Deductive & Inductive Reasoning
Here we have collected exercises that will allow you to move to a higher level and work not only with concepts but also with inferences. Our minds may take different inference routes: deductive or inductive. Let’s explore how they work!
Objective:
To teach participants to identify errors in reasoning using the rules of deductive reasoning.
Number of participants:
2 to 25
Age range:
14+
Duration:
60 minutes
How to conduct:
Briefly explain to participants that deductive reasoning moves from the general to the specific. It is a way of thinking that does not generate new knowledge, meaning it does not build hypotheses or discover anything. The peculiarity of this method is that it guarantees the truth of the conclusion if the premises are true (no other method of reasoning offers such a guarantee). This means that if we reason according to the rules and start from true statements, we will definitely reach a true conclusion.
Explain deductive reasoning using the following example, comprising two premises from which a conclusion follows:
1. Show participants a Venn diagram confirming that the conclusion drawn is the only possible and correct one within this line of reasoning. To do this, highlight the concepts present in the reasoning: planets, objects in motion, and Earth.
- First premise: "All planets are in motion."
- Second premise: "Earth is a planet."
- Conclusion: "Earth is in motion."
1. Show participants a Venn diagram confirming that the conclusion drawn is the only possible and correct one within this line of reasoning. To do this, highlight the concepts present in the reasoning: planets, objects in motion, and Earth.
Explanation for the facilitator:
According to the first premise – 'All planets move' – the set of planets must be included in the set of objects in motion. The second premise tells us that 'Earth is a planet,' so Earth must be included in the set of planets. The conclusion 'Earth is in motion’ necessarily follows from the previous reasoning. There is no other way to draw a Venn diagram for this line of reasoning. Deduction guarantees a 100% valid conclusion provided we have verified everything and followed the rules of deductive reasoning."
2. But what does "follow the rules" mean? Let’s move to the next level – identifying errors in reasoning – using the following example:
In the diagram, we can see that given the correct representation of the two premises (all planets are in the set of objects in motion, and Earth is in the set of objects in motion), the conclusion "Earth is a planet" is disproved, as the set "Earth" might not be included in the set "planets". Thus the inference in this example is invalid.
3.Ask participants to form pairs or groups of three and give them the task: "Find invalid deductive inferences and prove their invalidity using Venn diagrams."
According to the first premise – 'All planets move' – the set of planets must be included in the set of objects in motion. The second premise tells us that 'Earth is a planet,' so Earth must be included in the set of planets. The conclusion 'Earth is in motion’ necessarily follows from the previous reasoning. There is no other way to draw a Venn diagram for this line of reasoning. Deduction guarantees a 100% valid conclusion provided we have verified everything and followed the rules of deductive reasoning."
2. But what does "follow the rules" mean? Let’s move to the next level – identifying errors in reasoning – using the following example:
- First premise: "All planets are in motion."
- Second premise: "Earth is in motion."
- Conclusion: "Earth is a planet."
In the diagram, we can see that given the correct representation of the two premises (all planets are in the set of objects in motion, and Earth is in the set of objects in motion), the conclusion "Earth is a planet" is disproved, as the set "Earth" might not be included in the set "planets". Thus the inference in this example is invalid.
3.Ask participants to form pairs or groups of three and give them the task: "Find invalid deductive inferences and prove their invalidity using Venn diagrams."
- Premise 1: "All doctors with a rating of 5 stars on RateMDs are real professionals."
- Premise 2: "Dr. Smith has a rating of 5 stars on RateMDs."
- Conclusion: "Therefore, Dr. Smith is a real professional."
- Premise 1: "All real professionals on RateMDs have a rating of 5 stars."
- Premise 2: "Dr. Smith has a rating of 5 stars on RateMDs."
- Conclusion: "Therefore, Dr. Smith is a real professional.
- Premise 1: "All rabbits love cabbage."
- Premise 2: "Socrates loves cabbage."
- Conclusion: "Socrates is a rabbit."
- Premise 1: "All bloggers earn a lot of money."
- Premise 2: "Some people on social media are bloggers."
- Conclusion: "Some people on social media earn a lot of money."
Examples 2, 3, and 4 are incorrect. If the first two premises are depicted correctly, the conclusion can be refuted, as it is possible to create a Venn diagram with correct premises and an incorrect conclusion. Show the correct answers, discuss participants’ solutions, and ask participants what they have learned about reasoning while doing the tasks.
Objective:
To teach participants to identify errors in reasoning using the rules of inductive reasoning.
Number of participants:
2 to 25
Age range:
14+
Duration:
45 minutes
How to conduct:
Briefly explain to participants that inductive reasoning moves from the specific to the general. Unlike a conclusion made by deductive reasoning, a conclusion drawn through inductive reasoning will be probability-based. Depending on how much data or facts we have, the probability of the conclusion can increase and approach 100%. However, an inductive conclusion will never be 100% certain. Therefore, we use words like "probably," "most likely," and "possibly" as markers of inductive reasoning.
Provide an example of inductive reasoning:
First premise: "Earth is in motion."
Second premise: "Mars is in motion."
Third premise: "Jupiter is in motion."
Fourth premise: "Earth, Mars, and Jupiter are planets in the Solar System."
Conclusion: "Probably all planets in the Solar System are in motion."
Explain that in this example, we are dealing with incomplete induction. To make a 100% inductive conclusion about all the planets in the Solar System, we would have to list them all. When we draw a conclusion based on several specific elements of a group, the conclusion about all elements of this group will be probable but not certain.
Give an assignment: arrange the premises in the correct order and make an inductive conclusion based on them:
First premise: "Earth is in motion."
Second premise: "Mars is in motion."
Third premise: "Jupiter is in motion."
Fourth premise: "Earth, Mars, and Jupiter are planets in the Solar System."
Conclusion: "Probably all planets in the Solar System are in motion."
Explain that in this example, we are dealing with incomplete induction. To make a 100% inductive conclusion about all the planets in the Solar System, we would have to list them all. When we draw a conclusion based on several specific elements of a group, the conclusion about all elements of this group will be probable but not certain.
Give an assignment: arrange the premises in the correct order and make an inductive conclusion based on them:
- Premise: "Alan loves mathematics."
- Premise: "Alan, Sophie, and Charles are Catherine’s friends."
- Premise: "Sophie loves mathematics."
- Premise: "Charles loves mathematics."
- Conclusion: ?
Ответ
- Premise: "Copper, iron, and aluminum are metals."
- Premise: "Copper is conductive."
- Premise: "Iron is conductive."
- Premise: "Aluminum is conductive."
- Conclusion: ?
- Premise: "Jack is a bully."
- Premise: "Jack, Roger, and Maurice are boys."
- Premise: "Roger is a bully."
- Premise: "Maurice is a bully."
- Conclusion: ?
After participants complete the assignment, show them the correct answers.
After presenting the correct answers, emphasize the conclusions that should have been reached.
Provide the following explanation: in the first case, we state in the conclusion: "ALL of Catherine's friends love mathematics." In this example, we are dealing with complete induction. Despite what was mentioned earlier about the probabilistic nature of inductive conclusions, when we can list all the objects of one group (and in this case, we know Catherine has three friends and can list them all), the conclusion may be 100% true.
In the second example, we add the word "probably" because our premises list only four metals – far from all the metals known to science. Therefore, we cautiously conclude the conductivity of all metals, not asserting this fact 100%.
In the third example, participants, by analogy with the example about metals, can conclude, "Probably, all boys are bullies."
Emphasize this point and discuss another feature of induction – the fallacy of hasty generalization – which people often encounter in everyday reasoning. (This error is characteristic of popular induction. The difference from the example with metals is that scientific induction was used in that case). To make a 100% conclusion about all boys, we would have to list all young males, which is obviously impossible. Therefore, to avoid hasty generalizations, inductive reasoning adds: "Probably, some boys are bullies."
Conduct a reflection and ask what participants have learned about reasoning from this exercise.
After presenting the correct answers, emphasize the conclusions that should have been reached.
Provide the following explanation: in the first case, we state in the conclusion: "ALL of Catherine's friends love mathematics." In this example, we are dealing with complete induction. Despite what was mentioned earlier about the probabilistic nature of inductive conclusions, when we can list all the objects of one group (and in this case, we know Catherine has three friends and can list them all), the conclusion may be 100% true.
In the second example, we add the word "probably" because our premises list only four metals – far from all the metals known to science. Therefore, we cautiously conclude the conductivity of all metals, not asserting this fact 100%.
In the third example, participants, by analogy with the example about metals, can conclude, "Probably, all boys are bullies."
Emphasize this point and discuss another feature of induction – the fallacy of hasty generalization – which people often encounter in everyday reasoning. (This error is characteristic of popular induction. The difference from the example with metals is that scientific induction was used in that case). To make a 100% conclusion about all boys, we would have to list all young males, which is obviously impossible. Therefore, to avoid hasty generalizations, inductive reasoning adds: "Probably, some boys are bullies."
Conduct a reflection and ask what participants have learned about reasoning from this exercise.