Short Little Lessons in Logic: The Foundation of Propositions




Lesson 5: The Foundation of Propositions

What you’ll learn in this lesson:

  • Background on propositions
  • The metaphysics of abstract objects
  • How symbols work

In the last lesson we saw that premises and conclusion in logic are made up of statements. Statements make up the syntactic structure of an argument. But if you’ve ever tried to argue with someone (most of us have), you may have noticed that many times the person you’re talking to doesn’t really understand what you’re claiming or you don’t understand what they’re claiming. Sometimes you may say something like, "Let me rephrase that so it’s clearer." or "Can you say that using different words so I get a better idea of what you mean?" In other words, you get bogged down in the syntax of the terms you’re using and haven’t really gotten to the meaning of the claims you intend to make.

Here’s an example. Suppose someone wanted to declare that a person defending themselves in a trial isn’t going to win. They may say to you:

1. "The epistemic position of the defense side of the litigation lacks justificatory veracity."

This certainly seems to be a declarative statement but the terms may be entirely unfamiliar to you so you don’t really understand what the person is saying. You ask the person to clarify what they mean and they say in response,

2. "The defendant failed to make her case."

This is much clearer and now you can better understand what the claim is. But the problem is that the person exchanged the second statement for the first. She essentially changed the declaration by changing the sentence she used. Didn’t she just change the claim by using different words? Is this allowed? The short answer to this last question is "Yes!" In this lesson and the next lesson, we’ll see why.

The Metaphysics of Propositions

Okay, the title of this section certainly is a mouthful but the ideas here aren’t really all that difficult. In fact, you use propositions each day, many times a day and may not even know it. To explain what propositions are and how they work, we’re going to take a brief detour and look at some relevant concepts in a discipline in philosophy called ‘metaphysics’. Metaphysics is the study of being and existence. It examines the nature of reality like what a relation is (the "on top of" relation for example), what it means to exist or not exist, what it means for something to have an identity or a nature (the "human nature" for example), and similar subjects. The term comes from the Greek words ‘meta’ which means over or above and ‘phusike’—the term for ‘matter’. So it’s a study of the foundations of anything that exists and is an important foundation for scientific study which explores how the physical world operates.

To get a better grasp on how propositions function in logical arguments, we’re going to look at how symbols work and use the metaphysics of what are called ‘abstract objects’ to illustrate the concepts.

Not everyone agrees with the concept of abstract objects which was first articulated by Plato centuries ago. But it’s not important that you accept or even understand all the concepts involving abstract objects. We want to focus on the nature of symbols in this discussion and that’s the part that is relevant for understanding propositions.

What is this:


If you said, "Well, that’s the number seven" you’d be wrong. That’s actually the numeral seven. This means that this is one example of, or a symbol of, something else. You could write the same symbol ‘7’ on a piece of paper or you may have a calendar on your wall or on your computer that have a bunch of ‘7s’ on them. You even could draw 7 dots on a piece of paper and that is a different way to symbolize what the numeral is symbolizing. But what are they the symbol of? What do each of these particular symbols have in common?

They all symbolize or represent or point to the same thing: the number 7!

The idea here is that all the symbols–the 7 above, the 7 on your calendar, the 7 dots on the piece of paper–all are representations of something else: the number. In metaphysics, the number 7 is called an abstract object and the numeral is an instance or particular of that number. For our purposes, we can say that the numeral (particular or instance) is a symbol; it points to, or represents the number. But why make this distinction? Why not just say that all the individual 7s are what 7 is? Why do we need a distinction between the numerals and the number? Philosophers use another metaphysical concept called properties to explain why.

Suppose you took a red marker and drew a large number 7 on a piece of paper that was about 10 inches tall. That numeral 7, philosophers say, has the following properties. It is:

  • Red
  • 10 inches tall
  • Made up of certain chemicals (those of the ink of the marker)
  • Located somewhere in space (on your desk or in your home or school which you could determine with GPS for example)

Now let’s ask if those properties are true of the number 7?

  • Is the number 7 red?
  • Is the number 7 10 inches tall?
  • Is the number 7 made up of chemicals or located somewhere in space?

The obvious answer to each of these questions is no. We don’t think of the number 7 as having those properties even though the numeral 7 you wrote on that paper definitely does. So what properties does the number 7 have? How about these:

  • It’s odd (as opposed to even)
  • It’s greater than 6 but less than 8
  • It can be used in mathematical relations like addition (7 + 2 = 9)

Notice that none of these are true of the numeral 7 you drew on the piece of paper. So some philosophers make a distinction between the abstract object that is the number 7 and the various symbols we use to represent or point to that object, the numeral 7s that we write, type, and speak. In addition to the language of abstract objects and particulars, philosophers call this the type/token distinction (also controversial). A ‘token’ is the individual ‘instance’ or thing like a numeral 7. The ‘type’ is the thing the token represents or is an instance of.

Again, it’s not critical that you fully accept (or understand) the idea of abstract objects or the metaphysics of properties that I’ve described here. What is important is that there at least appears to be a distinction between a symbol and the thing that symbol represents. This allows us to use a wide variety of symbols to represent the exact same thing and this is the value of the distinction. As we’ll see in the next lesson, this is the same distinction that exists between statements and propositions—a statement represents or is a symbol of the proposition. This is why we could replace sentence 1 above with sentence 2. And this is where the magic of logical arguments really begins!

Here’s a short video that explains the concepts above again. Hearing the terms again and seeing the visuals may help some learners better grasp the ideas. If the above makes sense to you, the video will just reinforce your learning.

With this background, we’re ready to look at propositions specifically and see how they function in arguments. They’re related to statements and we’ll explore that relationship as well.

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