Categorical Propositions - Science Assignment Help

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Categorical Propositions

Up to this point in the class, we have learned about the different parts that make up arguments, general ways to measure the strength or weaknesses of these parts, and a specific set of tools that help us to identify weak arguments called logical fallacies. Now we will begin the difficult task of learning how to construct arguments.

As a reminder, think back to the definition of an argument. An argument is a series of propositions that aim to support a conclusion. This is different than simply stating facts. Facts don't need to be supported with evidence because there's no controversy, they're simply true. Arguments, on the other hand, always attempt to prove that a conclusion is true. There is no clear standard for identifying an argument, except to ask yourself, is this person stating something as a fact or are they trying to convince me or someone else that they're right?

The first set of argument that you will be identifying and constructing appear to simply be statements of facts. However, their entire purpose is to prove that they are indeed facts. In order to get a better sense of what I'm attempting to convey, we need to begin by taking a step back and thinking about the following question: how do we know what we know is true or false. Rather than pointing to examples from Asian American studies and the experiences of Asian Americans, which many of us are still learning about, let's begin with some general examples to help us begin thinking about this very complex question.

At some point during primary school, many of us learned about the differences between mammals and other kinds of species throughout the animal kingdom. We might have learned, for example, that whales are mammals. How do we know that whales are mammals apart from the fact that it was told to us in a science textbook? One way that we know that whales are mammals is through deductive reasoning.

In a deductive argument, the argument explicitly assumes that the conclusion is true. Really try to commit the following two statements to memory. In a deductive argument, (1) the premises are intended to provide such strong support for the conclusion that, if the premises are true, then it would be impossible for the conclusion to be false. And this is why (2) an argument that successfully guarantees that the conclusion is true is called deductively valid. It's rare to be able to be one-hundred percent certain about anything. Deductive arguments provide a tool for proving something to be as close to certain as possible; in other words, deductive arguments help us to establish facts. These two rules about validity are going to be your touchstone for the next few lessons, they're how you know a deductive argument is either very strong or very weak.

In a deductive argument, we begin with facts that we do know. For example, when it comes to mammals and whales, we start with what we know about both. And from that, we can deduce a new fact about the two. One fact we might know is that all mammals breathe by means of lungs and we also know that whales breathe by means of lungs. From what we know about the two, we can then deduce that whales are in fact mammals.

If we were to diagram this deductive argument using the method of argument analysis that we learned in lesson 4, it would look like the following:

1. All mammals breathe by means of lungs.
2. Whales breathe by means of lungs.
3. Whales are mammals.
1 + 2
?
3

This is an example of the inference pattern known as the categorical syllogism. A syllogism refers to various types of deductive arguments that have two premises and a conclusion. One of those types called a categorical syllogism contains premises and a conclusion that are categorical propositions. And categorical propositions are exactly what they sound like. They are statements proving that all or some parts of something in one category are part of another category.

It might sound really simple to begin this way. After all, there are comparatively fewer debates about categorical propositions than ones that contain much more complex variables. Apart from controversies around things like race, gender, or sexuality, there are simply more ways to prove categorical knowledge than abstract knowledge. Yet categorical syllogisms provide the foundation for more complex critical thinking. It's like arithmetic in math. We can only begin thinking about complex issues that contain multiple types of propositions and inferences because this knowledge is built on top of what we categorically know.

At what point can the temperature of the ocean change due to global warming so that sea life, including whales, can continue to exist? As we know, whales are incredibly intelligent. Are there any similarities or even a relationship between how whales think and human thinking? The only reason we can begin exploring these hotly debated questions is because of the categorical knowledge that we've accumulated about whales and things that aren't whales. In this lesson, we will examine the basic structure of categorical propositions.

Components of Categorical Propositions

(Note: In the following section, there are going to be quite a few concepts and definitions, so be sure to take notes. New information continually builds on top of previous information, so if you get lost, stop and read the previous paragraphs until you get back on track.)

A categorical proposition can be understood as an assertion about the relations between classes. Obviously we're not talking about socio-economic classes, we're talking about classes in terms of classification, as in species and genera (plural of genus). This is easy to see in the example above. Whales are a species of animal; mammals are the genus to which that species belongs. The proposition "Whales are mammals" says that the first class is included in the second. Therefore, every categorical proposition says that a certain relationship exists between two classes.

All categorical propositions have parts or components that make up the larger proposition. Since these are arguments that attempt to prove relationships between classes, the parts aim at helping us identify exactly what classes are being focused on. The parts of the proposition that refer to the classes are called the terms of the proposition, and there are two terms: the subject and the predicate, symbolized by S and P. In our example, the subject is "whales" and the predicate is "mammals." Some of you learned about subject and predicate during elementary school when you learned about grammar. The subject tells us what the sentence is about and the predicate tells us some thing or some things about the subject. In logic, the subject tells us what the proposition is arguing about and the predicate tells us what other class it is being included within.

In the proposition "Whales breathe by means of lungs," the phrase "breathes by means of lungs" indicates a property that some objects have. But for any property there's a class of things that have that property. In this example, things that breathe by means of lungs. So we would rewrite the proposition as "Whales are things that breathe by means of lungs," in order to make it clear that we are talking about two classes.

The subject and predicate are not always single words. In fact, that's rarely the case. More often, one or both of the terms is a complex phrase. But phrases can designate classes of things just as well as individual words can. In each of the following examples, the subject and predicate terms are set off by parentheses.

1. (Computers) are (electronic machines that can be programmed to follow a sequence of instructions).
2. (Soldiers who have won the Medal of Honor) are (heroes).
3. (Commodities such as corn and wheat) are (economic goods subject to the law of supply and demand).

In 1, the subject is a single word, but the predicate is not. The opposite is true in 2. And in 3 neither term is a single word. But all three have the have the same form: Ss are Ps.

In addition to the subject and predicate, there is a third element of categorical propositions, indicated by the words "is" or "are." This element-called the copula-links subject and predicate. In all the examples so far, the copula has been affirmative. We said S is P. But the copula can also be negative, as in the propositions:

4. Whales are not fish.
5. Copper is not a precious metal.
6. Thailand is the only country in Southeast Asia that has never been colonized.

In terms of classes we can make both the affirmative statement that S is included in P and the negative statement that S is excluded from P. The affirmative or negative character of a proposition is called its quality.

On the other hand, quantity is another component of a categorical proposition and the final one that we'll be discussing. It's also less obvious than the others. The subject of "Whales are mammals" is "whales," and it's clear that we are talking about all of them. But sometimes we make statements about only some members of a class: Some whales are fish-eating carnivores, some politicians are crooks, some of Bong Joon-ho's films are not about sci-fi monsters. In ordinary language, we don't often say "all" or "some" explicitly; the context makes it clear which we mean. But in critical thinking, the difference is crucial and it needs to be explicit.

A proposition with the form "All S are P" is universal in quantity. A proposition with the form "Some S are P" is particular. The distinction between universal and particular also applies to negative as well as affirmative propositions. Thus, "Some good tablet computers are not Apple products" is a particular negative proposition. "No freshman is a varsity player" is a universal negative proposition. Notice that the word "no" does double duty: It indicates both the negative quality and the universal quantity of the proposition.

A categorical proposition, then, has four components: (1) a subject term; (2) a predicate term; (3) a copula, which is either affirmative or negative in quality; (4) one or more words indicating quantity, universal or particular. The quality and quantity, taken to together, determine the logical form of a proposition; the subject and predicate determine its content. Thus, the two statements "All whales are mammals" and "All snakes are reptiles" have the same logical form-affirmative and universal-although their content is quite different.

Since there are two possible qualities, and two possible quantities, there are altogether just four standard logical forms for categorical propositions, no matter how complex their subject and predicate terms may be. These four forms are:

Each of these standard forms has a traditional label that we will use as a shorthand reference to the form. The two affirmative forms are A and I, the first two vowels of the Latin word affirmo ("I affirm"). The negative forms are E and O, the vowels in nego ("I deny").

There is an astounding number of mathematical and algorithmic equations that have been produced by these four letters. There is so much that the logical relationship between these four letters can tell us about the natural world. However, for this class, these components, including the quality and quantity, of categorical propositions helps to create a vocabulary for discussing the more complex types of deductive arguments that you will be introduced to shortly. The next lesson will walk us through how categorical propositions produce categorical syllogisms, the most basic form of deductive argument.

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