Types of Statements and Conversions

Statements can be classified into four types or patterns, as follows:

All A are B.
No A is B.
Some A are B.
Some A are not B.
Of course, the language used in arguments is much more complex than those four statements. As a result, part of your task as a critical thinker will be to restate claims to fit one of those patterns, composed of two terms subject and complement) connected by a state-of-being verb (some form of "to be," such as "is," "are," "were," "will be," and so on). The restatement can reduce the complexity of the original sentence as much as you like, providing it creates no change or confusion in the meaning. "All horses eat hay," for example, can be restated as "All horses are eaters." This may sound a little odd, but it is important to replace verbs expressing action with state of being verbs.

Take the claim, "I like good food." Does that mean "Everything I like is good food," or does it mean "All good food is something I like"? The correct restatement is the latter, "All good food is something I like." This has the form, "All A are B," where "good food" is A and "something I like" is B. But had we failed to convert the original verb, "like," to a state-of-being verb, we might have mistakenly assigned "I like" to A and "good food" to B, producing the equivalent of "Everything I like is good food," which we have already determined to be an erroneous restatement.

But we can also restate the claim, "I like good food," as "I am a good-food-liker." This sounds clumsy, but it, too, accurately represents the original idea. So how do we decide whether to restate this as "All good food is something I like" or "I am good-food-liker"? Both have the form "All A are B," but they are otherwise completely different. B in the former focuses on the food, "something I like," for example, while B in the latter focuses on the person, "good-food-liker." Is one a better choice than the other?

The problem here is really a problem of language, not logic. Without a context, we cannot tell whether the point of the claim is about the food itself ("something"), or about the person ("food-liker"). Faced with this choice when dealing with an argument, you would need to choose the form that has the same focus as the rest of the argument. Thus, "I like good food, and this is good" would involve the "something I like" food-focused version, while "I like good food, and people who like good food are good people" would involve the "good-food-liker" person-focused version.

As the previous example suggests, accurately expressing the sequence of terms in a statement is often very important. Two of the four types of claims, however, can switch the order of their terms without altering their logical implications. These two valid conversions are:

No A is B is equivalent to No B is A.
Some A are B is equivalent to Some B are A.
In other words, "No deciduous trees are evergreens" is equivalent to "No evergreen trees are deciduous," and "Some apples are red" is equivalent to "Some red things are apples." These may sound obvious, but language can be confusing, so remember that those are the only two possible valid conversions.

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Exercises for Statements and Conversions

1. "Most of the units involved in the invasion suffered no casualties" is what pattern of claim?

  All A are B.
  No A is B.
  Some A are B.
  Some A are not B.

2. Which of the following is not a valid conversion?

  "The people in the stands didn't catch any foul balls" and "The people who caught foul balls were not in the stands."
  "Some of our customers are not satisfied with this product" and "Some of the people who are satisfied with this product are not our customers."
  "Sometimes I cry when I'm lonely" and "Sometimes I'm lonely when I cry."
  "The proof is not the pudding" and "The pudding is not the proof."

3. Which of the following is not a valid conversion?

  "Only those without a valid driver's license should be arrested when stopped for a minor traffic violation" and "Those arrested for a minor traffic violation should not have a valid driver's license."
  "Some of those without a valid driver's license are arrested when stopped for a minor traffic violation" and "Some of those arrested when stopped for a minor traffic violation are without a valid driver's license."
  "Only those with a criminal record should be arrested when stopped for a minor traffic violation" and "All those arrested when stopped for a minor traffic violation should have a criminal record."
  "Only those without a valid driver's license should be arrested when stopped for a minor traffic violation" and "All those with a criminal record should be arrested when stopped for a minor traffic violation."

1. "Most of the units involved in the invasion suffered no casualties" is what pattern of claim? You answered:

  All A are B.

There are three problems here: converting "suffered" to a state-of-being verb, and interpreting "most of the units" and "no casualties." We can begin by restating the claim as "Most units are no casualty sufferers." At this point, it should be clear that neither "most of the units" nor "no casualty sufferers" can be restated as "All A," so this answer cannot be supported. Try again!

1. "Most of the units involved in the invasion suffered no casualties" is what pattern of claim? You answered:

  No A is B.

There are three problems here: converting "suffered" to a state-of-being verb, and interpreting "most of the units" and "no casualties." We can begin by restating the claim as "Most units are no casualty sufferers." Since "No A is B" can be converted to "No B is A," we can switch the positions of the terms, and attach the negative to either one. But that doesn't get rid of "most," so this answer cannot be supported. Try again!

1. "Most of the units involved in the invasion suffered no casualties" is what pattern of claim? You answered:

  Some A are B.

There are three problems here: converting "suffered" to a state-of-being verb, and interpreting "most of the units" and "no casualties." We can begin by restating the claim as "Most units are no casualty sufferers." Since "most" is more than "none" and less than "all," it is a non-universal qualification, like "some." To make "Some units are no casualty sufferers" into "Some A are B," however, B would have to include the negative "no." This is possible, but unless there are reasons to treat a negative inclusively, begin with the less complex assumption that the negative should not be part of the term itself.

1. "Most of the units involved in the invasion suffered no casualties" is what pattern of claim? You answered:

  Some A are not B.

Correct!!

There are three problems here: converting "suffered" to a state-of-being verb, and interpreting "most of the units" and "no casualties." We can begin by restating the claim as "Most units are no casualty sufferers." Since "most" is more than "none" and less than "all," it is a non-universal qualification, like "some." That leaves the equivalent of "Some units are no casualty sufferers," or "Some A and not B."

2. Which of the following is not a valid conversion? You answered:

  "The people in the stands didn't catch any foul balls" and "The people who caught foul balls were not in the stands."

Always begin by restating the claims with a state-of-being verb. Here, "The people in the stands didn't catch any foul balls" would become "People in stands are not ball catchers"; and "The people who caught foul balls were not in the stands" would become "The ball catchers are not people in the stands." If "people in the stands" is A and "ball catchers" is B, these can be reduced to "A is not B" and "B is not A," and the conversion can be more easily be seen as valid: "No A is B" and "No B is A."

2. Which of the following is not a valid conversion? You answered:

  "Some of our customers are not satisfied with this product" and "Some of the people who are satisfied with this product are not our customers."

Correct!!

Though it might sound good, "Some A are not B" cannot be converted to "Some B are not A," which is the case here.

2. Which of the following is not a valid conversion? You answered:

  "Sometimes I cry when I'm lonely" and "Sometimes I'm lonely when I cry."

Always begin by restating a claim with a state-of-being verb. "Sometimes I cry when I'm lonely" then becomes "Some crying times are lonely times," and "Sometimes I'm lonely when I cry" becomes "Some lonely times are crying times." If "crying times" are A and "lonely times" are B, this can be seen as the valid conversion of "Some A are B" to "Some B are A."

2. Which of the following is not a valid conversion? You answered:

  "The proof is not the pudding" and "The pudding is not the proof."

If "proof" is A and "pudding" is B, then this would be "A is not B" and "B is not A," which is just another way of saying "No A is B" and "No B is A," one of the two valid conversions you can make. Try again!

3. Which of the following is not a valid conversion? You answered:

  "Only those without a valid driver's license should be arrested when stopped for a minor traffic violation" and "Those arrested for a minor traffic violation should not have a valid driver's license."

Always begin by restating the claim with a state-of-being verb. Here, "Only those without a valid driver's license should be arrested when stopped for a minor traffic violation" could be restated as "Only not licensed are arrested"; and "Those arrested for a minor traffic violation should not have a valid driver's license" could be restated as "All arrested are not licensed." As covered above, "Only A is B" is equivalent to "All B is A," so "Only not licensed are arrested" would also become "All arrested are not licensed." We can further restate these claims as "No arrested is licensed," which itself can be validly converted to "No licensed is arrested," since "No A is B" and "No B is A" are valid conversions.

3. Which of the following is not a valid conversion? You answered:

  "Some of those without a valid driver's license are arrested when stopped for a minor traffic violation" and "Some of those arrested when stopped for a minor traffic violation are without a valid driver's license."

The two valid conversions are "No A is B = No B is A" and "Some A is B = Some B is A." Here we have the latter, where A is "those without a valid driver's license" and B is "arrested when stopped for a minor traffic violation." Notice that the verb here is "are," but in the other options it is "should be." That's because you must have a state-of-being verb, such as "are," before performing a valid conversion. The other options for this question are not conversions but equivalences: "Only A are B" is not one of the four forms of a claim, but it is equivalent to one of them, namely "All B are A."

3. Which of the following is not a valid conversion? You answered:

  "Only those with a criminal record should be arrested when stopped for a minor traffic violation" and "All those arrested when stopped for a minor traffic violation should have a criminal record."

Always begin by restating the claim with a state-of-being verb. Here, "Only those with a criminal record should be arrested when stopped for a minor traffic violation" can be restated as "Only criminals are arrested"; and "All those arrested when stopped for a minor traffic violation should have a criminal record" can be restated as "All arrested are criminals." As covered above, "Only A is B" is equivalent to "All B is A," so "Only criminals are arrested" would also become "All arrested are criminals." So this is a valid conversion.

3. Which of the following is not a valid conversion? You answered:

  "Only those without a valid driver's license should be arrested when stopped for a minor traffic violation" and "All those with a criminal record should be arrested when stopped for a minor traffic violation."

Correct!!

Always begin by restating the claim with a state-of-being verb. Here, "Only those with a criminal record should be arrested when stopped for a minor traffic violation" can be restated as "Only criminals are arrested"; and "All those with a criminal record should be arrested when stopped for a minor traffic violation" can be restated as "All criminals are arrested." As covered above, "Only A is B" is equivalent to "All B is A," so "Only criminals are arrested" would also become "All arrested are criminals." And "All A are B" cannot be converted to "All B are A," so this is not a valid conversion.

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