by an exclusive "or." That means you can have one or the other, but not both. (You boss's "guarantee" takes care of the other possibility, "neither.") Thus, if you do not get "a raise," you must get "a promotion but no raise." In that last phrase, the "but" is a conjunction, like "and," so for the phrase to be true, both elements must be true: "promotion" and "no raise.

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