“Chapter 10. Translating Categorical Statements” in “Critical Thinking, Logic, and Argument”
Chapter 10 Translating Categorical Statements
10.1 Three Issues for Translation of Statements
Few statements in ordinary English look like standard form categorical statements. But a surprisingly large number of statements can be translated into standard form categorical statements. Just trying to translate statements into categorical logic will demonstrate how limited these statements are in everyday use. However, if we are to try to direct claims into formal structures to test for validity, we have to be quite rigid and specific about how we formulate statements. The statements have to preserve a logical structure. There are three main issues that come up in these translations: the problem of empty terms, problems related to the natural world versus fictional terms, interpretations of “some,” and direct singular reference.
What About Empty Terms?
In the modern interpretation of categorical statements, the universal categoricals “All S are P” and “No S are P” make no claims about whether anything exists. The statement “All cats have fleas” is understood to be simply about the relation between being a cat and having fleas and to make the claim that for everything in the universe of discourse, if that thing is a cat, then that thing has fleas. It is not considered part of the job of the statement to say whether or not there are any cats. By contrast, the two particular categoricals “Some S are P” and “Some S are not P” are taken to make claims about the actual existence of at least one thing that is S. This distinction is observed in the Venn diagrams mentioned in Chapter 9 by the fact that the two universal categoricals are graphed only by shading out areas known to be empty, whereas the two particular categoricals are graphed by placing an X in an area to show that something exists in the corresponding class.
Figure 10.1 Universal and particular statements. Artwork by Jessica Tang.
This feature of the modern interpretation is admittedly to some degree at odds with how we normally speak. Usually when we say something like “All diamonds are hard,” we assume that there are some things that are diamonds and that all of them are hard things. But modern categorical logic treats the universal statements as conditional claims, so “All diamonds are hard” is understood as “If something is a diamond, then it is hard,” or more accurately “For all the things that there are, if a thing is a diamond, then that thing is a hard thing.”
Example 1 gives us a rule that is true of anyone found cheating on the test, but as far as we know right now, there is no one in that class. So we really understand empty terms as conditional statements: “If someone is found cheating on the test, then they will receive a failing grade.” That rule is true even if no one cheats (the class is empty).
Example 2 gives us a similar rule, since at any given time, no students might need to use the bathroom. But if they do, they must use a hall pass. So awkwardly, this would translate to “All bathroom-needing students are students who need a hall pass.” There might not be anyone who needs to go to the bathroom, but if they do, they need a hall pass.
Let’s look at example 3. As we have shown, a categorical statement can be expressed as an “if A, then B” form (a conditional). If 3 says, “If this being is a unicorn, then it has a horn,” but keep in mind that nothing is a unicorn (so anything that actually asserts there are will be false), then the whole statement is true. Why? Remember the cat on the mat? If the antecedent is false, then the whole statement is true, since the only way for the statement to be false is if it goes from true to false. In the case of universal categoricals, this has the unintuitive result that in cases where the subject term is known to be empty in advance, as in the case of the term “unicorn,” then “All unicorns are white” will be true in a trivial way—true because there aren’t any unicorns. In fact, “No unicorns are white” will be true for exactly the same reason.
There are some problems here, though. When we are talking about using categorical sentences such as “Not all Albertans are steelworkers,” we do understand that there are some Albertans who are not steelworkers. What this means is that in natural language, we often assume that classes have members even when we don’t make this assumption using categorical logic. But this feature of usage is due to a fact about context; we understand sentences in the context of the beliefs and knowledge we have about the world. So when we assume that there are Albertans, this is due not to the meaning of the universal categorical but to an independent piece of knowledge we bring to the task of how best to translate the sentence. This means that we need to pay attention to context when we are translating ordinary sentences into categorical form, and this will have a bearing on fictional contexts, as we will see in a moment.
What About When Terms Are Necessarily Connected?
The second problem is that universal claims often assert a necessary or conceptual connection between the terms in question. Compare these two claims:
What is the relationship between having a billion dollars and being rich? Is it anything like the relationship between winning a billion dollars and being a duck? Assuming a billion dollars means you are rich, 1 seems true. If it were true that you won a billion dollars, then you would be rich. But what about your duck status? It seems like there is no way for that to be true even if you win the billion dollars. This is because there’s no overlap between the classes of things: (a) people who could win a billion dollars and (b) ducks (who were once people?). There is just no relation of meaning or any other kind of connection of necessity between the classes.
This is why it is important to pay attention to the specific grammar of claims. Many universal claims appear to involve both subjunctive claims (claims that involve possibility) and indicative claims (claims that are objective or certain). Compare these two claims:
We know that all diamonds are hard because this is a fact about the nature of diamonds (indicative). But we can also express this using the subjunctive, such that if anything were a diamond, it would be hard. This shows us that the statement “All diamonds are hard” is true not only of all the actual diamonds but of all possible diamonds as well.
An indicative sentence is one that when uttered makes a truth claim—that is, it’s either true or false.
The fact that many universal claims tell us about necessity and possibility is due to a central feature of our ordinary conception of the world. We have a conception of the possible, of what could be true, that is part of our conception of actual things. It is part of our understanding of how things actually are that things could be different than they are in certain ways but not others. For example, if you see a red apple, it is possible that there is a weird light shining on it and it is actually green. We are always considering ways in which the world could be different in order to understand how it is.
We must understand how the world isn’t in order to understand how it is. Part of this is just about the nature of the causal fabric of the world. We don’t make traffic bridges out of butter. This isn’t by choice, really; it is because we can’t. Butter doesn’t have the right tensile strength necessary for it to be adequate bridge-building material.
Claims such as “If we had spent the day sunning on the beach, we would have been tanned by the sun” make sense in our ordinary language even though they aren’t currently true. Relationships of possibility are hard to express properly in categorical logic. Sometimes what we have to do is restrict the universe of discourse in order to allow our claims to work. For example, imagine we are planning a holiday, and imagine first that we plan to go to Mexico, and then at the last second we instead go to Greece; we have no trouble thinking about what would be true if we were in Mexico while we are in Greece.
What About Fictional Terms?
Can we use unicorns in our categorical statements even though they don’t properly exist? We have no trouble thinking that there are facts about what unicorns are like even though there aren’t any unicorns. Properly understood, these may simply be facts about what is appropriately (or commonly) said or imagined about unicorns, since there are no unicorns to make those claims true.
Here we have to think about what the universe of discourse is. There is an implicit universe of discourse such that we are talking about the stories told by J. R. R. Tolkien. Even though there aren’t any hobbits, isn’t this argument valid? If all hobbits have hairy feet and some hobbits garden, doesn’t it follow that some gardeners have hairy feet? This is valid, but depending on whether we can have a fictional universe of discourse, it is unsound. If you are interested in this area of logic, philosophers have different schools of thought about the metaphysical status of fictional entities. This doesn’t need to be dealt with for our purposes. It is a difficult subject, and the mathematical and logical resources needed to deal with those difficulties are well beyond the scope of this text.
Empty classes, subjunctive possibilities, and fictional objects are not issues for our modern interpretation of universal categoricals. We will not assume the existence of members of the classes being described. This is important because as we move toward constructing syllogisms with categorical statements, we need to remember that we cannot infer from the existence of a class that there are mem-bers in that class. You cannot infer as a matter of logic that there are some things in a class (a universal affirmative or negative). For example, from the truth of the statement “All unicorns have a horn in their foreheads,” you cannot infer that there are some unicorns. Of course, sometimes the context makes it clear that there are members of the classes in question, but then the inference from, say, “All cats are mammals” to “Some mammals are cats” is not grounded in the universal claim alone but also depends on our knowledge that there are cats.
10.2 Interpretations of “Some”
Unfortunately, categorical logic only has a meager stock of quantifiers (“all,” “no,” and “some”). Categorical logic doesn’t do a good job of distinguishing more from less, and thus the differences between exactly one, a few, some, many, and almost all are not clear. If there is exactly one member of S that is P, then the statement “Some S are P” is true, and the same is true if almost all S are P. Compare these examples:
Hopefully the translation of 1 is clear. It is making a claim about the kind of thing an apple is. It is claiming that the class of apples is a subset of the class of fruits. Thus we can represent 1 as “All apples are fruit.”
But how should we translate 2? Should we treat this as the universal claim “All dogs are funny” or as the particular one “Some dogs are funny”? To see which to do, we need to look at the context in which the statement occurs. We need to ask what kind of argument is being made and whether that argument depends on a universal or particular treatment for that claim. Consider for example the argument “Dogs are funny; funny things fall down a lot, so dogs fall down a lot.” If we translate this to particular affirmative statements:
- 1. Some funny things fall down a lot.
- 2. Some dogs are funny.
- 3. Thus some dogs fall down a lot.
Remember that we translate “some” as at least one—which means there might be only one. Premise 1 tells us that there is at least one funny thing that falls down a lot and premise 2 tells us that there is at least one dog that is funny. And the conclusion tells us that there is at least one dog that falls down a lot. Can we infer that? Do we know that the “funny thing” in premises 1 and 2 is the same thing, i.e., the dog in the conclusion? We cannot know that for sure, thus we cannot assert this conclusion, and the syllogism is invalid (it is possible for premises 1 and 2 to be true and the conclusion false). But if one of the premises were translated as a universal claim, the conclusion would follow, because we could make the connection through dogs with the statement “All dogs are funny.” We will see examples of arguments of these sorts ahead. By giving careful translations, we can extend the range of arguments that can be successfully translated into categorical form. It is usually clear in a particular context what the best translation is.
10.3 Direct Singular Reference
What do we do when a categorical statement is referring to a specific individual thing? Maybe we are picking out something by a proper name or identifying “this apple” or “that chair.” English contains many kinds of noun phrases that allow reference to individuals and groups of individuals. Proper names, determiners, and demonstratives all play roles in fixing the reference in noun phrases. If we say, “Nushi is tall,” you understand us to be referring to a particular person named Nushi even though there are many people named Nushi.
Definite reference | Indefinite reference |
|---|---|
that cat | a cat |
my house | houses |
this old computer | most old computers |
The difference here is between making reference to a specific individual creature or thing and referring to any member of a class (not a specific one). Sometimes we need to rely on context, since the use of “a” in a phrase such as “I want to buy a shirt” means that you want to buy one specific shirt, but you haven’t picked one out yet. If you told a salesperson “I want to buy a shirt,” they would know you mean something very different from “I want to buy that shirt.” At the same time, if you say, “I want that shirt in a large,” pointing to a small shirt, then you are referring to a specific type of shirt but not to a particular instance of that type.
Translating specific claims into categorical statements can be difficult. The universal categoricals make no reference to individuals—they say something about classes. The particular categoricals make reference to individuals, but they only refer indefinitely. “Some Manitobans are nurses,” for example, tells you that one or more Manitobans is a nurse, but it gives you no information about which Manitobans are nurses. This makes it difficult to translate arguments in which both premises are about the same individual or individuals (recall the dogs that are funny and fall down). Consider two interpretations of a categorical argument:
Version 1: Original argument | Version 2: Particular interpretation (invalid) | Version 3: Universal interpretation (valid) |
|---|---|---|
P1: Some Manitobans are nurses. P2: They are very dedicated. C: Some Manitobans are very dedicated. | P1: Some nurses are very dedicated people. P2: Some Manitobans are nurses. C: Some Manitobans are very dedicated people. | P1: All nurses are very dedicated people. P2: Some Manitobans are nurses. C: Some Manitobans are very dedicated people. |
In version 1, it is clear that “they” refers to the group of Manitoban nurses and that the point of the argument is to say that since they are very dedicated and Manitobans, it follows that some Manitobans are very dedicated people. But in categorical logic, we cannot keep the “they” in premise 2. It is not a proper subject term.
So we have two options for translation: some (particular) or all (universal). In version 2, we run into difficulty. The first premise does not tell you that the nurses who are dedicated are Manitoban nurses, and the second premise does not tell you that the nurses who are Manitobans are dedicated nurses, and so the argument is invalid. But if you translate the first premise as “All nurses are very dedicated people” (since in the context you know that you are talking about the dedicated Manitoban nurses and can thus reasonably restrict the universe of discourse to them), the argument is valid.
10.4 Proper Names
Translating statements containing proper names also requires special treatment. To properly translate proper names, we need to treat the name as referring to a special class that contains all and only the things named. So to translate “Kristin is a professor,” we say, “All persons identical to Kristin are a professor,” and since there is one and only one person in the class of persons identical to Kristin, this gets us most and perhaps all of what we want. It tells us that “for all the things that there are, if a thing is a member of the class of people identical to Kristin, then that thing is a professor.” Remember, though, that on the modern interpretation universals do not tell us that there is anything that is a member of the class of people identical to Kristin. If we need to assert the existence of Kristin, then we need to translate the statement as “Some professors are people identical to Kristin.”
Version 1: Original argument | Version 2: Particular only interpretation (invalid) | Version 3: Universal and particular interpretation (valid) |
|---|---|---|
P1: Kristin is a professor. P2: Kristin is a parent. C: Some professors are parents. | P1: Some people identical to Kristin are professors. P2: Some people identical to Kristin are parents. C: Some professors are parents. | P1: Some professors are people identical to Kristin. P2: All people identical to Kristin are parents. C: Some professors are parents. |
In version 2, we don’t know that the “some” picked out in premise 1 and 2 are the same, so we cannot draw the conclusion based on the premises. In version 3, premise 1 tells us that there is at least one professor who is a person identical to Kristin. The second premise tells us that if there is a person identical to Kristin, then that person is a parent. Between those two premises, we have the information that there is at least one parent who is a professor. Thus, version 3 is valid, since P1 and P2 cannot both be true while the conclusion is false.
As in the case involving the Manitoban nurses, we cannot translate both premises as either universal or particular categoricals, because in neither case will the argument be valid. Instead, we need to make one premise universal and the other particular. Version 3 is valid because the conclusion, which says that there is at least one professor who is a parent, is made true by the combination of the two premises.
The point to keep in mind when translating sentences into categorical form is that there is quite a bit of information implicitly available in an argument informally presented, and you always lose some of that information in the translation, so it is important to make sure that you don’t lose the information that you need for assessing the validity of the argument. Very often, the important information that must be preserved in the translation is information that keeps track of specific individuals mentioned in the argument.
10.5 Translating an Informal Statement
Why do we translate ordinary sentences with such precision in categorical logic? This is because in order to evaluate the argument with formal procedures, we must regiment the argument. This also provides us with tools of analysis that can be applied in other domains of analysis—clarifying how claims of quantity and class relate to each other is important in everyday reasoning contexts. In the case of categorical logic, we are trying to control for the effects of background and contextual knowledge required to understand certain claims. This also brings to our attention the assumptions we rely on when making arguments. Using a standard form or a template allows us to see how far we must go to clarify sentences that are normally expressed in English without regimentation. This section goes through informal categorical translations of the four statement types.
Example 3 is a bit tricky. You can break it down by asking, What are the two classes of things being discussed? Red smarties are a class of things we are talking about, and then we have “he only likes” to deal with. This can be transformed into a class of things: “smarties liked by him.” Then you can try the all statement in both directions: “All red smarties are smarties liked by him” or “All smarties liked by him are red smarties.” Which one seems right? The answer, or the “clue” of this sentence, lies in the use of “only.” If we say, “All red smarties are smarties liked by him,” we are saying that he definitely likes red smarties, but that leaves open the possibility that he could like other smarties, which isn’t what example 3 is saying. Example 3 says he only likes red smarties. Thus, if we put it in the other direction, “All smarties liked by him are red smarties,” we rule out him liking other smarties and capture the use of “only” properly.
Example 4 is similar in scope, since it also uses “only.” We can do the same exercise of reversal to see if we are translating properly. Our two classes are “people who will be hired today” and “engineers.” Let’s try this with the “if _____, then _____” structure: “If people are hired today, then they will be engineers,” or “If they are an engineer, they will be hired today.” The second version makes it seem like all engineers are people who will be hired. This can’t be what it is saying, since we can’t hire all engineers. What it is actually saying is that if a person gets hired today, that person will be an engineer, which means that “all people to be hired today are engineers.”
Example 3 deserves some discussion. At first blush, this might seem like an “all” statement, since it begins with “every.” But this would mean translating it as “All Baptists are non-drinkers.” To understand why we don’t prefer this, it is important to look at the class of people being discussed. The term we are talking about is “drinker,” and the complement to that term is “non-drinker”; it refers to everything in the universe of discourse that is not a drinker. And in the universe of discourse, everything either falls into a category or its complement (it has or lacks a property). You could technically say, “All Baptists are non-drinkers,” however, when offered the choice, you should use the term itself and not its complement. Ask yourself, Am I using a term that points to anything else in the universe of discourse, or am I picking out a specific class? This is what points us to the proper translation of 3, which is “No Baptists are drinkers.” Thus, we find a rule for translating E statements:
When there is a choice, you should always use the affirmative form of the predicate rather than its complement (use “are _____” rather than “are non-_____”), so that the negations are as much as possible expressed by the form of the categorical rather than by the predicates.
The same goes for “No cats are vegetarians.” We could say, “All cats are non-vegetarians,” but then we’d be using the complement term as a predicate, which is not preferred.
Notice how “many,” “most,” “some,” “a few,” and “at least one” all translate the same way as “some.” Because of this, we have to be very careful in figuring out how these statements fit into arguments (as demonstrated above with the cases of Kristin as a parenting professor and dedicated Manitoban nurses).
Example 1 builds on things we have already talked about in terms of reference. When we say, “Not all animals can fly,” we know that there are animals that do and don’t fly. Translating this sentence to “Some animals are not fliers” only works because the subject term “animals” has actual reference. If it is an empty subject term, or if it is unknown whether the subject term class has members, the translation would be illegitimate. Consider: “Not all unicorns are white” cannot necessarily be translated to “Some unicorns are not white” unless we restrict the universe of discourse specifically to a world in which unicorns do exist. Otherwise, what 1–4 demonstrate is that classes should express positive properties and the relation of “not” should be represented by where you put the X on the graph (O statement).
10.6 Steps in Translations
- 1. Rephrase the subject and predicate terms so that they refer to classes. Many sentences in English have adjectives as their grammatical predicates. These should be rewritten as noun phrases; thus “Some clowns are funny” becomes “Some clowns are funny people,” “All oceans are large” becomes “All oceans are large bodies of water,” and so on.
- 2. If the verb in the statement is not the copula, rewrite the verb or verb phrase so that it takes the copula noun-phrase form (conjugation of “to be,” meaning “are”). Use the copula and a noun phrase that captures the sense of the verb (in short use these forms: “are [noun phrase]” or “are not [noun phrase]”); thus “Fish swim” becomes “All fish are swimmers,” “Some newlyweds take vacations” becomes “Some newlyweds are people who take vacations,” and so on. Do not use the complement of classes in your translations of non-phrases (e.g., do not translate “fish swim” as “No fish are non-swimmers”).
- 3. Insert the right quantifiers. Pay close attention to the context, and make sure to get the quantity of the categorical right. Thus “Dogs are mammals” is universal. It is a definitional or classificatory claim and so should be written as “All dogs are mammals,” but “Bankers are conservatives” should be written as “Some bankers are conservatives” because it is implicitly a claim about what most or at least many bankers are like and is not a universal law or definitional claim about all bankers. When in doubt, look at the argument and ask yourself which translation is most well suited to the context of the argument being made.
- 4. Finally, treat statements about individuals as universal claims about the unit class in question. So “President Bush is a Christian” would be written as “All people identical to President Bush are Christians,” and “Ottawa is the capital of Canada” would be rewritten as “All places identical with Ottawa are places identical with the capital of Canada,” and “This beer doesn’t taste good” would become “No things identical with this beer are good tasting things.”
Key Takeaways
- • Translation is necessary in order to regiment the argument by helping us control for the effect of background and contextual knowledge required to understand certain claims.
- • Three problems that arise with translations: empty terms, interpretations of “some,” and direct singular reference.
- • When translating, use the affirmative form of the predicate rather than its complement.
- • There are four steps in translations: rephrase the subject and predicate terms, rewrite the verb so that it takes the copula noun-phrase form, insert the right quantifiers, and treat statements about individuals as universal claims.
Exercises
Part I. Categorical Statement Practice
Identify the form (A, E, I, or O) of these statements and put them in standard categorical form.
- 1. Only doctors are surgeons.
- 2. Mustangs are Fords.
- 3. Students often bike to school.
- 4. There are polar bears in Canada.
- 5. Some polar bears do not live in Canada.
- 6. If not you, I’ll have no friend.
- 7. Everything worth doing is worth doing well.
- 8. Paris is beautiful.
- 9. This swamp isn’t beautiful.
Part II. Categorical Arguments in Standard Form
Identify the three statements (the premises and conclusion) in these arguments, translate them appropriately, and put the premises and conclusion into the standard form.
- 1. Bananas are delicious, but rotten bananas are not, so some bananas are not rotten.
- 2. Stephen Harper is the prime minister, and Stephen Harper is anglophone, so some prime ministers are anglophone.
- 3. (In the TV show Buffy the Vampire Slayer): Angel is a vampire with a soul, and no one with a soul is totally evil, so some vampires are not totally evil.
- 4. Willow branches are weak, and the weak always fail, so some willow branches fail.
- 5. The melting point of tin is 232° C, and some of my pots are tin, so they melt at 232° C.
- 6. The monsters under your bed are afraid when your teddy is in your bed, and your teddy is here in bed with you, so no monsters will come out from under your bed tonight. (Hint: Remember that you need to translate this using only three terms so you will need to be creative.)
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