Introduction to formal methods (Dr. Van Cleave)

By Matthew Van Cleave

What are formal methods of evaluation and why do we need them?

Earlier we introduced the concept of validity and the informal test of validity.  According to that test, in order to determine whether an argument is valid we ask whether we can imagine a scenario where the premises are true and yet the conclusion is false.  If we can, then the argument is invalid; if we can’t then the argument is valid.  The informal test relies on our ability to imagine certain kinds of scenarios as well as our understanding of the statements involved in the argument.  Because not everyone has the same powers of imagination or the same understanding, this informal test of validity is neither precise nor objective.  For example, while one person may be able to imagine a scenario in which the premises of an argument are true while the conclusion is false, another person may be unable to imagine such a scenario.  As a result, the argument will be classified as invalid by the first individual, but valid by the second individual.  That is a problem because we would like our standard of evaluation of arguments (i.e., validity) to be as precise and objective as possible, and it seems that our informal test of validity is neither.  It isn’t precise because the concept of being able to imagine x is not precise—what counts as imagining x is not something that can be clearly specified.  What are the precise success conditions for having imagined a scenario where the premises are true and the conclusion is false?  But the informal test of validity also isn’t objective since it is possible that two different people who applied the imagination test correctly could come to two different conclusions about whether the argument is valid.  As I noted before, this is partly because people’s understanding of the statements differ and partly because people have different powers of imagination. 

The goal of a formal method of evaluation is to eliminate any imprecision or lack of objectivity in evaluating arguments.  As we will see, logicians have devised a number of formal techniques that accomplish this goal for certain classes of arguments.  What all of these formal techniques have in common is that you can apply them without really having to understand the meanings of the concepts used in the argument.  Furthermore, you can apply the formal techniques without having to utilize imagination at all.  Thus, the formal techniques we will survey in this chapter help address the lack of precision and objectivity inherent in the informal test of validity.  In general, a formal method of evaluation is a method of evaluation of arguments that does not require one to understand the meaning of the statements involved in the argument.  Although at this point this may sound like gibberish, after we have introduced the formal methods, you will understand what it means to evaluate an argument without knowing what the statements of the argument mean.  By the end of this chapter, if not before, you will understand what it means to evaluate an argument by its form, rather than its content. 

However, I will give you a sense of what a formal method of evaluation is in a very simple case right now, to give you a foretaste of what we will be doing in this chapter.  Suppose I tell you:

This statement is a conjunction because it is a complex statement that is asserting two things:

These two statements are conjoined with an “and.”  So the conjunction is really two statements that are conjoined by the “and.”  Thus, if I have told you that it is both sunny and warm today, it follows logically that it is sunny today.  Here is that simple argument in standard form:

This is a valid inference that passes the informal test of validity.  But we can also see that the form of the inference is perfectly general because it would work equally well for any conjunction, not just this one.  This inference has a particular form that we could state using placeholders for the statements, “it is sunny today” and “it is warm today”:

We can see that any argument that had this form would be a valid argument.  For example, consider the statement:

That statement is a conjunction of two statements that we can capture explicitly in the first premise of the following argument:

Regardless of whether you know what the statements in the first premise mean, we can still see that the inference is valid because the inference has the same form that I just pointed out above.  Thus, you may not know what “Kant” is (one of the most famous German philosophers of the Enlightenment) or what a “deontologist” or “Pietist” is, but you can still see that since these are statements that form a conjunction, and since the inference made has a particular form that is valid, this particular inference is valid.  That is what it means for an argument to be valid in virtue of its form.  In the next section we will delve into formal logic, which will involve learning a certain kind of language.  Don’t worry: it won’t be as hard as your French or Spanish class.