Why We Define the Deductive/Inductive Distinction the Way We Do within the Field of Logic

Note: This post is entirely optional – for the interested only.

In logic courses, deductive reasoning is defined as a type of inference in which the premises are intended to guarantee the truth of the conclusion, whereas inductive reasoning gives premises that are intended only to provide evidence for the truth of the conclusion (I discussed this definition in more depth in an earlier post on the inductive/deductive distinction).

However, it is possible that you have heard someone define induction as reasoning from the specific to the general and deductive inference as going from the general to the specific (I will call this the "old definition"). The purpose of this post is to give the reasoning for why the distinction is not defined that way in the field of logic today.

Definitions are tricky things to argue about because different people can define the same word differently and because different definitions can serve different purposes (I got involved in an argument recently, for example, about whether Pop-Tarts are "sandwiches").

However, there are several ways that we can argue about which definitions are best, which I will outline below. If you are interested in this question, you are welcome to read the below. However, do not feel obligated at all, as this goes well beyond what is required for an intro logic course.

Overview of the Argument:

Since this is a semantic issue, there may not be an objective ‘fact’ about what definition (of any term) is ‘right.’ However, some semantic decisions can be better than others for reasons including the following:

1. Historical definitions of the term.

2. How authorities in the field define the term.

3. How it is most useful to define the term.

4. What definition best captures pre-theoretic intuitions about what the term means (the ‘concept’).

5. How people commonly use the term. 

The rest of this entry will attempt to show that all five of these support the new definition, at least within the field of logic.

Part 1: Historical Definitions

It has been said that Aristotle defined inductive reasoning as reasoning that goes from the specific to general (Groarke, n.d.; Kneale & Kneale, 1962). However, Aristotle’s word for this was epagōgē (ἐπαγωγή), which was translated as “induction” in the work of Cicero (etymonline, n.d.). It is not clear, therefore, that the “old definition” (especially not Aristotle’s version) was a definition of the same term at all. Cicero defined "induction" as having to do with reasoning based on similar cases (Fortenbaugh, 1996, p. 2), which is not synonymous with generalization. 

Philosophers by the nineteenth century were using the modern definition (that inductive inferences are based on probabilistic reasoning). However, in 1865, logician Charles Sanders Peirce came up with a different way of categorizing the terms “deductive,” “inductive,” and “abductive” according to their forms (Burch, 2018). He interprets induction as generalization. His distinctions are fascinating, but they did not become the dominant ones used in logic today (Burch, 2018).

In the early 20^th century, the successes formal developments within deductive logic inspired many to attempt to create formal principles of inductive reasoning, understood as probabilistic inference (Hawthorne, 2018). Induction defined as probabilistic inference generally, not limited to generalization, has become the dominant usage of the term, as verified in the next section.

Part 2: How Authorities in the Field Define the Term

In my house, I have a pile of 18 logic books. I have looked through how each of them makes the deductive/inductive distinction. Here is how they sort out:

The books on the left define it in the new way (deductive arguments attempt to give conclusive support for their conclusions, while inductive arguments try to give probabilistic evidence for their conclusions). The two on the (far) right do not define the distinction at all. The one in the middle is the only one that does not define it this way. It defines inductive arguments as asserting that the future will resemble the past, learning from experience. However, it then gives typical examples of the new definition of inductive reasoning (induction by enumeration, analogy, statistical reasoning, etc.) and goes on to explicitly reject the old definition, calling it a “misconception” and giving counter-examples (Kahane & Cavender, 2006). So this jury of scholars voted 16 to 0 (with two abstentions).

The next section outlines reasons this logical terminology may have come to be preferred.

Part 3: The Most Useful Way to Define the Term

Perhaps the reason that most philosophers have resolved upon the new version of the distinction is because it is more useful. Here is a visual of how the two ways carve up the taxonomy of common argument forms:

Old Way:

Deductive: Statistical syllogism Universal instantiation

Inductive: Enumerative Induction (aka inductive generalization)* Mathematical Induction**

Neither: Propositional Logic Most Predicate Logic (including Aristotelian syllogisms) Argument from Analogy Appeals to Authority Inference to the Best Explanation

*One could argue for including the universal generalization rule (from predicate logic) as a second type of “inductive” argument in this column. However, that rule is not really an argument but an inference rule that requires the use of a special arbitrary constant (sometimes indicated with a dot over it) and does not typically occur as an argument in daily life.
The principle of mathematical induction is a form of inference that occurs within mathematical proofs and not much in daily life.

New Way:

Deductive: Propositional Logic Predicate Logic (Including Aristotelian Syllogisms) Mathematical Inference

Inductive: Inductive Generalizations Argument from Analogy Appeal to Authority Inference to the Best Explanation Statistical Syllogism

This chart may help to clarify why the new way is more useful than the old way within the field of logic. Since universal generalizations and mathematical induction are generally used only in formal proofs, and rarely in daily life, in typical reasoning contexts, the old classification does not make a distinction that we did not have available simply by talking about inductive generalizations. Furthermore, the new version divides all of logic into two important categories, which strikes me as the real intention behind the deductive/inductive distinction within the field of logic.

Furthermore, the distinction made by the new version is more important. The new version makes a distinction that is crucial since the two types of reasoning have different standards for evaluating the quality of the inference (validity versus strength). The distinction of whether one is generalizing is less important in terms of the concept used to evaluate the strength of the reasoning (which is the more important skill within the field of logic).

Finally, in the old version, the majority of inferences would be classified as neither deductive nor inductive. Contrast this with the new definition, which divides reasoning into two roughly equal categories based on an important commonality (allowing us to evaluate them based on unified standards). Because the newer version provides a more valuable way of dividing up two important types of logical inferences, one can perhaps see why the old definition has been replaced overwhelmingly in the study of logic.

Part Four: The Pre-Theoretic Notion and Common Usage

I cannot speak for others in terms of what they feel is the “true” meaning of the concept, many people who give the old definition may have something like the following types of arguments in mind:

Deductive:

All dolphins are mammals.

Flipper is a dolphin.

Therefore, Flipper is a mammal.

Inductive:

Every crow I have seen has been black.

Therefore, all crows are black.

They may feel that the first type of inference is deductive because the information in the conclusion is ‘contained’ in some sense, within the premises. They may feel that the second one is inductive because the information in the conclusion goes beyond what is contained in the premises. In this sense, inductive inference is ampliative (Haack, 1992).

However, this intuition is not what is captured by the old definition. Here are some counterexamples to the idea that a non-ampliative argument must go from general to specific and that an ampliative one must go from the specific to the general.

Is this argument deductive or inductive?

All dogs are mammals.

All mammals are animals.

Therefore, all dogs are animals.

According to the new definition, this argument is stereotypically deductive, but it does not go from the general to the specific. It does not go from the specific to general either, so according to the old definition, it is neither deductive nor inductive.

Here’s another example: Is this inductive or deductive?

97% of all people enjoy ice cream.

Therefore, Mike probably enjoys ice cream.

According to the old definition, this would qualify as deductive (since the premises are general and the conclusion is specific), though few in logic would call this argument deductive. Rather this argument strikes many as paradigmatically inductive since the conclusion is intended only to be supported with evidence rather than proven. This probabilistic inference is what is captured by the new definition.

In fact, according to the old definition, nearly the entire field of formal logic would not qualify as deductive, including propositional logic (which is frequently called “deductive logic” in university courses). Take, for example, one of the paradigm instances of a deductively valid argument form, modus ponens:

If P then Q

P

Therefore, Q

Though this form is valid, according to the old definition it would not use deductive logic, since its premises and conclusion have the same level of generality. This again violates standard intuitions about the distinction today. Furthermore, to call this argument non-deductive would also require divorcing the concept of validity from deduction.

Aside: Some supporters of the old definition will defend modus ponens as deductive by saying that the inference from the general form to each of its instances is from the general to the specific. However, to say that the inference from the form to each of its instances is deductive that is not to say that the form is deductive nor that its instances are deductive. It would be to say that the real argument is the second order inference. Such a response presupposes we have to see propositional logic, as well as most Aristotelian syllogisms, as really second-order logic; which would be a very controversial move (many philosophers of logic do not accept second order logic as logic at all).

To call Aristotelian syllogisms, the whole field of truth-functional logic and much of predicate logic not deductive is, in light of the hundreds of the content of hundreds of courses called “deductive logic” that cover all of those topics, would be in plain contradiction to standard professional use within the field of logic.

I bring these examples up not just to show the implications of the old definition, but to show how this usage is in violation of how we use the terms ‘deductive’ and ‘inductive’ routinely in logic today.

There are many in the fields of rhetoric and science, I am told, that define induction as generalization (DeLaplante, 2009). If this definition better satisfies the purposes of those disciplines, then it makes sense for them to use it that way. This can create an ambiguity within a word, with different fields meaning different things by the same word. This can be confusing, but it allows disciplines the freedom to define things in ways that are most useful within their own field (I have learned recently that scientists have defined the word "berry" in a way that included cucumbers and eggplants but excludes strawberries and blackberries. One could argue that this word is used ambiguously as well). The purpose of this post is to explain why we defined the term the way we do within the field of logic.

In summary, I have attempted to make a case that defining the deductive/inductive distinction in terms of the argument’s attempt to be valid versus strong (those terms are defined in another post) is a more useful distinction than the “old definition” and that perhaps this explains why it is the distinction overwhelmingly used by scholars in the field of logic today.

References:

Burch, R. (2018). Charles Sanders Peirce. The Stanford Encyclopedia of Philosophy. Retrieved from https://plato.stanford.edu/entries/peirce/

DeLaplante, K. (2009, November 15). Induction and scientific reasoning [Video file]. Retrieved January 2, 2020 from https://www.youtube.com/watch?v=w-bm-Cxg40E

Etymonline. Induction. Retrieved from https://www.etymonline.com/search?q=induction

Fortenbaugh, W. (1996). Cicero, On Invention 1.51-77: Hypothetical Syllogistic and the Early Peripatetics. New Brunswick, NJ: Rutgers University, 2. Retrieved from https://orb.binghamton.edu/cgi/viewcontent.cgi?article=1188&context=sagp

Groarke, L. F. (n.d.). Aristotle’s Logic. In Internet Encyclopedia of Philosophy. Retrieved from http://www.iep.utm.edu/aris-log/#H11

Haack, S. (1992). Philosophy of Logics. New York, NY: Cambridge University Press, 12.

Hawthorne, J. (2018). Inductive logic. The Stanford Encyclopedia of Philosophy. Retrieved from https://plato.stanford.edu/entries/logic-inductive/#InduLogiInduProb

Kahane, H. & Cavender, N. (2006). Logic and Contemporary Rhetoric: The Use of Reason in Everyday Life. Belmont, CA: Thomson Higher Education, 42.

Kneale, W. & Kneale, M. (1962). The Development of Logic. New York, NY: Oxford University Press, 36.