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Short Little Lessons in Logic: Symbolization

Lesson 8: Symbolization

What you'll learn in this lesson:

  • Using variables to better analyze form
  • More on the difference between structure and meaning
  • How to symbolize an argument

As we learned in the lessons in Module 1 of this course, statements, written as sentences, are symbols that represent an idea or meaning called propositions. In this lesson, we're going to take that idea further and show how we can use symbols within statements to better identify the structure of an argument and help us mentally separate the structure from the meaning. This is important because when analyzing an argument using our two-step process, we can get distracted with the content of the argument and neglect looking at the structure first. For example, consider this truth claim:

  1. Global warming is a myth and is not harming the planet

Most people who have considered global warming most likely have an opinion about the truth value of that statement. The statement also might evoke some kind of an emotional reaction and in reading it, you might be inclined to get very angry or to be someone comforted by the claim. If you disagree, it can be tempting to jump right into arguing that the claim is false and you might forget to see whether the argument its a part of has the right structure. We can write the same statement symbolically this way:

  1. ~G and ~H

That's much less interesting but it also allows us to look at the claim without the emotional reaction that can come with it. Emotions are important and we're not saying that they should be ignored. But emotionally reacting to a statement too early can cause us to lose sight of the argument as a whole and could lead to committing a variety of fallacies---something we'll talk about in a future lesson. By focusing on the structure of the claim first, we can see how it functions in an argument and determine whether the argument is one we should pay attention to. This is an important discipline in being a critical thinker. We want to focus on the rationality of the argument and then worry about the truth value of the claims the argument is making.

Symbolization in Logic

Think of symbolizing as part of the grammar or formula of an argument. In a language like English, a lot can be communicated even with poor grammar. But similar to mathematics, in logic, the grammar of an argument is essential. Symbolizing helps us abstract the proper forms from the improper ones to make analysis and reusing those forms easier. You may have encountered symbolization in a math course you took at some point in your education. You may have been given a test with a question that went something like this:

Find the value of X in this formula: 3 + Y = 7.

It's likely that you can look at this formula and just know that Y is 4 or you solve this by subtracting 3 from 7 and learn that Y represents the number 4. In either case Y is a symbol for the number 4. We can put this another way: 4 replaces Y in this formula. This replacement works because the formula simple addition with an abstract structure we can represent like this: X + Y = Z. We can replace X, Y, and Z with numbers and know that the addition relation will always work in the same way. Philosophers say that the formula 3 + 4 = 7 is an instance or token of an abstract formula or type of the formula X + Y = Z.

We can use this replacement model is logic. Suppose we want to argue that carrots are edible. We could develop the following argument:

  1. All vegetables are edible
  2. A carrot is a vegetable
  3. Therefore carrots are edible

This argument follows a very specific form in categorical logic (something we'll study later on in the course) called "Barbara". In order to see this form, we can write the argument this way:

  • Let V stand for Vegetable

  • Let E stand for Edible

  • Let C stand for Carrot

  • All V are E

  • C is a V

  • Therefore C is an E

Now this isn't exactly in the Barbara form and we'll see why later on. But we can see the symbolic nature of the argument better using the symbols. We can replace any words for V, E, and C and the argument will still be in the Barbara form and is considered to be in the right structure:

Let V stand for Automobile Let E stand for Engine Let C stand for Volkswagen Golf

  1. All automobiles are things that have an engine
  2. The Volkswagen Golf is an automobile
  3. Therefore the Volkswagen Golf is a thing that has an engine

We can even put nonsensical terms as replacements for the variables (the letters):

Let V stand for Toadstool Let E stand for Frignibbit Let C stand for Jabberwocky

  1. All toadstools are frignibbits
  2. The jabberwocky is a toadstool
  3. Therefore the jabberwocky is a frignibbit

The Barbara form is the abstract type and the arguments above are an instances or tokens of that Barbara type with the variables replaced with actual statements that have a truth value. We'll learn later that there are a limited number of forms that are considered 'right' (the term is 'valid' in logic) and a bunch of forms that are wrong and shouldn't be used. By symbolizing our statements we can more easily see the form the argument is in and analyze it. Symbolization also makes creating complex arguments much easier.

This gives us an overview of the value of symbolizing. Now we need to learn how to construct these argument forms and how they're used in logical analysis. We'll do that over the next few lessons.

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