Deduction

§4.1

Deduction is the strongest type of inference—the type in which it is impossible for the conclusion to be false when the premises are true. This is sometimes expressed by saying that, in a deduction, the premises necessitate the conclusion or that the conclusion follows necessarily from them. It is this feature of necessitation that Aristotle, the first logician, focused on when defining deduction.10 We will discuss shortly what it is about the premises of deductive arguments that makes the conclusions follow necessarily, but first let’s make a few observations and introduce a few terms. Unlike the sorts of inferences that will be discussed in the remaining sections, deductive inferences do not differ from one another in strength. Either an inference is deductive, or it is not. However, there are some arguments that may seem like deductions when they are not. To differentiate between the genuine deductions and the imposters, logicians call the genuine ones valid deductions and the imposters invalid. (We will turn soon to examples of each.) All valid deductive arguments have equally strong inferences, but this doesn’t mean that the arguments as a whole are equally strong, since their premises may still differ in epistemic status. Thus, the two things to do in evaluating a deductive argument are to determine whether it is valid and to assess the premises. If the deduction is valid and the premises are certain then the argument will make the conclusion certain. If it is valid and all but one of the premises are certain, then it will elevate the conclusion to the epistemic status of the remaining premise. If more than one of the premises is uncertain, then the argument will give us less reason to believe the conclusion than we have to believe the least certain of its premises. Now let’s consider how the premises of deductive arguments necessitate their conclusions, by looking at some examples. (For the time being ignore the symbolic representation.)

Maps 16-18:

All birds are animals. All parrots are birds. All parrots are animals.

All animals need food. All men are animals. All men need food.

All B are C All A are B All A are C

Let’s focus on the first of these examples. Notice that it is not because of anything special about the subject matter that the premises necessitate the conclusion. If we changed the argument to be about Volkswagens, cars, and Jettas, rather than birds, animals, and parrots, the premises would still necessitate the conclusion. The same holds if we changed the argument to make it about musicians, matadors, and marriage counselors. In this last case, both premises would be false, but if they were true (that is, if all musicians were matadors and all matadors, marriage counselors), then the conclusion would have to be true: all musicians would have to be marriage counselors. What makes the deduction work doesn’t have to do with the subject matter of the propositions involved, but with their structure and interrelation. This is called the form of the argument. Notice now that the argument immediately to the right of the one we have been considering has the same logical form. (If you don’t see this immediately, try replacing the phrase “need food” with “are things that need food” or “are food-needers”.) In order to focus on the forms of deductive arguments rather than their content, Aristotle introduced the practice of using letters to stand for the subjects and predicates of the propositions, leaving behind only words like “some”, “all”, “no”, “is” and “not” (and their variants). Thus, we can arrive at the symbolic representation, presented in the right above, of the form of the arguments we’ve been discussing. Here are some other forms of deductive arguments. (Again, the arguments next two each other share the same form which is represented symbolically on the right.)

Maps 19-21:

Maps 22-24:

Maps 25-27:

Aristotle went through all possible combinations of two premises and a conclusion that could be made with propositions of the form “All/Some A are/aren’t B” and figured out which were valid deductions. It turned out that all of the valid forms could be restated in terms of the four that we’ve already looked at. Why, in each of these cases, do the premises necessitate the conclusion? It is because one premise is a universal proposition saying something about everything of a certain sort and the other premise tells us that something else is that sort of thing. The conclusion then applies the universal premise about all things of the sort to the thing that the other premise tells us is a member of that sort. If the universal premise is really true, and the thing in question really belongs to the relevant sort, then the universal premise will, of course, have to apply to that thing.

Though most deductive arguments involve the application of universal propositions to particular cases, not all do. Some, which were focused on by later Greek and Medieval logicians, involve the application of hypothetical statements to actual cases or the application of statements about alternative possibilities to cases in which some of the alternatives have been ruled out. Here are some examples:

Map 28-29:

Map 30-31:

Map 32-33:

These argument forms involve complex propositions that are built up from simpler ones. When representing the arguments symbolically, we use lowercase letters to represent the simpler propositions.

Thanks to Aristotle and later logicians, the way deductions of different sorts work is well understood, and the various valid and invalid forms have been catalogued. But even among people who have not studied their works it is relatively rare to find sustained disagreement about whether an argument is valid. People are generally quite good at recognizing whether or not the premises follow necessarily from the conclusion.

That said, we do sometimes argue invalidly, particularly when we are not paying attention. Here are the three most common invalid argument forms (or “deductive fallacies”):

Map 34-35:

Map 36-37:

Map 38-39: