categorical logic critical thinking

6 Categorical Logic

I. venn diagrams 1.

Pictures and diagrams can be very useful in presenting information or assisting reasoning. In this module we shall focus on Venn diagram. They are used to represent classes of objects. We can also use them to evaluate the validity of certain types of arguments.

Venn diagrams are named after the British logician John Venn (1834-1923), a fellow of Gonville and Caius college at Cambridge University. He was also a philosopher and mathematician, a pioneer of logic and probability theory.

II. Basic Notation

1. a class is defined by its members.

Let us start with the concept of a class. A class or a set is simply a collection of objects. These objects are called members of the set. A class is defined by its members. So for example, we might define a class C as the class of black hats. In that case, every black hat in the world is a member of C, and anything that is not a black hat is not a member of C. If something is not a member of a class, we can also say that the object is outside the class.

Note that a class can be empty. The class of men over 5 meters tall is presumably empty since nobody is that tall. The class of plane figures that are both round and square is also empty since nothing can be both round and square. A class can also be infinite, containing an infinite number of objects. The class of even number is an example. It has infinitely many members, including 2, 4, 6, 8, and so on.

2. Classes are represented by circles

As you can see in the diagram above, the class of black hats, C, is represented by a circle. We normally use circles to represent classes in Venn diagrams, though sometimes we also use bounded regions with different shapes, such as ovals.
We can write the name of the class, e.g. “C”, or “Class C”, next to the circle to indicate which class it is.
The area inside the circle represents those things which are members of the class.
The area outside the circle represents those things which are not members of the class, e.g. green hats, keys, cakes, etc.
A Venn diagram is usually enclosed by a rectangular box that represents everything in the world.

3. Use shading to indicate an empty class

Let us now consider what shading means:

To indicate that a class is empty, we shade the circle representing that class. So the diagram above means that class A is empty.

In general, shading an area means that the class represented by the area is empty. So the second diagram above represents a situation where there isn’t anything which is not a member of class A.

However, even though shading indicates emptiness, a region that is not shaded does not necessarily indicate a non-empty class. As we shall see in the next tutorial, we use a tick to indicate existence. So in the second diagram above, the circle marked A is not shaded. This does not imply that there are things which exist which are members of A. If the area is blank, this means that we do not have any information as to whether there is anything there.

III. Everything and nothing

1. intersecting circles.

Now let us consider a slightly more complicated diagram where we have two intersecting circles. The left circle represents class A. The right one represents class B.

Let us label the different bounded regions:

Region 1 represents objects which belong to class A but not to B.
Region 2 represents objects which belong to both A and B.
Region 3 represents objects which belong to B but not A.
Region 4, the area outside the two circles, represents objects that belong to neither A nor B.

Exercise #1

So for example, suppose A is the class of apples, and B is the class of sweet things. In that case what does region 2 represent?

Exercise #2

Furthermore, which region represents the class that contains sour lemons that are not sweet?

2. Everything and nothing

Continuing with our diagram, suppose we now shade region 1. This means that the class of things which belong to A but not B is empty. Or more simply, every A is a B. ( It might be useful to note that this is equivalent to saying that if anything is an A, it is also a B. ) This is an important point to remember. Whenever you want to represent “every A is B”, shade the area within the A circle that is outside the B circle.

What if we shade the middle region where A and B overlaps? This is the region representing things which are both A and B. So shading indicates that nothing is both A and B. If you think about it carefully, you will see that “Nothing is both A and B” says the same thing as “No A is a B” and “No B is an A”. Make sure that you understand why these claims are logically equivalent!

Incidentally, we could have represented the same information by using two non-overlapping circles instead.

IV. Exercises

See if you can explain what each diagram represents.

V . Three circles

So far we have been looking at Venn diagrams with two circles. We now turn to Venn diagrams with three circles. The interpretation of these diagrams is the same as before, with each circle representing a class of objects, and the overlapping area between the circles representing the class of objects that belong to all the classes.

As you can see from the diagram below, with three circles we can have eight different regions, the eighth being the region outside the circles. The top circle represents the class of As, whereas the circles on the left and the right below it represent the class of Bs and Cs respectively. The area outside all the circles represents those objects which are not members of any of these three classes.

Now that you know what each of the region represents, you should know how to use shading to represent situations where “Every X is Y”, or “No X is Y”. As before, shading an area indicates that nothing exists in the class that is represented by the shaded region.

Look at the sentences in the diagram below. Ask yourself which region should be shaded to represent the situation described by the sentence. Then click that sentence and check the answer.

VI. Existence

We have seen how to use shading to indicate that there is nothing in the class represented by the shaded region. We now see how to use ticks to indicate existence. The basic idea is that when a tick is present in a region, it indicates that there is something in the class represented by the region. So for example, in the diagram below, we have a tick outside the circles. Since the area outside the circle represents the class of things that are neither A, nor B, nor C, the diagram is saying that something exists that is neither A nor B nor C:

There are two important points to remember :

A tick in a region says that there is something in the class represented by the region. It does not say how many things there are in that class. There might be just one, or perhaps there are many.
A region without a tick does not represent an empty class. Without a tick, a blank region provides no information as to whether anything exists in the class it represents. Only when a region is shaded can we say that it represents an empty class.

What about the following diagram? What does it represent?

The diagram above does NOT say “something is A”. Actually it says something more specific, namely that “something is A but is not B and not C”. If you have given the wrong answer, you might be thinking that the tick indicates that there is something in the class represented by the A circle. But here we use a tick to indicate existence in the class represented by the smallest bounded region that encloses the tick . In the top diagram of this page the smallest bounded area that encloses the tick is the area outside the three circles. In the diagram above, although circle A does enclose the tick, it is not the smallest bounded area that does that. That smallest region is the colored one in this diagram :

Now see if you can determine what these diagrams indicate.

Notice in the last diagram above, the two ticks indicate that there are two different things. What if you just want to say “Something is C but not A”? The way to do this is to put a tick across two bounded regions, as follows:

E xercise #5

The interpretation of this diagram employs the same rule as before. What the tick indicates is that there is something in the smallest closed region (the colored area) that encloses the tick. Of course, the bigger C circle also completely encloses the tick, but it is not the smallest bounded region that does that. So the tick does not mean that “something is C”.

Notice that the tick does not tell us whether there is anything that is B, because it is not completely enclosed by the B circle.

See if you can explain why these diagrams represent:

So far we have used ticks to cut across only two bounded regions. But of course there are other possibilities:

What do you think this means? Applying the same rule of interpretation as before, we see that the smallest closed region that encloses the big tick would have to be the combined three regions which the tick spreads across. This combined region represents things which are either B or C (or both), but which are not A. So what the diagram says is that there is something of this kind.

So what if we just want to represent the fact that something is A? Here is one way to draw the diagram: Notice that the tick cuts across all the different regions within the A circle, and is completely enclosed by it.

We can now combine what we have learnt about ticks and shading together. Suppose we start with the information that something is both A and C. We therefore draw the following diagram :

Now suppose we are also told that every C is a B. So we add the additional information by shading the appropriate area, and end up with this diagram :

How should this be interpreted and what should we conclude? Half of the green tick is in a shaded region. What does that mean? Give yourself a minute to think about it before you read on …

The answer is actually quite simple. The tick indicates that something is both A and C, and it occupies two separate regions. The left hand side region represents things that are A, B and C. The right hand side region represents things that are A and C but not B. Since the tick crosses these two regions, it indicates that there is something either in the class represented by the left region or in the class represented by the right region (or both of course). Shading tells us that there is nothing in the class represented by the right region. So whatever that exists according to the tick must be in the class represented by the left region. In other words, we can conclude that something is A, B, and C. In effect then, shading “moves” the tick into the left region since it tells us that there is nothing on the right. The above diagram is therefore equivalent to the following one :

So here is a general principle you should remember:

A truncated tick within a region R counts as a complete tick in R if part of the tick is in R and all other parts not in R are in shaded regions.

Is the Statement “something is either B or C” true according to the diagram?

Is the diagram consistent with the statement “Everything is B or not C, or both”?

Exercise #3

Is the diagram consistent with the statement “Something is B and not A”?

Exercise #4

Is the diagram consistent with the statement “Everything is A or C”?

Exercise #5

What does the diagram tell us?

VII. Syllogism

We now see how Venn diagrams can be used to evaluate certain arguments. There are many arguments that cannot be analysed using Venn diagrams. So we shall restrict our attention only to arguments with these properties:

The argument has two premises and a conclusion.
The argument mentions at most three classes of objects.
The premises and the conclusion include only statements of the following form: Every X is Y, Some X is Y, No X is Y. Here are two examples :

(Premise #1) Every whale is a mammal.

(Premise #2) Every mammal is warm-blooded.

(Conclusion) Every whale is warm-blooded.

(Premise #1) Some fish is sick.

(Premise #2) No chicken is a fish.

(Conclusion) No chicken is sick.

These arguments are sometimes known as syllogisms . What we want to determine is whether they are valid . In other words, we want to find out whether the conclusions of these arguments follow logically from the premises. To evaluate validity, we want to check whether the conclusion is true in a diagram where the premises are true. Here is the procedure to follow:

Draw a Venn diagram with 3 circles.

Represent the information in the two premises.

Draw an appropriate outline for the conclusion. Fill in the blank in “If the conclusion is true according to the diagram, the outlined region should.”

See whether the condition that is written down is satisfied. If so, the argument is valid. Otherwise not.

1. Example #1

Let us apply this method to the first argument on this page :

Step 1 : We use the A circle to represent the class of whales, the B circle to represent the class of mammals, and the C circle to represent the class of warm-blooded animals.

Step 2a : We now represent the information in the first premise. (Every whale is a mammal.)

Step 2b : We now represent the information in the second premise. (Every mammal is warm-blooded.)

Step 3 : We now draw an outline for the area that should be shaded to represent the conclusion. (Every whale is warm-blooded.) This is the red outlined region. We write: “If the conclusion is true according to the diagram, the outlined region should be shaded.”

Step 4 : Since this is indeed the case, this means that whenever the premises are true, the conclusion must also be true. So the argument is valid.

2. Example #2

Let’s go through another example:

Every A is B.

Some B is C.

Therefore, some A is C.

We now draw a Venn diagram to represent the two premises:

In the diagram above, we have already drawn a Venn diagram for the three classes and encode the information in the first two premises. To carry out the third step, we need to draw an outline for the conclusion. Do you know where the outline should be drawn?

3. Example #3

Some A is B.

Every B is C.

Step 1 : Representing the first premise.

Step 2 : Representing the second premise.

Step 3 : Add an outline for conclusion

VIII. Limitations of Venn diagrams

Although Venn diagrams can help us reason about classes of objects, they also have many limitations. First of all, the diagrams can become too complicated to deal with if we are reasoning about many classes of objects. So far in our tutorials we have considered Venn diagrams with at most three circles. It is possible to add more bounded regions if we are dealing with more than three classes, but then the resulting diagrams will become rather difficult to handle and interpret. It is very easy to make mistakes when we encode information in such diagrams.

The other problem with Venn diagrams is that they have limited expressive power . What this means is that there are many pieces of information that cannot be accurately represented. For example, our system of notation allows us to talk about classes of objects, but not particular individual objects. For example, to say that a and b are cats and c and d are not, we might have to introduce new symbols, using dots to represent individuals, as in the diagram below:

However, even with this new notation, there are still other pieces of information that cannot be represented, such as:

Either Felix is a cat or it is a dog.
If Peter is taller than Mary then Peter is older than Mary.

Perhaps it might be possible to introduce additional new symbols to represent such ideas. But then the system of Venn diagrams will get really complicated and difficult to use. So now that we know the limitations of Venn diagrams, we should be in a better position to know when they are useful and when they are not.

Is this a suitable Venn diagram for showing the relationships between four sets of objects?

In the second diagram, there are four overlapping rectangles. Which area corresponds to those items which are A, B and D, but not C?

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Introduction to Logic and Critical Thinking

(10 reviews)

Matthew Van Cleave, Lansing Community College

Publisher: Matthew J. Van Cleave

Language: English

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Reviewed by "yusef" Alexander Hayes, Professor, North Shore Community College on 6/9/21

Formal and informal reasoning, argument structure, and fallacies are covered comprehensively, meeting the author's goal of both depth and succinctness. read more

Comprehensiveness rating: 5 see less

Formal and informal reasoning, argument structure, and fallacies are covered comprehensively, meeting the author's goal of both depth and succinctness.

Content Accuracy rating: 5

The book is accurate.

Relevance/Longevity rating: 5

While many modern examples are used, and they are helpful, they are not necessarily needed. The usefulness of logical principles and skills have proved themselves, and this text presents them clearly with many examples.

Clarity rating: 5

It is obvious that the author cares about their subject, audience, and students. The text is comprehensible and interesting.

Consistency rating: 5

The format is easy to understand and is consistent in framing.

Modularity rating: 5

This text would be easy to adapt.

Organization/Structure/Flow rating: 5

The organization is excellent, my one suggestion would be a concluding chapter.

Interface rating: 5

I accessed the PDF version and it would be easy to work with.

Grammatical Errors rating: 5

The writing is excellent.

Cultural Relevance rating: 5

This is not an offensive text.

Reviewed by Susan Rottmann, Part-time Lecturer, University of Southern Maine on 3/2/21

Comprehensiveness rating: 4 see less

I reviewed this book for a course titled "Creative and Critical Inquiry into Modern Life." It won't meet all my needs for that course, but I haven't yet found a book that would. I wanted to review this one because it states in the preface that it fits better for a general critical thinking course than for a true logic course. I'm not sure that I'd agree. I have been using Browne and Keeley's "Asking the Right Questions: A Guide to Critical Thinking," and I think that book is a better introduction to critical thinking for non-philosophy majors. However, the latter is not open source so I will figure out how to get by without it in the future. Overall, the book seems comprehensive if the subject is logic. The index is on the short-side, but fine. However, one issue for me is that there are no page numbers on the table of contents, which is pretty annoying if you want to locate particular sections.

Content Accuracy rating: 4

I didn't find any errors. In general the book uses great examples. However, they are very much based in the American context, not for an international student audience. Some effort to broaden the chosen examples would make the book more widely applicable.

Relevance/Longevity rating: 4

I think the book will remain relevant because of the nature of the material that it addresses, however there will be a need to modify the examples in future editions and as the social and political context changes.

Clarity rating: 3

The text is lucid, but I think it would be difficult for introductory-level students who are not philosophy majors. For example, in Browne and Keeley's "Asking the Right Questions: A Guide to Critical Thinking," the sub-headings are very accessible, such as "Experts cannot rescue us, despite what they say" or "wishful thinking: perhaps the biggest single speed bump on the road to critical thinking." By contrast, Van Cleave's "Introduction to Logic and Critical Thinking" has more subheadings like this: "Using your own paraphrases of premises and conclusions to reconstruct arguments in standard form" or "Propositional logic and the four basic truth functional connectives." If students are prepared very well for the subject, it would work fine, but for students who are newly being introduced to critical thinking, it is rather technical.

It seems to be very consistent in terms of its terminology and framework.

Modularity rating: 4

The book is divided into 4 chapters, each having many sub-chapters. In that sense, it is readily divisible and modular. However, as noted above, there are no page numbers on the table of contents, which would make assigning certain parts rather frustrating. Also, I'm not sure why the book is only four chapter and has so many subheadings (for instance 17 in Chapter 2) and a length of 242 pages. Wouldn't it make more sense to break up the book into shorter chapters? I think this would make it easier to read and to assign in specific blocks to students.

Organization/Structure/Flow rating: 4

The organization of the book is fine overall, although I think adding page numbers to the table of contents and breaking it up into more separate chapters would help it to be more easily navigable.

Interface rating: 4

The book is very simply presented. In my opinion it is actually too simple. There are few boxes or diagrams that highlight and explain important points.

The text seems fine grammatically. I didn't notice any errors.

The book is written with an American audience in mind, but I did not notice culturally insensitive or offensive parts.

Overall, this book is not for my course, but I think it could work well in a philosophy course.

Reviewed by Daniel Lee, Assistant Professor of Economics and Leadership, Sweet Briar College on 11/11/19

This textbook is not particularly comprehensive (4 chapters long), but I view that as a benefit. In fact, I recommend it for use outside of traditional logic classes, but rather interdisciplinary classes that evaluate argument read more

Comprehensiveness rating: 3 see less

To the best of my ability, I regard this content as accurate, error-free, and unbiased

The book is broadly relevant and up-to-date, with a few stray temporal references (sydney olympics, particular presidencies). I don't view these time-dated examples as problematic as the logical underpinnings are still there and easily assessed

Clarity rating: 4

My only pushback on clarity is I didn't find the distinction between argument and explanation particularly helpful/useful/easy to follow. However, this experience may have been unique to my class.

To the best of my ability, I regard this content as internally consistent

I found this text quite modular, and was easily able to integrate other texts into my lessons and disregard certain chapters or sub-sections

The book had a logical and consistent structure, but to the extent that there are only 4 chapters, there isn't much scope for alternative approaches here

No problems with the book's interface

The text is grammatically sound

Cultural Relevance rating: 4

Perhaps the text could have been more universal in its approach. While I didn't find the book insensitive per-se, logic can be tricky here because the point is to evaluate meaningful (non-trivial) arguments, but any argument with that sense of gravity can also be traumatic to students (abortion, death penalty, etc)

No additional comments

Reviewed by Lisa N. Thomas-Smith, Graduate Part-time Instructor, CU Boulder on 7/1/19

The text covers all the relevant technical aspects of introductory logic and critical thinking, and covers them well. A separate glossary would be quite helpful to students. However, the terms are clearly and thoroughly explained within the text,... read more

The content is excellent. The text is thorough and accurate with no errors that I could discern. The terminology and exercises cover the material nicely and without bias.

The text should easily stand the test of time. The exercises are excellent and would be very helpful for students to internalize correct critical thinking practices. Because of the logical arrangement of the text and the many sub-sections, additional material should be very easy to add.

The text is extremely clearly and simply written. I anticipate that a diligent student could learn all of the material in the text with little additional instruction. The examples are relevant and easy to follow.

The text did not confuse terms or use inconsistent terminology, which is very important in a logic text. The discipline often uses multiple terms for the same concept, but this text avoids that trap nicely.

The text is fairly easily divisible. Since there are only four chapters, those chapters include large blocks of information. However, the chapters themselves are very well delineated and could be easily broken up so that parts could be left out or covered in a different order from the text.

The flow of the text is excellent. All of the information is handled solidly in an order that allows the student to build on the information previously covered.

The PDF Table of Contents does not include links or page numbers which would be very helpful for navigation. Other than that, the text was very easy to navigate. All the images, charts, and graphs were very clear

I found no grammatical errors in the text.

Cultural Relevance rating: 3

The text including examples and exercises did not seem to be offensive or insensitive in any specific way. However, the examples included references to black and white people, but few others. Also, the text is very American specific with many examples from and for an American audience. More diversity, especially in the examples, would be appropriate and appreciated.

Reviewed by Leslie Aarons, Associate Professor of Philosophy, CUNY LaGuardia Community College on 5/16/19

This is an excellent introductory (first-year) Logic and Critical Thinking textbook. The book covers the important elementary information, clearly discussing such things as the purpose and basic structure of an argument; the difference between an argument and an explanation; validity; soundness; and the distinctions between an inductive and a deductive argument in accessible terms in the first chapter. It also does a good job introducing and discussing informal fallacies (Chapter 4). The incorporation of opportunities to evaluate real-world arguments is also very effective. Chapter 2 also covers a number of formal methods of evaluating arguments, such as Venn Diagrams and Propositional logic and the four basic truth functional connectives, but to my mind, it is much more thorough in its treatment of Informal Logic and Critical Thinking skills, than it is of formal logic. I also appreciated that Van Cleave’s book includes exercises with answers and an index, but there is no glossary; which I personally do not find detracts from the book's comprehensiveness.

Overall, Van Cleave's book is error-free and unbiased. The language used is accessible and engaging. There were no glaring inaccuracies that I was able to detect.

Van Cleave's Textbook uses relevant, contemporary content that will stand the test of time, at least for the next few years. Although some examples use certain subjects like former President Obama, it does so in a useful manner that inspires the use of critical thinking skills. There are an abundance of examples that inspire students to look at issues from many different political viewpoints, challenging students to practice evaluating arguments, and identifying fallacies. Many of these exercises encourage students to critique issues, and recognize their own inherent reader-biases and challenge their own beliefs--hallmarks of critical thinking.

As mentioned previously, the author has an accessible style that makes the content relatively easy to read and engaging. He also does a suitable job explaining jargon/technical language that is introduced in the textbook.

Van Cleave uses terminology consistently and the chapters flow well. The textbook orients the reader by offering effective introductions to new material, step-by-step explanations of the material, as well as offering clear summaries of each lesson.

This textbook's modularity is really quite good. Its language and structure are not overly convoluted or too-lengthy, making it convenient for individual instructors to adapt the materials to suit their methodological preferences.

The topics in the textbook are presented in a logical and clear fashion. The structure of the chapters are such that it is not necessary to have to follow the chapters in their sequential order, and coverage of material can be adapted to individual instructor's preferences.

The textbook is free of any problematic interface issues. Topics, sections and specific content are accessible and easy to navigate. Overall it is user-friendly.

I did not find any significant grammatical issues with the textbook.

The textbook is not culturally insensitive, making use of a diversity of inclusive examples. Materials are especially effective for first-year critical thinking/logic students.

I intend to adopt Van Cleave's textbook for a Critical Thinking class I am teaching at the Community College level. I believe that it will help me facilitate student-learning, and will be a good resource to build additional classroom activities from the materials it provides.

Reviewed by Jennie Harrop, Chair, Department of Professional Studies, George Fox University on 3/27/18

While the book is admirably comprehensive, its extensive details within a few short chapters may feel overwhelming to students. The author tackles an impressive breadth of concepts in Chapter 1, 2, 3, and 4, which leads to 50-plus-page chapters that are dense with statistical analyses and critical vocabulary. These topics are likely better broached in manageable snippets rather than hefty single chapters.

The ideas addressed in Introduction to Logic and Critical Thinking are accurate but at times notably political. While politics are effectively used to exemplify key concepts, some students may be distracted by distinct political leanings.

The terms and definitions included are relevant, but the examples are specific to the current political, cultural, and social climates, which could make the materials seem dated in a few years without intentional and consistent updates.

While the reasoning is accurate, the author tends to complicate rather than simplify -- perhaps in an effort to cover a spectrum of related concepts. Beginning readers are likely to be overwhelmed and under-encouraged by his approach.

Consistency rating: 3

The four chapters are somewhat consistent in their play of definition, explanation, and example, but the structure of each chapter varies according to the concepts covered. In the third chapter, for example, key ideas are divided into sub-topics numbering from 3.1 to 3.10. In the fourth chapter, the sub-divisions are further divided into sub-sections numbered 4.1.1-4.1.5, 4.2.1-4.2.2, and 4.3.1 to 4.3.6. Readers who are working quickly to master new concepts may find themselves mired in similarly numbered subheadings, longing for a grounded concepts on which to hinge other key principles.

Modularity rating: 3

The book's four chapters make it mostly self-referential. The author would do well to beak this text down into additional subsections, easing readers' accessibility.

The content of the book flows logically and well, but the information needs to be better sub-divided within each larger chapter, easing the student experience.

The book's interface is effective, allowing readers to move from one section to the next with a single click. Additional sub-sections would ease this interplay even further.

Grammatical Errors rating: 4

Some minor errors throughout.

For the most part, the book is culturally neutral, avoiding direct cultural references in an effort to remain relevant.

Reviewed by Yoichi Ishida, Assistant Professor of Philosophy, Ohio University on 2/1/18

This textbook covers enough topics for a first-year course on logic and critical thinking. Chapter 1 covers the basics as in any standard textbook in this area. Chapter 2 covers propositional logic and categorical logic. In propositional logic, this textbook does not cover suppositional arguments, such as conditional proof and reductio ad absurdum. But other standard argument forms are covered. Chapter 3 covers inductive logic, and here this textbook introduces probability and its relationship with cognitive biases, which are rarely discussed in other textbooks. Chapter 4 introduces common informal fallacies. The answers to all the exercises are given at the end. However, the last set of exercises is in Chapter 3, Section 5. There are no exercises in the rest of the chapter. Chapter 4 has no exercises either. There is index, but no glossary.

The textbook is accurate.

The content of this textbook will not become obsolete soon.

The textbook is written clearly.

The textbook is internally consistent.

The textbook is fairly modular. For example, Chapter 3, together with a few sections from Chapter 1, can be used as a short introduction to inductive logic.

The textbook is well-organized.

There are no interface issues.

I did not find any grammatical errors.

This textbook is relevant to a first semester logic or critical thinking course.

Reviewed by Payal Doctor, Associate Professro, LaGuardia Community College on 2/1/18

This text is a beginner textbook for arguments and propositional logic. It covers the basics of identifying arguments, building arguments, and using basic logic to construct propositions and arguments. It is quite comprehensive for a beginner book, but seems to be a good text for a course that needs a foundation for arguments. There are exercises on creating truth tables and proofs, so it could work as a logic primer in short sessions or with the addition of other course content.

The books is accurate in the information it presents. It does not contain errors and is unbiased. It covers the essential vocabulary clearly and givens ample examples and exercises to ensure the student understands the concepts

The content of the book is up to date and can be easily updated. Some examples are very current for analyzing the argument structure in a speech, but for this sort of text understandable examples are important and the author uses good examples.

The book is clear and easy to read. In particular, this is a good text for community college students who often have difficulty with reading comprehension. The language is straightforward and concepts are well explained.

The book is consistent in terminology, formatting, and examples. It flows well from one topic to the next, but it is also possible to jump around the text without loosing the voice of the text.

The books is broken down into sub units that make it easy to assign short blocks of content at a time. Later in the text, it does refer to a few concepts that appear early in that text, but these are all basic concepts that must be used to create a clear and understandable text. No sections are too long and each section stays on topic and relates the topic to those that have come before when necessary.

The flow of the text is logical and clear. It begins with the basic building blocks of arguments, and practice identifying more and more complex arguments is offered. Each chapter builds up from the previous chapter in introducing propositional logic, truth tables, and logical arguments. A select number of fallacies are presented at the end of the text, but these are related to topics that were presented before, so it makes sense to have these last.

The text is free if interface issues. I used the PDF and it worked fine on various devices without loosing formatting.

1. The book contains no grammatical errors.

The text is culturally sensitive, but examples used are a bit odd and may be objectionable to some students. For instance, President Obama's speech on Syria is used to evaluate an extended argument. This is an excellent example and it is explained well, but some who disagree with Obama's policies may have trouble moving beyond their own politics. However, other examples look at issues from all political viewpoints and ask students to evaluate the argument, fallacy, etc. and work towards looking past their own beliefs. Overall this book does use a variety of examples that most students can understand and evaluate.

My favorite part of this book is that it seems to be written for community college students. My students have trouble understanding readings in the New York Times, so it is nice to see a logic and critical thinking text use real language that students can understand and follow without the constant need of a dictionary.

Reviewed by Rebecca Owen, Adjunct Professor, Writing, Chemeketa Community College on 6/20/17

This textbook is quite thorough--there are conversational explanations of argument structure and logic. I think students will be happy with the conversational style this author employs. Also, there are many examples and exercises using current events, funny scenarios, or other interesting ways to evaluate argument structure and validity. The third section, which deals with logical fallacies, is very clear and comprehensive. My only critique of the material included in the book is that the middle section may be a bit dense and math-oriented for learners who appreciate the more informal, informative style of the first and third section. Also, the book ends rather abruptly--it moves from a description of a logical fallacy to the answers for the exercises earlier in the text.

The content is very reader-friendly, and the author writes with authority and clarity throughout the text. There are a few surface-level typos (Starbuck's instead of Starbucks, etc.). None of these small errors detract from the quality of the content, though.

One thing I really liked about this text was the author's wide variety of examples. To demonstrate different facets of logic, he used examples from current media, movies, literature, and many other concepts that students would recognize from their daily lives. The exercises in this text also included these types of pop-culture references, and I think students will enjoy the familiarity--as well as being able to see the logical structures behind these types of references. I don't think the text will need to be updated to reflect new instances and occurrences; the author did a fine job at picking examples that are relatively timeless. As far as the subject matter itself, I don't think it will become obsolete any time soon.

The author writes in a very conversational, easy-to-read manner. The examples used are quite helpful. The third section on logical fallacies is quite easy to read, follow, and understand. A student in an argument writing class could benefit from this section of the book. The middle section is less clear, though. A student learning about the basics of logic might have a hard time digesting all of the information contained in chapter two. This material might be better in two separate chapters. I think the author loses the balance of a conversational, helpful tone and focuses too heavily on equations.

Consistency rating: 4

Terminology in this book is quite consistent--the key words are highlighted in bold. Chapters 1 and 3 follow a similar organizational pattern, but chapter 2 is where the material becomes more dense and equation-heavy. I also would have liked a closing passage--something to indicate to the reader that we've reached the end of the chapter as well as the book.

I liked the overall structure of this book. If I'm teaching an argumentative writing class, I could easily point the students to the chapters where they can identify and practice identifying fallacies, for instance. The opening chapter is clear in defining the necessary terms, and it gives the students an understanding of the toolbox available to them in assessing and evaluating arguments. Even though I found the middle section to be dense, smaller portions could be assigned.

The author does a fine job connecting each defined term to the next. He provides examples of how each defined term works in a sentence or in an argument, and then he provides practice activities for students to try. The answers for each question are listed in the final pages of the book. The middle section feels like the heaviest part of the whole book--it would take the longest time for a student to digest if assigned the whole chapter. Even though this middle section is a bit heavy, it does fit the overall structure and flow of the book. New material builds on previous chapters and sub-chapters. It ends abruptly--I didn't realize that it had ended, and all of a sudden I found myself in the answer section for those earlier exercises.

The simple layout is quite helpful! There is nothing distracting, image-wise, in this text. The table of contents is clearly arranged, and each topic is easy to find.

Tiny edits could be made (Starbuck's/Starbucks, for one). Otherwise, it is free of distracting grammatical errors.

This text is quite culturally relevant. For instance, there is one example that mentions the rumors of Barack Obama's birthplace as somewhere other than the United States. This example is used to explain how to analyze an argument for validity. The more "sensational" examples (like the Obama one above) are helpful in showing argument structure, and they can also help students see how rumors like this might gain traction--as well as help to show students how to debunk them with their newfound understanding of argument and logic.

The writing style is excellent for the subject matter, especially in the third section explaining logical fallacies. Thank you for the opportunity to read and review this text!

Reviewed by Laurel Panser, Instructor, Riverland Community College on 6/20/17

This is a review of Introduction to Logic and Critical Thinking, an open source book version 1.4 by Matthew Van Cleave. The comparison book used was Patrick J. Hurley’s A Concise Introduction to Logic 12th Edition published by Cengage as well as... read more

Competing with Hurley is difficult with respect to comprehensiveness. For example, Van Cleave’s book is comprehensive to the extent that it probably covers at least two-thirds or more of what is dealt with in most introductory, one-semester logic courses. Van Cleave’s chapter 1 provides an overview of argumentation including discerning non-arguments from arguments, premises versus conclusions, deductive from inductive arguments, validity, soundness and more. Much of Van Cleave’s chapter 1 parallel’s Hurley’s chapter 1. Hurley’s chapter 3 regarding informal fallacies is comprehensive while Van Cleave’s chapter 4 on this topic is less extensive. Categorical propositions are a topic in Van Cleave’s chapter 2; Hurley’s chapters 4 and 5 provide more instruction on this, however. Propositional logic is another topic in Van Cleave’s chapter 2; Hurley’s chapters 6 and 7 provide more information on this, though. Van Cleave did discuss messy issues of language meaning briefly in his chapter 1; that is the topic of Hurley’s chapter 2.

Van Cleave’s book includes exercises with answers and an index. A glossary was not included.

Reviews of open source textbooks typically include criteria besides comprehensiveness. These include comments on accuracy of the information, whether the book will become obsolete soon, jargon-free clarity to the extent that is possible, organization, navigation ease, freedom from grammar errors and cultural relevance; Van Cleave’s book is fine in all of these areas. Further criteria for open source books includes modularity and consistency of terminology. Modularity is defined as including blocks of learning material that are easy to assign to students. Hurley’s book has a greater degree of modularity than Van Cleave’s textbook. The prose Van Cleave used is consistent.

Van Cleave’s book will not become obsolete soon.

Van Cleave’s book has accessible prose.

Van Cleave used terminology consistently.

Van Cleave’s book has a reasonable degree of modularity.

Van Cleave’s book is organized. The structure and flow of his book is fine.

Problems with navigation are not present.

Grammar problems were not present.

Van Cleave’s book is culturally relevant.

Van Cleave’s book is appropriate for some first semester logic courses.

Chapter 1: Reconstructing and analyzing arguments

1.1 What is an argument?
1.2 Identifying arguments
1.3 Arguments vs. explanations
1.4 More complex argument structures
1.5 Using your own paraphrases of premises and conclusions to reconstruct arguments in standard form
1.6 Validity
1.7 Soundness
1.8 Deductive vs. inductive arguments
1.9 Arguments with missing premises
1.10 Assuring, guarding, and discounting
1.11 Evaluative language
1.12 Evaluating a real-life argument

Chapter 2: Formal methods of evaluating arguments

2.1 What is a formal method of evaluation and why do we need them?
2.2 Propositional logic and the four basic truth functional connectives
2.3 Negation and disjunction
2.4 Using parentheses to translate complex sentences
2.5 “Not both” and “neither nor”
2.6 The truth table test of validity
2.7 Conditionals
2.8 “Unless”
2.9 Material equivalence
2.10 Tautologies, contradictions, and contingent statements
2.11 Proofs and the 8 valid forms of inference
2.12 How to construct proofs
2.13 Short review of propositional logic
2.14 Categorical logic
2.15 The Venn test of validity for immediate categorical inferences
2.16 Universal statements and existential commitment
2.17 Venn validity for categorical syllogisms

Chapter 3: Evaluating inductive arguments and probabilistic and statistical fallacies

3.1 Inductive arguments and statistical generalizations
3.2 Inference to the best explanation and the seven explanatory virtues
3.3 Analogical arguments
3.4 Causal arguments
3.5 Probability
3.6 The conjunction fallacy
3.7 The base rate fallacy
3.8 The small numbers fallacy
3.9 Regression to the mean fallacy
3.10 Gambler's fallacy

Chapter 4: Informal fallacies

4.1 Formal vs. informal fallacies
4.1.1 Composition fallacy
4.1.2 Division fallacy
4.1.3 Begging the question fallacy
4.1.4 False dichotomy
4.1.5 Equivocation
4.2 Slippery slope fallacies
4.2.1 Conceptual slippery slope
4.2.2 Causal slippery slope
4.3 Fallacies of relevance
4.3.1 Ad hominem
4.3.2 Straw man
4.3.3 Tu quoque
4.3.4 Genetic
4.3.5 Appeal to consequences
4.3.6 Appeal to authority

Answers to exercises Glossary/Index

Ancillary Material

About the book.

This is an introductory textbook in logic and critical thinking. The goal of the textbook is to provide the reader with a set of tools and skills that will enable them to identify and evaluate arguments. The book is intended for an introductory course that covers both formal and informal logic. As such, it is not a formal logic textbook, but is closer to what one would find marketed as a “critical thinking textbook.”

About the Contributors

Matthew Van Cleave , PhD, Philosophy, University of Cincinnati, 2007. VAP at Concordia College (Moorhead), 2008-2012. Assistant Professor at Lansing Community College, 2012-2016. Professor at Lansing Community College, 2016-

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Pursuing Truth: A Guide to Critical Thinking

Chapter 4 propositional logic.

Categorical logic is a great way to analyze arguments, but only certain kinds of arguments. It is limited to arguments that have only two premises and the four kinds of categorical sentences. This means that certain common arguments that are obviously valid will not even be well-formed arguments in categorical logic. Here is an example:

I will either go out for dinner tonight or go out for breakfast tomorrow.
I won’t go out for dinner tonight.
I will go out for breakfast tomorrow.

None of these sentences fit any of the four categorical schemes. So, we need a new logic, called propositional logic. The good news is that it is fairly simple.

4.1 Simple and Complex Sentences

The fundamental logical unit in categorical logic was a category, or class of things. The fundamental logical unit in propositional logic is a statement, or proposition 5 Simple statements are statements that contain no other statement as a part. Here are some examples:

Oklahoma Baptist University is in Shawnee, Oklahoma.
Barack Obama was succeeded as President of the US by Donald Trump.
It is 33 degrees outside.

Simple sentences are symbolized by uppercase letters. Just pick a letter that makes sense, given the sentence to be symbolized, that way you can more easily remember which letter means which sentence.

Complex sentences have at least one sentence as a component. There are five types in propositional logic:

Conjunctions
Disjunctions
Conditionals
Biconditionals

4.1.1 Negations

Negations are “not” sentences. They assert that something is not the case. For example, the negation of the simple sentence “Oklahoma Baptist University is in Shawnee, Oklahoma” is “Oklahoma Baptist University is not in Shawnee, Oklahoma.” In general, a simple way to form a negation is to just place the phrase “It is not the case that” before the sentence to be negated.

A negation is symbolized by placing this symbol ‘ $\neg$ ’ before the sentence-letter. The symbol looks like a dash with a little tail on its right side. If $\textrm{D}$ = ‘It is 33 degrees outside,’ then $\neg \textrm{D}$ = ‘It is not 33 degrees outside.’ The negation symbol is used to translate these English phrases:

it is not the case that
it is not true that
it is false that

A negation is true whenever the negated sentence is false. If it is true that it is not 33 degrees outside, then it must be false that it is 33 degrees outside. if it is false that Tulsa is the capital of Oklahoma, then it is true that Tulsa is not the capital of Oklahoma.

When translating, try to keep the simple sentences positive in meaning. Note the warning on page 24, about the example of affirming and denying. Denying is not simply the negation of affirming.

4.2 Conjunction

Negations are “and” sentences. They put two sentences, called conjuncts, together and claim that they are both true. We’ll use the ampersand (&) to signify a negation. Other common symbols are a dot and an upside down wedge. The English words that are translated with the ampersand include:

nevertheless

For example, we would translate the sentence ‘It is raining today and my sunroof is open’ as ‘ $\textrm{R} \& \textrm{O}$ .’

4.3 Disjunction

A disjunction is an “or” sentence. It claims that at least one of two sentences, called disjuncts, is true. For example, if I say that either I will go to the movies this weekend or I will stay home and grade critical thinking homework, then I have told the truth provided that I do one or both of those things. If I do neither, though, then my claim was false.

We use this symbol, called a “vel,” for disjunctions: $\vee$ . The vel is used to translate - or - eitheror - unless

4.4 Conditional

The conditional is a common type of sentence. It claims that something is true, if something else is also. Examples of conditionals are

“If Sarah makes an A on the final, then she will get an A for the course.”
“Your car will last many years, provided you perform the required maintenance.”
“You can light that match only if it is not wet.”

We can translate those sentences with an arrow like this:

$F \rightarrow C$
$M \rightarrow L$
$L \rightarrow \neg W$

The arrow translates many English words and phrases, including

provided that
is a sufficient condition for
is a necessary condition for
on the condition that

One big difference between conditionals and other sentences, like conjunctions and disjunctions, is that order matters. Notice that there is no logical difference between the following two sentences:

Albany is the capital of New York and Austin is the capital of Texas.
Austin is the capital of Texas and Albany is the capital of New York.

They essentially assert exactly the same thing, that both of those conjuncts are true. So, changing order of the conjuncts or disjuncts does not change the meaning of the sentence, and if meaning doesn’t change, then true value doesn’t change.

That’s not true of conditionals. Note the difference between these two sentences:

If you drew a diamond, then you drew a red card.
If you drew a red card, then you drew a diamond.

The first sentence must be true. if you drew a diamond, then that guarantees that it’s a red card. The second sentence, though, could be false. Your drawing a red card doesn’t guarantee that you drew a diamond, you could have drawn a heart instead. So, we need to be able to specify which sentence goes before the arrow and which sentence goes after. The sentence before the arrow is called the antecedent, and the sentence after the arrow is called the consequent.

Look at those three examples again:

The antecedent for the first sentence is “Sarah makes an A on the final.” The consequent is “She will get an A for the course.” Note that the if and the then are not parts of the antecedent and consequent.

In the second sentence, the antecdent is “You perform the required maintenance.” The consequent is “Your car will last many years.” This tells us that the antecedent won’t always come first in the English sentence.

The third sentence is tricky. The antecedent is “You can light that match.” Why? The explanation involves something called necessary and sufficient conditions.

4.4.1 Necessary and Sufficient Conditions

A sufficient condition is something that is enough to guarantee the truth of something else. For example, getting a 95 on an exam is sufficient for making an A, assuming that exam is worth 100 points. A necessary condition is something that must be true in order for something else to be true. Making a 95 on an exam is not necessary for making an A—a 94 would have still been an A. Taking the exam is necessary for making an A, though. You can’t make an A if you don’t take the exam, or, in other words, you can make an a only if you enroll in the course.

Here are some important rules to keep in mind:

‘If’ introduces antecedents, but Only if introduces consequents.
If A is a sufficient condition for B, then $A \rightarrow B$ .
If A is a necessary condition for B, then $B \rightarrow A$ .

4.5 Biconditional

We won’t spend much time on biconditionals. There are times when something is both a necessary and a sufficient condition for something else. For example, making at least a 90 and getting an A (assuming a standard scale, no curve, and no rounding up). If you make at least a 90, then you will get an A. If you got an A, then you made at least a 90. We can use a double arrow to translate a biconditional, like this:

$N \rightarrow A$

For biconditionals, as for conjunctions and disjunctions, order doesn’t matter.

Here are some English phrases that signify biconditionals:

it and only if
when and only when
just in case
is a necessary and sufficient condition for

4.6 Translations

Propositional logic is language. Like other languages, it has a syntax and a semantics. The syntax of a language includes the basic symbols of the language plus rules for putting together proper statements in the language. To use propositional logic, we need to know how to translate English sentences into the language of propositional logic. We start with our sentence letters, which represent simple English sentences. Let’s use three borrowed from an elementary school reader:

We then build complex sentences using the sentence letters and our five logical operators, like this:

We can make even more complex sentences, but we will soon run into a problem. Consider this example:

\[ T \mathbin{\&} J \rightarrow S\]

We don’t know this means. It could be either one of the following:

Tom hit the ball, and if Jane caught the ball, then Spot chased it.
If Tom hit the ball and Jane caught it, then Spot chased it.

The first sentence is a conjunction, $T$ is the first conjunct and $M \rightarrow S$ is the second conjunct. The second sentence, though, is a conditional, $T \mathbin{\&}M$ is the antecdent, and $S$ is the consequent. Our two interpretations are not equivalent, so we need a way to clear up the ambiguity. We can do this with parentheses. Our first sentence becomes:

\[ T \mathbin{\&} (J \rightarrow S) \]

The second sentence is:

\[ (T \mathbin{\&} J) \rightarrow S\]

If we need higher level parentheses, we can use brackets and braces. For instance, this is a perfectly good formula in propositional logic:

\[ [(P \mathbin{\&} Q) \vee R] \rightarrow \{[(\neg P \leftrightarrow Q) \mathbin{\&} S] \vee \neg P\} \] 6

Every sentence in propositional logic is one of six types:

Conjunction
Disjunction
Conditional
Biconditional

What type of sentence it is will be determined by its main logical operator. Sentences can have several logical operators, but they will always have one, and only one, main operator. Here are some general rules for finding the main operator in a symbolized formula of propositional logic:

If a sentence has only one logical operator, then that is the main operator.
If a sentence has more than one logical operator, then the main operator is the one outside the parentheses.
If a sentence has two logical operators outside the parentheses, then the main operator is not the negation.

Here are some examples:

Informally, we use ‘proposition’ and ‘statement’ interchangeably. Strictly speaking, the proposition is the content, or meaning, that the statement expresses. When different sentences in different languages mean the same thing, it is because they express the same proposition. ↩︎

It may be a good formula in propositional logic, but that doesn’t mean it would be a good English sentence. ↩︎

PHIL102: Introduction to Critical Thinking and Logic

Course introduction.

Time: 40 hours
College Credit Recommended ($25 Proctor Fee) -->
Free Certificate

The course touches upon a wide range of reasoning skills, from verbal argument analysis to formal logic, visual and statistical reasoning, scientific methodology, and creative thinking. Mastering these skills will help you become a more perceptive reader and listener, a more persuasive writer and presenter, and a more effective researcher and scientist.

The first unit introduces the terrain of critical thinking and covers the basics of meaning analysis, while the second unit provides a primer for analyzing arguments. All of the material in these first units will be built upon in subsequent units, which cover informal and formal logic, Venn diagrams, scientific reasoning, and strategic and creative thinking.

Course Syllabus

First, read the course syllabus. Then, enroll in the course by clicking "Enroll me". Click Unit 1 to read its introduction and learning outcomes. You will then see the learning materials and instructions on how to use them.

Unit 1: Introduction and Meaning Analysis

Critical thinking is a broad classification for a diverse array of reasoning techniques. In general, critical thinking works by breaking arguments and claims down to their basic underlying structure so we can see them clearly and determine whether they are rational. The idea is to help us do a better job of understanding and evaluating what we read, what we hear, and what we write and say.

In this unit, we will define the broad contours of critical thinking and learn why it is a valuable and useful object of study. We will also introduce the fundamentals of meaning analysis: the difference between literal meaning and implication, the principles of definition, how to identify when a disagreement is merely verbal, the distinction between necessary and sufficient conditions, and problems with the imprecision of ordinary language.

Completing this unit should take you approximately 5 hours.

Unit 2: Argument Analysis

Arguments are the fundamental components of all rational discourse: nearly everything we read and write, like scientific reports, newspaper columns, and personal letters, as well as most of our verbal conversations, contain arguments. Picking the arguments out from the rest of our often convoluted discourse can be difficult. Once we have identified an argument, we still need to determine whether or not it is sound. Luckily, arguments obey a set of formal rules that we can use to determine whether they are good or bad.

In this unit, you will learn how to identify arguments, what makes an argument sound as opposed to unsound or merely valid, the difference between deductive and inductive reasoning, and how to map arguments to reveal their structure.

Completing this unit should take you approximately 7 hours.

Unit 3: Basic Sentential Logic

This unit introduces a topic that many students find intimidating: formal logic. Although it sounds difficult and complicated, formal (or symbolic) logic is actually a fairly straightforward way of revealing the structure of reasoning. By translating arguments into symbols, you can more readily see what is right and wrong with them and learn how to formulate better arguments. Advanced courses in formal logic focus on using rules of inference to construct elaborate proofs. Using these techniques, you can solve many complicated problems simply by manipulating symbols on the page. In this course, however, you will only be looking at the most basic properties of a system of logic. In this unit, you will learn how to turn phrases in ordinary language into well-formed formulas, draw truth tables for formulas, and evaluate arguments using those truth tables.

Completing this unit should take you approximately 13 hours.

Unit 4: Venn Diagrams

In addition to using predicate logic, the limitations of sentential logic can also be overcome by using Venn diagrams to illustrate statements and arguments. Statements that include general words like "some" or "few" as well as absolute words like "every" and "all" – so-called categorical statements – lend themselves to being represented on paper as circles that may or may not overlap.

Venn diagrams are especially helpful when dealing with logical arguments called syllogisms. Syllogisms are a special type of three-step argument with two premises and a conclusion, which involve quantifying terms. In this unit, you will learn the basic principles of Venn diagrams, how to use them to represent statements, and how to use them to evaluate arguments.

Completing this unit should take you approximately 6 hours.

Unit 5: Fallacies

Now that you have studied the necessary structure of a good argument and can represent its structure visually, you might think it would be simple to pick out bad arguments. However, identifying bad arguments can be very tricky in practice. Very often, what at first appears to be ironclad reasoning turns out to contain one or more subtle errors.

Fortunately, there are many easily identifiable fallacies (mistakes of reasoning) that you can learn to recognize by their structure or content. In this unit, you will learn about the nature of fallacies, look at a couple of different ways of classifying them, and spend some time dealing with the most common fallacies in detail.

Completing this unit should take you approximately 3 hours.

Unit 6: Scientific Reasoning

Unlike the syllogistic arguments you explored in the last unit, which are a form of deductive argument, scientific reasoning is empirical. This means that it depends on observation and evidence, not logical principles. Although some principles of deductive reasoning do apply in science, such as the principle of contradiction, scientific arguments are often inductive. For this reason, science often deals with confirmation and disconfirmation.

Nonetheless, there are general guidelines about what constitutes good scientific reasoning, and scientists are trained to be critical of their inferences and those of others in the scientific community. In this unit, you will investigate some standard methods of scientific reasoning, some principles of confirmation and disconfirmation, and some techniques for identifying and reasoning about causation.

Completing this unit should take you approximately 4 hours.

Unit 7: Strategic Reasoning and Creativity

While most of this course has focused on the types of reasoning necessary to critique and evaluate existing knowledge or to extend our knowledge following correct procedures and rules, an enormous branch of our reasoning practice runs in the opposite direction. Strategic reasoning, problem-solving, and creative thinking all rely on an ineffable component of novelty supplied by the thinker.

Despite their seemingly mystical nature, problem-solving and creative thinking are best approached by following tried and tested procedures that prompt our cognitive faculties to produce new ideas and solutions by extending our existing knowledge. In this unit, you will investigate problem-solving techniques, representing complex problems visually, making decisions in risky and uncertain scenarios, and creative thinking in general.

Completing this unit should take you approximately 2 hours.

Study Guide

This study guide will help you get ready for the final exam. It discusses the key topics in each unit, walks through the learning outcomes, and lists important vocabulary terms. It is not meant to replace the course materials!

Course Feedback Survey

Please take a few minutes to give us feedback about this course. We appreciate your feedback, whether you completed the whole course or even just a few resources. Your feedback will help us make our courses better, and we use your feedback each time we make updates to our courses.

If you come across any urgent problems, email [email protected].

Certificate Final Exam

Take this exam if you want to earn a free Course Completion Certificate.

To receive a free Course Completion Certificate, you will need to earn a grade of 70% or higher on this final exam. Your grade for the exam will be calculated as soon as you complete it. If you do not pass the exam on your first try, you can take it again as many times as you want, with a 7-day waiting period between each attempt.

Once you pass this final exam, you will be awarded a free Course Completion Certificate .

Saylor Direct Credit

Take this exam if you want to earn college credit for this course . This course is eligible for college credit through Saylor Academy's Saylor Direct Credit Program .

The Saylor Direct Credit Final Exam requires a proctoring fee of $5 . To pass this course and earn a Credly Badge and official transcript , you will need to earn a grade of 70% or higher on the Saylor Direct Credit Final Exam. Your grade for this exam will be calculated as soon as you complete it. If you do not pass the exam on your first try, you can take it again a maximum of 3 times , with a 14-day waiting period between each attempt.

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Standard Form Categorical Propositions: Quantity, Quality, and Distribution

1. Traditional logic and the logic of the syllogism descended from Aristotelian logic. Bocheński writes, “[Aristotle] exercised a decisive influence on the history of logic for more than two thousand years, and even today much of the doctrine is traceable back to him.” I. M. Bocheński, A History of Formal Logic , (New York: Chelsea,1961), 40. ↩
2. Categorical propositions or statements are simple propositions. Other kinds of statements include compound statements such as the hypothetical (If… then …), the disjunctive (Either … or …), and the conjunctive (Both … and …) statements. ↩

‘[I]n [the] use of schematic letters like ‘S’ and ‘P’ you find, for example, in one and the same context the phrase ‘every S’, which requires that ‘S’ be read as a general term like ‘man’ and the phrase ‘the whole of S’, which requires that ‘S’ be a singular designation of a class taken collectively, like ‘the class of men’ obviously ‘man’ and ‘the class of men’ are wholly different sorts of expression. ↩

“[T]he extreme grammatical oddity” of “All S is P ” being translated from “Omne S est P ” of the historical Latin texts which ought be, according to him, “Every S is P, ” so that the whole class of the subject term is universally quantified and no distribution error occurs.

“All S is not P ”

“No S is P ”

“Some S is not P .”

“No lodestones are non-magnetic ore”

“ Some swans are not white”

“Some things that glitter are not gold.”

“Some S is not P. ”

“Not all honey badgers are solitary hunters”

“Some honey badgers are not solitary hunters.”

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PHILO-notes

Free Online Learning Materials

Categorical Logic: Terms and Propositions

As we may already know, our main goal in logic is to determine the validity of arguments.

And in categorical logic , we will employ the Eight (8) Rules of Syllogisms for us to be able to determine the validity of an argument. But since the 8 rules of syllogisms talk about the quantity and quality of terms and propositions, then it is but logical enough to discuss the nature of terms and propositions before we delve into the discussion on the 8 rules of syllogisms. In what follows, I will discuss the nature of terms and propositions.

First of all, logicians define a term as an idea expressed in words either spoken or written. Of course, an idea is understood as the mental representation of something. Hence, when one says, for example, “a table”, then we have at term, that is, a “table”.

Classification of Terms

There are four (4) classifications of terms in terms of quantity, namely: singular, collective, particular, and universal.

A singular term is one that stands for only one definite object.

A c ollective term is one that is applicable to each and every member of a class taken as a whole but not to an individual taken singly.

1) Orchestra

A particular term is one that refers to an indefinite number of individuals or groups. Some signifiers of a particular term are: some, a number of, several, almost all, a few of, practically all, at least one, not all , and the like. Hence, if a term is signified by at least one of these signifiers, then we conclude that that term is a particular one.

1) Some Asians

2) Almost all students

3) Several politicians

A universal term is one that is applicable to each and every member of a class. Some of the signifiers of a universal term are: no, all, each, every , and the like.

1) All Asians

2) Every politician

3) No student

A proposition , on the other hand, is a judgment expressed in words either spoken or written. When we say a judgment, it refers to the mental act of affirming or denying something.

1) President Trump is a good president.

2) President Trump is not a good president.

The first example above is an act of affirmation because the copula (or linking verb) is does not contain a negation sign “not”. The second example is an act of negation because the copula (or linking verb) is contains a negation sign “not”.

Kinds of Propositions used in Logic

There are two types of propositions used in logic, namely, categorical and hypothetical propositions. On the one hand, a categorical proposition is one that expresses an unconditional judgment. For example, we may say “The Japanese people are hard-working.” According to logicians, this proposition is a categorical one because it does not pose any condition. On the other hand, a hypothetical proposition is one that expresses a conditional judgment. For example, we may say “ If it rains today, then the road is wet.” Please note that in categorical logic we always use categorical propositions.

Elements of a Categorical Proposition

A categorical proposition has three elements, namely: Subject (S), Copula (C), and Predicate (P).

Quantity of a Categorical Proposition

In terms of quantity, a categorical proposition can be classified into two, namely: 1) particular and 2) universal.

A particular proposition is one that contains a particular subject term.

Some Asians are excellent basketball players.

A universal proposition is one that contains a universal subject term.

1) All men are mortal.

As we can see, it is the quantity of the subject that determines the quantity of the proposition. Thus, if the subject is particular , then the proposition is particular , and if the subject is universal , then the proposition is universal .

Now if the subject of the proposition does not contain a signifier, then the quantity of the proposition must be based on what the proposition denotes. Consider the example below:

Nuns are girls.

As we can see, the subject of the proposition does not contain a signifier. But if we analyze it, it would become clear that the proposition is universal. This is because there is not at least 1 nun that is not a girl. In other words, all nuns are girls. Let us consider another example:

Americans are rich.

Obviously, the example above denotes particularity because it’s not sound to assume that all Americans are rich. Of course, many Americans are rich, but reason tells us that not all of the Americans are rich. Hence, the above proposition can be translated as follows: “ Some Americans are rich”.

Quality of a Categorical Proposition

Categorical propositions can be either affirmative or negative .

A proposition is affirmative if the copula of the proposition does not contain a negation sign “ not ”.

Example: 1) Some students are brilliant.

A proposition is negative if the copula of the proposition contains a negation sign “ not ”.

Example: 1) Some students are not brilliant.

Four Basic Types of Categorical Propositions

If we combine the quantity and quality of propositions, the result is the four (4) types of categorical propositions, namely: 1) Universal Affirmative, 2) Universal Negative, 3) Particular Affirmative, and 4) Particular Negative. Logicians use the letter “ A ” to represent a universal affirmative proposition, “ E ” for universal negative, “ I ” for particular affirmative, and “ O ” for particular negative. Consider the examples below:

Universal Affirmative (A) : All men are mortal.

Universal Negative (E) : No men are mortal.

Particular Affirmative (I) : Some men are mortal.

Particular Negative (O) : Some men are not mortal.

Distribution of Terms

In a universal proposition, the subject term is distributed, while in a particular proposition subject term is undistributed. And in a negative proposition, the predicate term is distributed while in an affirmative proposition the predicate term remains undistributed. In other words, the subject terms of all universal propositions are always universal, while the subject terms of all particular propositions are always particular. And the predicate terms of all affirmative propositions are always particular, while the predicate terms of all negative propositions are always universal.

Translating Categorical Propositions into their Standard Form:

To avoid confusion when we analyze the 8 rules of syllogisms, it is helpful to translate categorical propositions into their standard form. Below are the standard forms of an A, E, I, and O propositions.

A proposition : All + subject + copula + predicate

E proposition : No + subject + copula + predicate

I proposition : Some + subject + copula + predicate

O proposition : Some + subject + copula + not + predicate

A : Every priest is religious.

Standard form : All priests are

E : Every priest is not religious.

Standard form : No priest is religious.

I : Almost all politicians are corrupt.

Standard form: Some politicians are corrupt.

O : Several politicians are not corrupt.

Standard form : Some politicians are not corrupt.

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The Mostly Persuasive Logic Behind the New Ban on Noncompetes

By Peter Coy

Opinion Writer

The Federal Trade Commission used two very different rationales to get to its near-total ban this week on noncompete agreements. One of them is a no-brainer. The other is provocative but not completely obvious. I guess I’d call it a brainer.

As you might have read, the F.T.C. commissioners on Tuesday voted 3 to 2 on a final rule against noncompete clauses in employment contracts, which limit the ability of an employee to quit and immediately go work for a rival. The commission determined that they are an “unfair method of competition.” The rule takes effect 120 days after its publication in the Federal Register, unless a court blocks it before then.

The easy prong of the ban for the F.T.C. to justify is the one that applies to nurses, hairdressers, truck drivers — actually, every kind of worker except for senior executives. For 99 percent of the American work force, the F.T.C. said, requiring workers to sign noncompete agreements as a condition of employment is “coercive and exploitative conduct.” The agency’s 570-page ruling cites articles in The Times and The Wall Street Journal in which workers came forward to say, in the F.T.C.’s words, that noncompete agreements “derailed their careers, destroyed their finances and upended their lives.” I agree. I wrote a piece in 2021 titled , “Why Are Fast Food Workers Signing Noncompete Agreements?”

But the “coercive and exploitative” rationale doesn’t work for senior executives, who aren’t so easy to coerce or exploit. They’re more likely to have lawyers look over contract offers. They typically have some power in the employment negotiation and know how to use it. Many won’t sign a noncompete agreement unless they get something in return, such as a sweetened pay package.

The F.T.C. defined senior executives as people earning more than $151,164 per year who are in a “policy-making position” and estimated that fewer than 1 percent of workers meet the description. Under the rule, existing noncompetes for senior executives can remain in force but most new ones are banned. The rule doesn’t apply to clauses that are related to the sale of a business.

For noncompetes involving senior executives, the F.T.C. fell back on another argument, which is that the agreements are “restrictive and exclusionary conduct” that harms competition in product, service and labor markets. (The F.T.C. says that this second argument also applies to other workers, but for them I think it’s overshadowed by the “coercive and exploitative” argument.)

This is a bit subtle. It requires you to think of the employer and the senior executive as being in cahoots rather than fighting each other. Together they cook up a noncompete that rewards the executive for agreeing to deprive other potential employers of her or his talents and depriving the customers of those other companies of potentially better products and services. In economists’ terms, noncompete signatories are maximizing their bilateral surplus at the expense of others.

The logic is that the company that can’t hire the executive might have better growth prospects, so holding it back is bad for society as a whole. Or, after leaving the old employer, the executive has to be (wastefully) inactive for six months or so to wait out what finance people call the garden leave. Or the new employer has to pay a large sum to buy out the noncompete clause — again, socially wasteful.

“There can be sizable gains from restricting these contracts,” Liyan Shi of Carnegie Mellon’s Tepper School of Business wrote in a 2023 article in the journal Econometrica.

As I said, this is an interesting and even persuasive argument. But it’s not simple to make.

“If this becomes the approach,” Sean Heather, the senior vice president for international regulatory affairs and antitrust at the U.S. Chamber of Commerce, asked me, will any contract that doesn’t take into account the interests of third parties be “no longer viable”?

Charles Tharp, a professor of the practice at Boston University’s Questrom School of Business, said that while banning the noncompete might benefit a future employer, it harms the current employer, so there’s no net benefit; it’s a wash.

But two other economists I contacted disagreed with Tharp and Heather. Evan Penniman Starr, an associate professor at the University of Maryland’s Smith School of Business who is an expert on noncompete agreements, wrote to me that governments shouldn’t always put third parties first, but shouldn’t ignore them either, citing smoking bans to protect third parties from secondhand smoke. As for Tharp’s point, he wrote, “If match quality is higher at the subsequent firm, it is not a wash. It’s an efficient move that would destroy value if it wasn’t made.”

Sandeep Vaheesan, the legal director of the Open Markets Institute, emailed me that companies could still retain senior executives through higher pay packages and fixed-term contracts. Noncompetes are a “stick,” he wrote. “Public policy should encourage employers to use carrots instead. The F.T.C. noncompete ban does exactly that.”

Vaheesan also sided with the F.T.C.’s argument that companies have other ways to protect themselves when a key employee leaves, such as trade secret protection and agreements that prohibit people from soliciting customers of the companies they used to work for.

There’s precedent for taking into account the interests of third parties, Starr told me. He cited an American Bar Association model rule on professional conduct that forbids restricting attorneys from working elsewhere not only because it harms the attorney but also because it “limits the freedom of clients to choose a lawyer.”

The strongest evidence against noncompete agreements is that Silicon Valley has thrived even though — or maybe even partly because — the state of California has long banned noncompete agreements in most circumstances, under a law passed in 1872. The prohibition does not seem to have discouraged companies from sharing valuable inside information with employees who might leave. And it has enabled the germination of ideas as people flit from company to company like pollinating honeybees.

“Noncompetes are a pain in the neck for us,” Dr. Stephen DeCherney, who is the chair of New York-based Helios Clinical Research, told me. “Overall I won’t be sorry to see them go.”

Still, this is going to be messy for a while. The U.S. Chamber of Commerce has filed a lawsuit against the F.T.C. to block the rule, arguing that the agency doesn’t have the power to issue such a ban and that even if it did, a categorical ban isn’t lawful. Eugene Scalia, who was President Donald Trump’s secretary of labor for a year and a half, also filed a lawsuit, this one on behalf of Ryan L.L.C., a tax services firm in Texas whose chief executive, Brint Ryan , is a Republican donor who has advised Trump.

Even if the F.T.C. wins on the legality of its rule, enforcing it is going to be tricky. Let’s say a company gets rid of its noncompete clause, but it imposes a nondisclosure agreement that’s so broad and strict that it has the same functional effect of preventing someone from taking a job elsewhere. According to the F.T.C., “such a term is a noncompete clause under the final rule.”

Arguing over what’s the same functional effect is going to keep a lot of lawyers busy. Same for nonsolicitation agreements and trade secret protection. “‘You can’t work for a competitor for a year’ is a pretty clear rule; ‘you can’t use our secrets at a competitor’ will mean more lawsuits,” Matt Levine, a columnist for Bloomberg Opinion, wrote Wednesday.

I admire the F.T.C. for looking at the entire economic landscape in evaluating the pros and cons of noncompete agreements, not just the interests of the employer and employee. It’s a bold step, though.

The Readers Write

You wrote that most right-to-work laws were passed in the 1940s and 1950s, when Southern states were solidly Democratic. True, but in the ’60s after the passage of the Civil Rights Act the Southern Democrats were wholly absorbed by the Republican Party. Right-to-work is an anti-union strategy implemented by the same power elite that discouraged workers in this most recent vote. Their failure is significant. When Southerners start thinking for themselves, I view that as a hopeful development.

Rebecca Bartlett Brattleboro, Vt.

I’m a 47-year union member enjoying my retirement with an old-fashioned, union-negotiated pension and lifetime medical coverage. As those Volkswagen workers told you, to a certain degree, it doesn’t matter who the president is when it comes to what union members are paid. But it does matter to all employees who the president appoints to critical agencies such as the National Labor Relations Board, the Occupational Safety and Health Administration, the Equal Employment Opportunity Commission, and many more. Those agencies have real day-to-day impact on workers’ lives and futures.

Jim Griffin King George, Va.

Concerning your newsletter on Donald Trump’s economic agenda: He is clearly advocating an isolationist strategy. One does not have to look far to see that isolationism is a dead-end street. Is there anything to love about North Korea’s economy? How about Brexit?

Bob Kerst San Francisco

Quote of the Day

“Got no diamond, got no pearl Still I think I’m a lucky girl I got the sun in the morning and the moon at night”

— Irving Berlin, “I Got the Sun in the Morning” (1946)

Peter Coy is a writer for the Opinion section of The Times, covering economics and business. Email him at [email protected] . @ petercoy

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2.17: Venn Validity for Categorical Syllogisms

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Matthew Van Cleave
Lansing Community College

A categorical syllogism is just an argument with two premises and a conclusion, where every statement of the argument is a categorical statement. As we have seen, there are four different types (forms) of categorical statement:

All S are P (universal affirmative) No S are P (universal negative) Some S are P (particular affirmative) Some S are not P (particular negative)

Thus, any categorical syllogism’s premises and conclusion will be some mixture of these different types of statement. The argument I gave at the beginning of section 2.13 was a categorical syllogism. Here, again, is that argument:

All humans are mortal
All mortal things die
Therefore, all humans die

As we can see now that we have learned the four categorical forms, each one of the statements in this syllogism is a “universal affirmative” statement of the form, “all S are P.” Let’s first translate each statement of this argument to have the “all S are P” form:

All humans are things that are mortal.
All things that are mortal are things that die.
ll humans are things that die.

In determining the validity of categorical syllogisms, we must construct a three category Venn diagram for the premises and a two category Venn diagram for the conclusion. Here is what the three category Venn looks like for the premises:

Screen Shot 2019-10-22 at 11.17.52 PM.png

We need a three category Venn for the premises since the two premises refer to three different categories. The way you should construct the Venn is with the circle that represents the “S” category of the conclusion (i.e., the category “humans”) on left, the circle that represents the “P” category of the conclusion (i.e., the category “things that die”) on the right, and the remaining category (“things that are mortal”) in the middle, as I have done above. Constructing your three category Venn in this way will allow you to easily determine whether the argument is valid.

The next thing we must do is represent the information from the first two premises in our three category Venn. We’ll start with the first premise, which says “all humans are things that are mortal.” That means that we must shade out anything that is in the “human” category, but that isn’t in the “things that are mortal” category, like this:

Screen Shot 2019-10-22 at 11.18.34 PM.png

The next thing we have to do is fill in the information for the second premise, all things that are mortal are things that die. That means that there isn’t anything that is in the category “things that are mortal” but that isn’t in the “things that die” category. So we must shade out all of the parts of the “things that are mortal” category the lie outside the “things that die” category, like this:

Screen Shot 2019-10-22 at 11.19.04 PM.png

The next thing we have to do is construct a two category Venn for the conclusion and then compare the information represented by the three category Venn for the premises to the two category Venn for the conclusion.

Screen Shot 2019-10-22 at 11.19.40 PM.png

The conclusion represents the information that there is nothing in the “humans” category that isn’t also in the “things that die” category. It also allows that there are things that die, but that aren’t humans. The premise Venn also includes this same information, since every part of the “humans” category that is outside the “things that die” category is shaded out. Thus, this argument passes the Venn test of validity and is thus valid since there is no information represented in the conclusion Venn that is not also represented in the premise Venn. Notice that it doesn’t matter that the premise Venn contains more information than the conclusion Venn. That is to be expected, since the premise Venn is representing a whole other category that the conclusion Venn isn’t. This is perfectly allowable. What isn’t allowable (and thus would make an argument fail the Venn test of validity) is if the conclusion Venn contained information that wasn’t already contained in the premise Venn. However, since this argument does not do that, it is valid.

Let’s try another one.

All pediatricians are doctors
All pediatricians like children
Therefore, all doctors like children.

The first step is to identify the three categories referred to in this categorical syllogism. They are:

Pediatricians Doctors Things that like children

The next step is to fill out the three category Venn for the premises and the two category Venn for the conclusion.

Screen Shot 2019-10-22 at 11.20.33 PM.png

This argument does not pass the Venn test of validity because there is information contained in the conclusion Venn that is not contained in the premise Venn. In particular, the conclusion says that there is nothing in the “doctors” category that is outside the “things that like children category.” However, the premises do not represent that information, since the section of the category “doctors” that lies outside of the intersection of the category “things that like children” is unshaded, thus representing that there can be things there.

Sometimes when filling in particular statements on a three category for the premises, you will encounter a problem that requires another convention in order to accurately represent the information in the Venn. Here is an example where this happens:

Some mammals are bears
Some two-legged creatures are mammals
Therefore, some two-legged creatures are bears

There are three categories referred to in this categorical syllogism:

Mammals Bears Two-legged creatures

As always, we will put the “S” term of the conclusion on the left of our three category Venn, the “P” term on the right, and the remaining term in the middle, as follows:

Screen Shot 2019-10-22 at 11.21.26 PM.png

Now we need to represent the first premise, which means we need to put an asterisk in the intersection of the “mammals” and “bears” categories. However, here we have a choice to make. Since the intersection of the “bears” and “mammals” categories contains a section that is outside the “two-legged creatures” category and a section that is inside the “two-legged creatures” category, we must choose between representing the particular as part of the “two-legged creatures” category or not.

But neither of these can be right, since the first premise says nothing at all about whether the thing that is both a bear and a mammal is two-legged! Thus, in order to accurately represent the information contained in this premise, we must adopt a new convention. That convention says that when we encounter a situation where we must represent a particular on our three category Venn, but the premise says nothing about a particular category, then we must put the asterisk on the line of that category as I have done below. When we do this, it will represent that the particular is neither inside the category or outside the category.

Screen Shot 2019-10-22 at 11.22.49 PM.png

We must do this same thing for the second premise, since we encounter the same problem there. Thus, when putting the asterisk in the intersection of the “two-legged creatures” and “mammals” categories, we cannot put the asterisk either inside or outside the “bears” category. Instead, we must put the asterisk on the line of the “bears” category. Thus, using this convention, we can represent the premise Venn and conclusion Venn as follows:

Screen Shot 2019-10-22 at 11.23.41 PM.png

Keeping in mind the convention we have just introduced, we can see that this argument fails the Venn test of validity and is thus invalid. The reason is that the conclusion Venn clearly represents an individual in the intersection of the “two-legged creatures” and “bears” categories, whereas the premise Venn contains no such information. Thus, the conclusion Venn contains information that is not contained in the premise Venn, which means the argument is invalid.

We will close this section with one last example that will illustrate an important strategy. The strategy is that we should always map universal statements before mapping particular statements. Here is a categorical syllogism that illustrates this point. This time I am going to switch to just using the capital letters S, P, and M to represent the categories. Recall that we can do this because the Venn test of validity is a formal evaluation method where we don’t have to actually understand what the categories represent in the world in order to determine whether the argument is valid.

Some S are M
All M are P
∴ Some S are P

If we think about mapping the first premise on our three category Venn, it seems that we will have to utilize the convention we just introduced, since the first premise is a particular categorical statement that mentions only the categories S and M and nothing about the category P:

Screen Shot 2019-10-22 at 11.24.35 PM.png

However, as it turns out, we don’t have to use this convention because when we map premise 2, which is a universal statement, this clears up where the asterisk has to go:

Screen Shot 2019-10-22 at 11.25.00 PM.png

We can see that once we’ve mapped the universal statement onto the premise Venn (on the left), there is only one section where the asterisk can go that is in the intersection of S and M. The reason is that once we have mapped the “all M are P” premise, and have thus shaded out any portion of the M category that is outside the P category, we know that that asterisk cannot belong inside the M category, given that it has to be inside the P category. When we apply the Venn test of validity to the above argument, we can see that it is valid since the conclusion Venn does not contain any information that isn’t already contained in the premise Venn. The conclusion simply says that there is some thing that is both S and P, and that information is already represented in our premise Venn. Thus, the argument is valid. The point of strategy here is that we should always map our universal statements onto our three category Venns before mapping our particular statements. The reason is that the universal can determine how we map our particular statements (but not vice versa).

Use the Venn test of validity to determine whether the following syllogisms are valid or invalid.

1. All M is P All M is S ∴ All S is P

2. All P is M All M is S ∴ All S is P

3. All M is P Some M is S ∴ Some S is P

4. All P is M Some M is S ∴ Some S is P

5. All P is M Some S is M ∴ Some S is P

6. All P is M Some S is not M ∴ Some S is not P

7. All M is P Some S is not M ∴ Some S is not P

8. All M is P Some M is not S ∴ Some S is not P

9. No M is P Some S is M ∴ Some S is not P

10. No P is M Some S is M ∴ Some S is not P

11. No P is M Some S is not M ∴ Some S is not P

12. No M is P Some S is not M ∴ Some S is not P

13. No P is M Some M is not S ∴ Some S is not P

14. No P is M No M is S ∴ No S is P

15. No P is M All M is S ∴ No S is P

16. No P is M All S is M ∴ No S is P

17. All P is M No S is M ∴ No S is P

18. All M is P No S is M ∴ No S is P

19. Some M is P Some M is not S ∴ Some S is not P

20. Some P is M Some S is not M ∴ Some S is P

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categorical proposition, in syllogistic or traditional logic, a proposition or statement, in which the predicate is, without qualification, affirmed or denied of all or part of the subject. Thus, categorical propositions are of four basic forms: "Every S is P," "No S is P," "Some S is P," and "Some S is not P." These forms are designated by the letters A, E, I, and O ...
6: Categorical Logic
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Categorical Logic: Terms and Propositions
As we may already know, our main goal in logic is to determine the validity of arguments. And in categorical logic, we will employ the Eight (8) Rules of Syllogisms for us to be able to determine the validity of an argument.But since the 8 rules of syllogisms talk about the quantity and quality of terms and propositions, then it is but logical enough to discuss the nature of terms and ...
Understanding categorical logic can be valuable for nurses ...
In summary, categorical logic is important for nurses because it enhances their critical thinking, communication, decision-making, and problem-solving skills, all of which are crucial for delivering high- quality patient care and ensuring patient safety.
The Mostly Persuasive Logic Behind the New Ban on Noncompetes
The F.T.C. defined senior executives as people earning more than $151,164 per year who are in a "policy-making position" and estimated that fewer than 1 percent of workers meet the description ...
2.17: Venn Validity for Categorical Syllogisms
Introduction to Logic and Critical Thinking 2e (van Cleave) 2: Formal Methods of Evaluating Arguments 2.17: Venn Validity for Categorical Syllogisms ... A categorical syllogism is just an argument with two premises and a conclusion, where every statement of the argument is a categorical statement. As we have seen, there are four different types ...