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Assignment ContentFor this assignment, you will prove the validity of a claim by presenting concrete information that can be used to persuade your audience. You will have an opportunity to argue for or against a specific business decision and provide evidence to support your position.Write a minimum 800 word argumentative essay on a business decision that you would like to propose. Include the following:Explain the business decision.Discuss why the business decision is valid or not valid based on your argument.Identify the logical structure of each argument within the essay by placing the number in bold at the beginning of each argument.For example (not to be included as part of your assignment):Scenario:Recently the city has shared there will be a water restriction to all homeowners for the next 12 months for conservation efforts. Residents will be allowed to water their yards one time per month. Argument:As a homeowner, I am in clear disagreement with the city’s new water restriction for homeowners. If I am unable to water the yard, (1) then my flowers will die, resulting in a negative return on investment. (2) The dry land may cause foundation issues to the property, resulting in potential repair costs. If I am unable to water the lawn, (3) the dry yard and soil will not provide a good curb appeal for the property, which will negatively affect resale value. Because of this, I do not agree with the water restriction imposed by the city. I propose a more lenient water conservation program, such as the one implemented by XYZ County. XYZ County’s water restriction program allowed residents to water twice per week.Cite at least 1 reference.What is on the Reference page must be cited in the paper.Format your assignment according to APA guidelines.
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Deductive Arguments I 9
Categorical Logic
Students will learn to . . .
1.Recognize the four types of categorical claims and the Venn diagrams that represent
them
2.Translate claims into standard form
3.Use the square of opposition to identify logical relationships between
corresponding categorical claims
4.Use conversion, obversion, and contraposition with standard form to make valid
arguments
5.Recognize and evaluate the validity of categorical syllogisms
The Science of Deduction and Analysis is one which can only be acquired by long and patient
study, nor is life long enough to allow any mortal to attain the highest possible perfection in it.
—From an article by Sherlock Holmes,
in A Study in Scarlet by Sir Arthur Conan Doyle
F
ortunately, the greatest detective was exaggerating in this quotation. While it may be that few
of us mortals will attain “the highest possible perfection” in “the Science of Deduction,” most of
us can learn quite a bit in a short time if we put our minds to it. In fact, you already have an
understanding of the basics from Chapter 2.* In this chapter and the next, you’ll learn two kinds
of techniques for making and evaluating deductive inferences—in other words, arguments.
If you flip through the pages of this chapter, you’ll see diagrams with circles and Xs, and in the
next chapter you will see weird symbols that remind some people of mathematics. These pages
may look intimidating, but there’s nothing complicated about them if you keep in mind that each
paragraph builds on those that Page 243 came before. If you try to skip ahead, you may have
trouble. We recommend reading slowly, carefully, and thoughtfully, and even taking a break now
and then.
You may remember from Chapter 2 that deductive arguments depend on the meanings of the
words that occur in their premises for their validity. In particular, it is words like “all,” “and,” “or,”
“if . . . then . . .” that carry this burden. We’ll see how this works as we go along. Recall also that
valid deductive arguments actually prove or demonstrate their conclusion—that is, if the
premises of such an argument are true, the conclusion cannot fail to be true as well. That said,
we’ll move on to our treatment of the first general type of deductive argument.
The first topic we’ll take up is categorical logic. Categorical logic is logic based on the relations
of inclusion and exclusion among classes (or “categories”) as stated in categorical claims. Its
methods date back to the time of Aristotle, and it was the principal form that logic took among
most knowledgeable people for more than 2,000 years. During that time, all kinds of bells and
whistles were added to the basic theory, especially by monks and other scholars during the
medieval period. So as not to weigh you down with unnecessary baggage, we’ll just set forth the
basics of the subject in what follows.
Like propositional logic, the subject of the next chapter, categorical logic is useful in clarifying
and analyzing deductive arguments. But there is another reason for studying the subject: There is
no better way to understand the underlying logical structure of our everyday language than to
learn how to put it into the kinds of formal terms we’ll introduce in these chapters.
To test your analytical ability, take a look at these claims. Just exactly what is the difference
between them?
(1)Everybody who is ineligible for Physics 1A must take Physical Science 1.
(2)No students who are required to take Physical Science 1 are eligible for Physics 1A.
■For over a hundred years, the symbol of “the Science of Deduction.” The smaller photo has Holmes and Watson
portrayed by Basil Rathbone and Nigel Bruce, the classic cast in over a dozen films. Benedict Cumberbatch and
Martin Freeman have starred in some excellent recent films about the famous detective.
Page 244 Here’s another pair of claims:
(3)Harold won’t attend the meeting unless Vanessa decides to go.
(4)If Vanessa decides to go, then Harold will attend the meeting.
You might be surprised at how many college students have a hard time trying to determine whether
the claims in each pair mean the same thing or something different. In this chapter and the next,
you’ll learn a foolproof method for determining how to unravel the logical implications of such
claims and for seeing how any two such claims relate to each other. (Incidentally, claims 1 and 2
do not mean the same thing at all, and neither do 3 and 4.) If you’re signing a lease or entering into
a contract of any kind, it pays to be able to figure out just what is said in it and what is not; those
who have trouble with claims like the ones above risk being left in the dark.
Studying categorical and truth-functional logic can teach us to become more careful and precise
in our own thinking. Getting comfortable with this type of thinking can be helpful in general, but
for those who will someday apply to law school, medical school, or graduate school, it has the
added advantage that many admission exams for such programs deal with the kinds of reasoning
discussed in these chapters.
Let’s start by looking at the four basic kinds of claims on which categorical logic is based.
CATEGORICAL CLAIMS
In logic, a category is a group or a class or a population; any bunch of things can serve as a
category for our purposes. Terms are noun phrases, like “dogs,” “cats,” “Christians,” “Arabs,”
“people who read logic books,” and so on. These terms are labels for categories (or classes or
populations, all of which for our purposes are the same thing). There are all kinds of ways to
express claims about categories, but we are interested in four standard-form types of sentences—
the names of which are simple: A, E, I, and O—sentences that result from putting terms in the
blanks of the following:
A: All _____ are _____.
(Example: All pianists are musicians.)
E: No _____ are _____.
(Example: No otterhounds are pianists.)
I: Some _____ are _____.
(Example: Some musicians are prodigies.)
O: Some _____ are not _____.
(Example: Some politicians are not criminals.*)
The phrases that go in the blanks are terms; the one that goes into the first blank is the subject
term of the claim, and the one that goes into the second blank is the predicate term. Thus,
“musicians” is the predicate term of the first example and the subject term of the third example. In
many of the examples and explanations that follow, we’ll use the letters S and P (for “subject” and
“predicate”) to stand for terms in categorical claims. And we’ll talk about the subject and
predicate classes, which are just the classes (or populations) the terms refer to. Page 245
But first, a caution: Only nouns and noun phrases will work as terms. An adjective alone, such as
“red,” won’t do. “All fire engines are red” is not a standard-form categorical claim, because “red”
by itself does not name a class of things.
Looking back at the standard-form structures just given, notice that each one has a letter to its left.
These are the traditional names of the four types of standard-form categorical claims. The claim
“All pianists are musicians” is an A-claim, and so are “All idolaters are heathens,” “All people
born after the year 2000 are millennials,” and any other claim of the form “All S are P.” The same
is true for the other three letters, E, I, and O, and the other three kinds of claims.
Venn Diagrams
Each of the standard forms has its own visual illustration in a Venn diagram, as shown
in Figures 1 through 4. Named after British logician John Venn, these diagrams graphically
represent the four standard-form categorical claim types.
The following are conventions that govern the diagrams:
1.Circles represent classes (“categories”).
2.A shaded area indicates that nothing is in it. It means that this part of a class (or classes, if the
area is where two or more circles overlap) is “empty.”
3.An X in an area indicates that at least one thing is in that part of a class or classes.
4.An area that is blank does NOT mean that it is empty. It means we have no information about
that part of the category or class.
Now let’s understand the diagrams:
1. Figure 1 is the diagram for A-claims. It shows that All S are P. It shows this by shading out
all the S area that is outside the P area, thus showing that if anything is an A it must be in
the P area.
2. Figure 2 is the diagram for E-claims. It shows that No S are P, by showing that the
category “SP” is empty.
3. Figure 3 is the diagram for I-claims. It shows that Some S are P, by placing an X in the
“SP” area. For our purposes, the word “some” means “at least one.”
4. Figure 4 is the diagram for O-claims. It shows that there is at least one thing that is an S
but which is outside the P area.
FIGURE 1 A-claim: All S are P.
FIGURE 2 E-claim: No S are P.
FIGURE 3 I-claim: Some S are P.
FIGURE 4 O-claim: Some S are not P.
Finally, notice that the A- and I-claims are affirmative, and the E- and O-claims are negative.
Although there are only four standard-form claim types, it’s remarkable how versatile they are. A
large portion of what we want to say can be rewritten, or “translated,” Page 246 into one or another
of them. Because this task is sometimes easier said than done, we’d best spend a little while making
sure we understand how to do it. And we warn you: A lot of standard-form translations are not
pretty—but it’s accuracy we seek, not style.
TRANSLATION INTO STANDARD FORM (INTRODUCTION)
The idea here is to turn an ordinary claim into a standard-form categorical claim that is equivalent.
We’ll say that two claims are equivalent claims if, and only if, they would be true in exactly the
same circumstances—that is, under no circumstances could one of them be true and the other false.
(You can think of such claims as “saying the same thing.”)
Lots of ordinary claims in English are easy to translate into standard form. A claim of the sort
“Every X is a Y,” for example, turns into the standard-form A-claim “All Xs are Ys.” And “Minors
are not eligible” turns into the E-claim “No minors are eligible people.”
All standard-form claims are in the present tense, but even so, we can use them to talk about the
past. For example, we can translate “There were creatures weighing more than four tons that lived
in North America” as “Some creatures that lived in North America are creatures that weighed more
than four tons.”
Translating Claims in Which the Word “Only” or the Phrase “The Only” Occurs
The word “only” is not only versatile (see the box on the next page), it can cause confusion when
it occurs in a claim you need to translate into standard form. An example:
ORIGINAL: Only sophomores are eligible candidates.
A careful reading and a moment’s thought will probably indicate that this should be an A-claim.
But we have to decide between
INCORRECT TRANSLATION: All sophomores are eligible candidates.
and
CORRECT TRANSLATION: All eligible candidates are sophomores.
These claims are very different, and only one of them say the same thing as our original statement.
Notice that the original says something about everyeligible Page 247 candidate; we’re saying
something about all eligible candidates. So eligible candidates form the subject class of the Aclaim. The correct translation is the claim at the bottom of the preceding page. Notice that the word
“only” introduced the class of sophomores in the original sentence. That provides us with a general
rule:
The word “only,” used by itself, introduces the predicate term of an A-claim.
Now look at this example:
ORIGINAL: The only people admitted are people over twenty-one.
Here, the class being restricted is the class of people admitted, right? Every one of them has to be
over twenty-one. So we’re talking about all people admitted:
TRANSLATION: All people admitted are people over twenty-one.
And this is always the case when a term in an A-claim is introduced by the phrase “the only”; it
works in the opposite way from how the word “only” works by itself:
The phrase “the only” introduces the subject term of an A-claim.
Note that, in accordance with these rules, we would translate both of these claims:
ORIGINAL: Only matinees are half-price shows.
and
ORIGINAL: Matinees are the only half-price shows.
as
TRANSLATION: All half-price shows are matinees.
Translating Claims About Times and Places
Sometimes statements that look to be about one thing need to be interpreted to be about something
else in order to make them work as standard form categorical claims. For example:
ORIGINAL: I always get nervous when I take logic exams.
This looks like a claim about the speaker, but it is best seen as a claim about times or occasions.
The speaker is saying, “Every time I take a logic exam is a time I get nervous,” which of course
translates into standard form:
TRANSLATION: All times I take logic exams are times I get nervous.
Page 248
The Most Versatile Word in English
Only one word can be placed anywhere in the following sentence (at the beginning, at the end,
or between any other two words) and still make sense:
I gave my son the money he needed yesterday.
That word is “only,” the most versatile word in the English language. Each placement, with a
hint as to the different meanings produced, is given below:
1.Only
I
gave
my
(Nobody else gave him any.)
son
the
money
he
needed
yesterday.
2.I
only
gave
my
son
the
money
(He wanted the car, too, but I would not give it to him.)
he
needed
yesterday.
3.I
gave
only
my
son
the
(His friend wanted money too, but I refused.)
he
needed
yesterday.
money
4.I
gave
my
only
son
the
(I have a daughter, but only the one son.)
money
he
needed
yesterday.
5.I
gave
my
son
only
(He’ll need more for tomorrow.)
the
money
he
needed
yesterday.
only
money
he
needed
yesterday.
7.I
gave
my
son
the
money
(Everybody else must have had their money.)
only
he
needed
yesterday.
8.I
gave
my
son
the
money
(He won’t need any money the rest of the week.)
he
only
needed
yesterday.
9.I
gave
my
son
the
(And already he’s needing more?)
money
he
needed
only
yesterday.
10.I
gave
my
son
the
money
(And I told him that was the first and last time.)
he
needed
yesterday
6.I
gave
my
son
(He didn’t need much. . . .)
the
only.
—Based on an idea from Marilyn Vos Savant, author of the “Ask Marilyn” column
in Parade Magazine
The word “whenever” can be a good clue that you have such a claim on your hands. For example:
Whenever Peg shows up, Dick gets anxious.
The word “whenever” tells us two things: that we probably have a claim about times or occasions,
and that the term it introduces will be the subject term. There can be Page 249 exceptions to this,
depending on how “whenever” is used, but this rule of thumb is always worth keeping in mind.
Places are handled much like times in categorical claims. Consider this:
ORIGINAL: It’s snowing everywhere in Massachusetts.
This is about snow, and it’s about today, but we’re best seeing it as about places: places where it’s
snowing and places in Massachusetts. Once you see it that way, it’s easily translated into standard
form:
TRANSLATION: All places in Massachusetts are places it’s snowing.
The word “wherever” is an indicator for places in the way that “whenever” is for times. When you
see it, it is likely that it introduces the subject term of an A-claim. For example:
The lamb goes wherever Bo Peep goes.
This clearly should translate as “All places Bo Peep goes are places the lamb goes.”
So, to recap the two rules of thumb for this section:
“Whenever” usually introduces the subject term of an A-claim about times;
“wherever” usually introduces the subject term of an A-claim about places.
Translating Claims About Specific Individuals
The next claims that can confuse efforts at translation are claims about one individual person,
thing, event, and so on. Consider the example, Aristotle is a logician.
It’s clear that this claim specifies a class, “logicians,” and places Aristotle as a member of that
class. The problem is that categorical claims are always about two classes, and Aristotle isn’t a
class. (We certainly don’t talk about some of Aristotle being a logician.) What we want to do is
treat such claims as if they were about classes with exactly one member—in this case, Aristotle.
One way to do this is to use the term “people who are identical with Aristotle,” which of course
has only Aristotle as a member. The important thing to remember about such claims can be
summarized in the following rule:
Claims about single individuals should be treated as A-claims or E-claims.
So our example
ORIGINAL: Aristotle is a logician.
Page 250 can be translated as:
TRANSLATION: All people identical with Aristotle are logicians.
Which is an A-claim. Similarly, “Aristotle is not left-handed” becomes the E-claim “No people
identical with Aristotle are left-handed people.” (Your instructor may prefer to leave the claim in
its original form and simply treat it as an A-claim or an E-claim. This avoids the awkward “people
identical with Aristotle” wording and is certainly okay with us.)
It isn’t just people that crop up in individual claims. For example, the preferred translation of
ORIGINAL: St. Louis is on the Mississippi.
is
TRANSLATION: All cities identical with St. Louis are cities on the Mississippi.
We warned you that some of these translations would not be pretty.
Translating Claims that Use Mass Nouns
Other claims that cause translation difficulty contain what are called mass nouns. Example:
ORIGINAL: Boiled okra is too ugly to eat.
This claim is about a kind of stuff. The best way to deal with it is to treat it as a claim
about examples of this kind of stuff. The present example translates into an A-claim
about all examples of the stuff in question:
TRANSLATION: All examples of boiled okra are things that are too ugly to eat.
An example such as
ORIGINAL: Most boiled okra is too ugly to eat.
translates into the I-claim,
TRANSLATION: Some examples of boiled okra are things that are too ugly to eat.
As we noted, it’s not possible to give rules or hints about every kind of problem you might run
into when translating claims into standard-form categorical versions. Only practice and discussion
can bring you to the point where you can handle this part of the material with confidence. The best
thing to do now is to turn to some exercises.
Page 251
Exercise 9-1
Translate each of the following into a standard-form claim. Make sure that each answer follows the
exact form of an A-, E-, I-, or O-claim and that each term you use is a noun or noun phrase that
refers to a class of things.

1.Every one of the senators is a politician.
2.Not all senators are politicians.
3.If somebody is a senator, then that person must be a politician.

4.You can be a senator only if you’re a politician.
5.The only politicians I know are senators.
6.Being a senator is all it takes to be a politician.

7.Being a politician is not enough to make you a senator.
8.You can be a senator only if you’re a politician.
9.Nobody who is a politician is also a senator.

10.A few senators are not politicians.
11.There are scholars who are philosophers.
12.There are no philosophers who are not scholars.

13.Philosophers are not the only scholars.
14.Philosophers are the only scholars.
15.Not every scholar is a philosopher.
Exercise 9-2
Follow the directions for the previous exercise. Remember that you’re trying to produce a claim
that’s equivalent to the one given; it doesn’t matter whether the given claim is actually true.

1.Every salamander is a lizard.
2.Not every lizard is …
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