Deduction

Deduction is the type of reasoning that we study in a course on logic, such as in my Logic 0: An Intuitive, Practical Introduction to Zeroth-Order Logic. It is the type of reasoning that is prevalent within a mathematical proof. It takes place in the formal inference from premises to a conclusion, which does not question the truth of the premises. 

Example: A Modus Ponens Argument

For example, consider the following argument. 

This argument, which has the Modus Ponens form, is valid. It is not possible for the conclusion to be false, if the premises are true. When making the deductive inference from the premises to the conclusion, it is essential to assume that the premises are true. Then under this assumption, it would be a logical contradiction to claim that the conclusion is false. The contradiction in claiming the negation of the conclusion is the key characteristic of a valid deduction. Therefore, we are absolutely certain that the premises logically guarantee the conclusion—assuming the premises are true. 

Are the Premises True? 

But, in this example, there is nothing to guarantee that the premises are in fact true. As a matter of fact the sun may not rise tomorrow. And even if the sun does rise, there is nothing to guarantee that as a matter of fact Julia and her father will build a sand castle on the seashore. Are the premises actually true? 

Our certainty about the conclusion entirely depends on how certain we are about the truth of the premises. And there is no way to be absolutely certain about the premises. It is a matter of degree. We certainly strongly anticipate that the sun will rise tomorrow, and would be extremely shocked if it did not. But it isn't outside of the realm of possibility. And as far as her father taking Julia to the seashore—regarding that we would have to know about promises made, and promises kept, along with a myriad of other factors. 

The Inference Is the Deduction

Note, however, that the questions regarding the truth of premises are not questions about the deduction itself—because it is the inference that is the deduction. The inference occurs precisely in the logical move from the truth of the premises (assumed) to the truth of the conclusion. Given the form of the premises and the conclusion, the inference is valid. That is the end of the story as far as pure deduction goes. 

Induction

Not all of our inferences are deductive. And that is not to say that some of our inference are invalid, or somehow illogical. We certianly do engage in such pseudo-reasoning, but the point is that many of our legitimate inferences are are non-deductive. We make a logical move from premises to a conclusion without it being purely a matter of the form of the propositions involved. 

Example: Julia's Likely Inference

Julia from the argument above might, for example, infer that she and her father are going to build a sand castle on the seashore tomorrow, because her father promised that they would do so. Her father promised, and she as a consequence believes it. 

Example: How We Know What Julia Believes

How do we know that Julia believes her father? Well, for example, we might see her packing a beach bag this evening before bedtime. She wouldn't be packing things for the beach, if she didn't believe that she and her father were going to the beach tomorrow, right? 

Note the difference between these two examples. The second example is an instance of Modus Tollens argument form. 

Let:

The form of the argument is:

If we do some substitution, we can see the Modus Tollens form very clearly. 

Let:

Then the argument is:

Our inference about what Julia believes was a deductive inference—not an inductive inference. 

Julia's inference, on the other hand, does not have any deductive structure. Her father promised, and as a direct consequence of this Julia believes what her father promised. This is what we call "induction." 

Induction Pervasive

Now, we do often use deductive inference, but induction is nonetheless pervasive, because most of our reasoning involves matters of fact. 

Our inference that Julia believes she is going to the beach tomorrow was a deductive inference from the premises to the conclusion. However, our inference that Julia is packing for the beach is probably an inductive inference. We see Julia packing some things in some bag, and infer that she is planning for a trip to the beach—especially in the context where we heard her father promise that they were going to the beach. But where is the formal deduction? 

Likewise, our other inference was inductive—that Julia wouldn't be packing for the beach, if she didn't believe she was going to the beach tomorrow. Perhaps we come to learn that the whole scenario we observed was a ritual. Every evening Julia's father promises tomorrow they will go to the seashore. Before bedtime Julia packs for the beach. The next day they never go. Our inference didn't even contemplate such a thing. Why were we so sure that it was a sign of her belief? It wasn't because we articulated a set of premises, and thereby deduced our conclusion. —And even if we had, there would be new premises to analyze. 

Critical Thinking

Whenever we reason about matters of fact we eventually find some inductive inference. Deductive reasoning only gets us so far. We could study formal logic our whole life and never come to a deductive understanding of the simplest scenarios from our everyday life subsumed as it is within matters of fact. So, the study of induction is very important. How do we apply rational inference to real-world situations couched in matters of fact? In academia, we usually refer to this field of study as "critical thinking" as opposed to purely deductive "logic." And critical thinking is at the core of any college curriculum. So, it is of the utmost importance to be able to fully describe what critical thinking is. Of course, much of critical thinking is learning how to construct arguments that have a great deal of deductive structure. But with an eye to applying reason to the real world of matters of fact, we always come sooner or later to inductive inferences. 

A Negative Definition of Induction

So, what exactly is induction? How do we define it? One way is in a negative fashion: 

In the study of critical thinking and logic, this definition holds up very well, as far as accuracy. But there is something inherently dissatisfying about such negative definitions. We would much prefer a positive definition of what induction actually is—not just what it is not. 

Example: Julia's Unlikely Inference 

To get a clear understanding of the accuracy of the above negative definition, consider the following example. Let's suppose that Julia really does believe that she and her father are going to build a sand castle on seashore tomorrow, because—as unlikely as it may seem—she reasoned to herself in this way: Father promised. And if father promised it, then it is likely so. The inference here is a deduction. 

Let:

The form of the argument is:

Placing the word "likely" in one of the component propositions, does not change the fact that the inference is a deduction. So, according to our negative defintion, this is not induction. 

Probability and Statistics Not Characteristic of Induction 

Similarly, using mathematical techniques of statistics and probability does not make an inference inductive. As mathemematical sciences, probability and statistics resort to deductive inference wherever possible. Theorems in these fields are proved as deductively as possible. And applications of the theorems are calculated as deductively as possible. 

Nevertheless, if an argument involves any matter of fact, induction will come into play at some point. 

Mill's Attempt at a Positive Definition

Due to its pervasive presence and importance within human reasoning, we would really like to understand the nature of induction better, and be able to give a better definition of it. But no one has constructed a positive definition of induction that holds up under scrutiny. The most ambitious attempt, and perhaps the most successful, is found in John Stuart Mill's A System of Logic: Ratiocinative and Inductive. However, even with eight editions worth of revisions, Mill's construal repeatedly met with scathing rebuke. 

Two-Fold Epistemological Problem

Any attempt to positively define induction comes up against a two-fold epistemological problem. First, any matter of fact could be other than what it is. In other words, any proposition describing a matter of fact could be false. From this fact we can deduce that it is not possible to absolutely know the truth of any matter of fact. Any certainty regarding a material fact must always be a matter of degree. But that is not all, because—secondly—we do not precisely know how we even come to have any degree of confidence in the truth of matters of fact. 

Hume's Enquiry

David Hume does a great job at elaborating upon the two-fold epistemological problem of induction in An Enquiry Concerning Human Understanding. So, for anyone who wants a better definition of induction than the negative definition above, I recommend reading Hume. And to get an initial sense of the nature of induction in light of Hume's Enquiry, the best place to start is in section 4, "Sceptical Doubts Concerning the Operations of the Understanding."