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Short Little Lessons in Logic: Truth Value

Lesson 4: Truth Value

What you'll learn in this lesson:

  • The definition of truth value
  • How truth value functions in logical arguments
  • The difference between the form of an argument and the truth value of the statements of an argument

We saw in the previous lesson that statements are declarative --- they make a claim that something is true. While the concept of truth can be a complex one in philosophy (for a deeper discussion of truth, see the Philosophy News article "What is Truth?" ), for our purposes we can simply define truth as the idea that what is being claimed is the way the world actually is.

What is truth value?

In logic, a statement's truth value is whether the claim being made represents the way the world actually is. Some statements do and others do not. So, any statement in logic can have one of two truth values at any given time: true or false. In logic, a statement must be either true or false; it can't be neither and it can't be both true and false at the same time. We add the qualifier at the same time because some statements can be true at one time and false at another. Other statements are always true or always false. Consider these examples:

  1. A triangle has four sides (always false)
  2. The Sagrada Familia cathedral is in Barcelona, Spain (always true)
  3. It is raining in Seattle (sometimes true and other times false depending on the time the statement is made)

Statement 3 is interesting because as it stands, it can be ambiguous. So in order to function better in an argument, we may need to specify it by saying something like, "It rained in Seattle on January 3rd 2018". That's clearer and gives us better information about what exactly the truth claim is.

The idea of a statement's truth value is pretty simple in terms of how it functions syntactically---the way it functions in the structure of the argument. As most of us know, determining whether a statement actually is true or false is another matter and can be very complex. The ambiguities of language, gathering all the right facts, making the proper interpretation of those facts and a host of other issues can make this determination difficult and this complexity is the source of most of our social and professional debates. The challenge of determining what is true is a concept in the discipline called 'epistemology' or the study of knowledge. You can read more about theories that attempt to tackle these problems in the Philosophy News article What is Knowledge?.

Truth value and form

We need to explore one final concept before we leave this topic. When constructing or analyzing logical arguments, we will learn later that the process involves a specific type of arrangement of statements (premises and conclusion) in order to have the right relationship between the terms of the argument. This is called the arguments form and it is the foundation of the logical relation we talked about in an earlier lesson. The form of an argument should be considered separately from the truth value of the statements of the argument. This means that creating or analyzing an argument is a two-step process: look at the form the argument takes, then consider the truth value of the claims of the argument. Consider this argument:

  1. All shrews are nocturnal
  2. This animal is nocturnal
  3. Therefore, surely this animal is a shrew

Now, you haven't learned anything about logical form yet but if you look closely, you may notice intuitively that something has gone wrong with the form of the argument. In other words, you can't get to or derive the conclusion (statement 3) from the premises (statements 1 and 2). Even if you believed statements 1 and 2, they don't give the right information in the right way to give you reason to believe statement 3. Notice also that you may not even know what a shrew is or what it means to be nocturnal. You really don't have to know the meaning of those terms to see that the argument's form isn't quite right. This illustrates how the form and the truth value of the statements can be analyzed separately.

What do you do if you don't know the truth value of the statements but you still want to analyze the form? Logicians take the approach of assuming the premises are true in order to analyze the form of the argument. In the example above, if you don't know what a shrew is or what it means to be nocturnal, you still can assume that statements 1 and 2 are true and see if the conclusion follows from the assumed truth of the premises (see the lesson on arguments if you're not clear on how the premises and the conclusion relate to each other).

In future lessons, you'll learn the mechanics of how to construct arguments yourself and learn some of the many forms and argument can take so don't worry too much at this point if some of this is unclear.

With this background in truth value and how truth value relates to statements, we can now turn to propositions which can be a bit more challenging but are critical for understanding how statements and truth value are used to construct arguments.

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