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Short Little Lessons in Logic: More on Propositions

Lesson 6: More on Propositions

What you'll learn in this lesson:

  • What a proposition is
  • The relationship between statements and propositions
  • How the words of an argument (the syntax) relates to the meaning of the premises and conclusion (the semantics)

In the previous lesson, we looked at some foundational concepts in the discipline of philosophy called metaphysics to better understand how propositions work. If you haven't studied that lesson yet (and have no background in philosophy), you might find it helpful to spend some time on that lesson first. In this lesson, we'll look specifically at propositions and how they function in arguments.

As we've been learning, premises and conclusions are the "components," the building blocks of arguments. These components are statements that make claims that certain things are true. One problem we saw in the last lesson is that we can change the statement of an argument and it can be unclear whether each statement works the same in terms of their logical force. For example, I could have this premise---a declarative statement---that makes a truth claim:

  1. Everybody loves Raymond

But I could also write that same idea this way:

  1. All persons that met Raymond are persons that love Raymond

It may be obvious that sentence 1 is clearer and easier to read. Hopefully you also see that both statements mean roughly the same thing. So which should we use? Generally we want to use the clearest and most straightforward language in our arguments so we can focus on the logical force of what we're arguing rather than spending time simplifying what we're saying. But it the case above (as we'll see in later lessons), sometimes it's necessary to write a statement in a specific way so we better can see the logical relationship between the terms. Sentence 2 is an example of how we might need to write a statement in what is called a "categorical syllogism" --- a concept we'll study later.

The question before us right now is whether we can use different sentences and still retain the logical force of the argument. This is where propositions comes in!

Statements are symbols for propositions

As we saw in the previous lesson, a symbol is something points to or represents something else. The numeral 7 is a pointer to the number 7. This allows us to use a variety of symbols like the numeral, or Morse code, or a set of objects (like lemons or flashlights) or anything that allows us to point to or represent the number 7. In logic, statements are the symbols that point to propositions. Statements are the words that point to the meaning.

Consider these two English sentences:

  1. Jan likes Pat
  2. Pat is liked by Jan

These two sentences clearly are different grammatically (or syntactically). Sentence 1 has three words and sentence 2 has five. Sentence 1 starts with Jan and sentence 2 starts with Pat. Sentence 1 is in the active voice and sentence 2 is in the passive voice. Even though the syntax of the sentences are different, we can say that the mean the same thing. Put in the language we've been using in this course, they refer to the same truth claim or represent something (possibly) true about the world. We can represent the same idea even using different languages:

  1. The moon has craters
  2. La luna tiene créteres

Both mean the same thing---they refer to the landscape of the object that orbits the planet earth and both make a truth claim about the same reality. But that truth is stated in English and Spanish. The "thing" that both sentences point to or symbolize is the meaning of the statements---a proposition. There is a lot of debate in metaphysics and linguistics about whether propositions are necessary or if they exist, how do they exist (are they 'abstract objects?' and can such objects exist?) and the like. We won't get bogged down in that debate. Instead we'll focus on how propositions function in arguments.

Truth Value Again

To do this, we'll need to adjust the definition of truth value we learned in an earlier lesson. In that lesson, we said that statements either are true or false. But you should now see that this is incomplete. Statements actually only symbolize or point to what is true or false. It is actually the meaning of a statement, the proposition, that is true or false. This is how we can talk about who Jan likes or the surface of the moon using a lot of different statements but always refer to the same underlying meaning.

Philosopher Peter van Iwagen has given us a helpful and widely cited definition of a proposition:

A proposition is a non-linguistic bearer of truth value

Okay, that sounds a little intimidating. What does it mean? We should have all the tools to unpack this pretty easily now.

  1. Non-linguistic. This just means that a proposition doesn't have the properties of being grammatical or syntactical---it's not the type of thing that is language-based. So what properties does a proposition have? That's the second part of the definition:
  2. Propositions are the kind of things that "bear truth value" or are true or false. The idea is that when you say or write a statement, you're symbolizing a proposition and it is the proposition that is true or false.

Here's an illustration that may help. The sphere's represent claims about the world and true propositions are those that "fall under" what we refer to as "reality" or the way the world actually is. False propositions don't fall under any category. They're just claims that don't represent anything in the world.

image showing the relation of propositions to truth

Why is this important?

The concept of propositions is relevant because it allows us to state or restate claims in an argument to make the argument clearer or to structure the argument so it can be put into logical form as long as the statement we make captures the same exact meaning or propositional content. The examples above all show this being done in different ways. Propositions help us then to avoid being too rigid about syntax (terms and grammar) and focus on semantics (meaning). In fact, philosophers spend a lot of time debating the meaning of a word or the structure of an idea. But the goal is to get to the root meaning of that term or idea. And in logic, this becomes the primary focus. After all, this is the goal of an argument: to get someone to believe something is true about the world.

With this background, in the next lesson, we'll turn again to statements so we can continue to refine the building blocks we need to construct arguments.

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