2.13: Short Review of Propositional Logic

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So far in this chapter we have learned a formal method for determining whether a certain class of arguments (i.e., those that utilize only truth functional operators) are valid or invalid. That method is the truth table test of validity.
We have also learned a formal method for proving arguments are valid or invalid (the method of proof). The other important skill we have learned in this chapter so far is translating sentences into propositional logic. Thus, there are three different skills that you should know how to do:

Translate sentences from English into propositional logic
Construct truth tables in order to determine whether an argument is valid or invalid
Construct proofs to prove an argument is valid

It is important to reiterate that truth tables are the only formal method that allow us to determine whether an argument is valid or invalid; proofs can only show that an argument is valid, but not that it is invalid. You might think that you can use proofs to show that an argument is invalid—for example, if you are unable to construct a proof for an argument, that means that the argument is invalid. However, this doesn’t follow. There could be many reasons why you are unable to construct a proof, including that you just aren’t skilled enough to construct proofs. But the fact that you aren’t skilled enough to find a proof for an argument wouldn’t mean that the argument is invalid, it would just mean that you weren’t skilled enough to show that it is valid! So we cannot use one’s inability to construct a proof for an argument to establish that the argument is invalid. Again, only the truth table test of validity can establish that an argument is invalid.

The study of propositional logic has given us a way of understanding what “formal” means in the phrase, “formal logic.” We can see this clearly with the truth table test of validity. After we translate an argument into propositional
logic using constants and the truth-functional connectives, we don’t need to know what the constants mean in order to know whether the argument is valid or invalid. We simply have to fill out the truth table in the mechanical way we have learned and then apply the truth table test of validity (which is also a mechanical procedure). Thus, once an argument has been translated into propositional logic, determining whether an argument passes the truth table test of validity is something a computer could easily do. The translation from English to symbolic format is not as easy for a computer to do because successfully doing so depends on understanding the nuances of English. Although today there are computer programs that are pretty good at doing this, it has taken many years to get there. In contrast, any simple computer program from half a century ago could easily construct and evaluate a truth table using the truth table test of validity because this doesn’t take any understanding—it is simply a mechanical procedure. There are many different programs, many of which are readily available on the web, that allow you to construct and evaluate truth tables.

In contrast, the informal test of validity (from chapter 1) requires that we understand the meaning of the statements involved in the argument in order for us to be try to imagine the premises as true and the conclusion as false. Since this test requires the use of our imagination, it clearly also requires that we understand the meanings of the statements in the argument. The truth table test of validity does not require any of this. Since the truth table method does not require understanding of the meaning of the statements involved in the argument, but only an awareness of their logical form, we refer to it as a formal logic. Formal logic is a kind of logic that looks only at the form, rather than the content (meaning) of the statements. We can easily see this by constructing an argument where the atomic propositions use silly, made-up words, such as those from Lewis Carroll’s “Jabberwocky”:

If toves are slithy, then the borogoves are mimsy
Borogoves are not mimsy
Therefore, toves are not slithy

If we translate “toves are slithy” at “T” and “borogoves are mimsy” as “B” then the form of this argument is clearly modus tollens, which is one of the 8 valid forms of inference:

T ⊃ B
~B
∴ ~T

We can thus see that this argument is valid even though we have no idea what “toves” or “borogoves” are or what “slithy” and “mimsy” mean. Thus, propositional logic, which includes the truth table test of validity, is a kind of formal logic, whereas the informal test of validity is not. There are other kinds of formal logic besides propositional logic. In the next section I will introduce another kind of formal logic: categorical logic.