Inductive Reasoning

The ability to think logically and analytically is natural and exclusive for humans. It is our biological nature and identity to reason. It is what separates us from all other leaving things. Reasoning is what makes us human!

The 17th Century mathematician and philosopher René Descartes made famous the saying "Cogito, ergo, sum!" which translates to "I think, therefore, I am", or "I exist because I can think". This basically summarizes the theory of existentialism in philosophy which argues that (we are aware that) we exist because because we have the ability to think analytically and logically. Other things that do not have the ability to think are not aware that they exist.

In this lecture, we shall study inductive reasoning.

Inductive reasoning is the process of reaching a general conclusion by examining specific examples.

Inductive reasoning is also interpreted as "bottom-up" logic because it starts with smaller pieces of information an then makes a higher level of conclusion. In inductive reasoning, premises are considered evidences to support the conclusion. The more or the stronger these evidences are, the more likely the conclusion is true. However, it does not guarantee that the conclusion is true even if all of the premises are true.

Example.

1. All swans I have seen are white. Therefore, all swans in the world are white.

2. Anna bought some cake. Anna invites me over to her house. Anna is wearing red. Therefore, it's Anna's birthday.

3. Mother left the house just half a minute ago. Rain suddenly started falling. Mother forgot to bring an umbrella. Therefore, mother will return to get an umbrella.

Here's another version in the modern remake as a TV Series:

Unfortunately though, Sir Doyle, the author, mistakenly called Sherlock Holme's methods as deductive in the original novels although they are clearly inductive.

Inductive reasoning can be classified into five types:

1. Generalization

2. Prediction

3. Causal Inference

4. Inference of Events

5. Analogy

Generalization is the most basic form of inductive reasoning. In generalization, observations taken from some elements of a set are considered as premises. Then, the observation is generalized for the entire set. There are two sub-classifications of generalization: anecdotal and statistical.

Anecdotal generalization is used when a conclusion about a population is generated from a non-statistical sample.

Example.

1. Our dog eats rice. Therefore, all dogs eat rice.

2. The Los Angeles Lakers won seven out of there ten last games. Therefore, they will win 70% of their games this season.

Anecdotal generalization is like saying, "this case is true, therefore, all other such cases are true."

Statistical generalization is a generalization about a population using a statistically-representative sample.

Example.

1. In a survey, 28% of would-be voters said they would vote for Candidate X for president, 47% said they would vote for Candidate Y, 24% said they would vote for Candidate Z while the remaining 1% said they are still undecided. Therefore, Candidate Y will win the elections.

2. Seventeen random test tube water samples taken from a link show an average of 2.3ppm of ammonia content in a lake. Therefore, the lake has 2.3ppm of ammonia content.

The help of quantified justifications makes statistical generalization a more formal and a more scientifically-accepted method of inductive reasoning.

Prediction is a type of inductive reasoning wherein a future event is concluded based on past or current events. The prediction may be a consequence (effect) or a subsequence (an event that follows although it is not necessarily caused by the premise).

Example.

1. The clouds are dark. Therefore, it will rain.

2. A comet is known to make pass near the Earth every 225 years. It made pass Earth in 1813. Therefore, it will make pass the Earth on 2038.

3. It's almost "BER" months! Therefore, Christmas songs will be played in malls.

Simply put, predictive reasoning is telling what will happen in the future based on what is known now.

Causal inference or causal reasoning is a type of inference wherein the cause of a particular occurrence of an effect is inferred. Causal inference does not guarantee truth because other factors might have intervened for the effect to occur.

Example.

1. My car won't start. Therefore, its battery is weak.

2. He has coughs, fever, and lost his sense of taste. Therefore, he is infected with CoViD-19.

Causal inference is making a guess of what caused a certain observed premise.

Inference of events is another type of inference in which the conclusion is an inference of current or past events based on current observations as premise. Inference of events is often confused with causal inference. The difference is that in inference of events, the conclusion is not a cause, or that a cause-effect relationship between the premise and conclusion is not being established.

Inference of events may either infer a past or a current event.

Example.

1. (Inference of a past event) Robert has just arrived home from abroad. Therefore, he was in the airport earlier.

2. (Inference of a past event) It's low tide this afternoon. Therefore, it was high tide in the morning.

3. (Inference of a current event) Robert said he will be here in a few hours from abroad. Therefore, he is flying in a plane now.

4. (Inference of a current event) There was lightning a few seconds ago. Therefore, it should be thundering now.

Inference of an event is like saying, "this is happening now, so this must have happened in the past" or "this is true so this must be currently happening" without the conclusion causing it to happen.

An analogy is a reasoning wherein a similarity to an observed premise is used in order to draw conclusion of something that is yet unknown. So there are two settings in place: a known and an unknown setting; and one fact that is existing in the known setting. The conclusion is that the same fact should exist in the unknown setting.

Analogy is like saying, "this is true in this situation, so the same should be true in another similar situation".

Example.

1. On Earth, phosphine is produced only by living organisms. There are traces of phosphine found on the atmosphere of Venus. Therefore, living organisms producing phosphine are in the atmosphere of Venus.

2. If a human wraps his or her arms on another, it's for him or her to show love to the other. A snake is wrapping his body on a frog. Therefore, the snake is showing love to the frog.

Inductive reasoning in mathematics is mainly used in making conjectures. A conjecture is a mathematical statement that is true for all known cases, but is not proven in general. In easier terms, we are making a conjecture when we "detect" a pattern in a series and then "guess" what the next term would be [1]. Take the following as an example.

To answer this and more logic test questions, click Psycho Tests' 20-Question Logical Reasoning Test here.

We also use the strategy of inductive reasoning for numerical sequences like the following:

3 7 13 25 49 ???

Question: What type of inductive reasoning was applied for the above examples?

1. What are the two types of reasoning? How do they differ?

2. What are the five classifications of inductive reasoning? How is each done? Give an example for each.

3. Why is reasoning important?

[1] Bhandari, P. (2022). Inductive Reasoning | Types, Examples, Explanation. Scribbr. [link]

[2] MasterClass (2021). What Is Inductive Reasoning? Learn the Definition of Inductive Reasoning With Examples, Plus 6 Types of Inductive Reasoning. MasterClass. [link]

[3] Aufmann, R. N., Lockwood, J., Nation, R. D., & Clegg, D. K. (2016). Mathematical excursions. Cengage Learning.