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How to Solve Math Problems Using Logical Reasoning

Logical reasoning systematically solves complex math problems.

How to Do a Simple Calorimeter Experiment

Logical reasoning is a useful tool in many areas, including solving math problems. Logical reasoning is the process of using rational, systemic steps, based on mathematical procedure, to arrive at a conclusion about a problem. You can draw conclusions based on given facts and mathematical principles. Once you master the skill in solving math problems, you can use logical reasoning in a wide array of real-world situations.

Read and understand the problem. For example, let’s say Bob grilled hot dogs to sell at the concession stand during the hockey game. By the end of the first period, Bob had sold one-third of the hot dogs. During the second period, Bob sold 10 more hot dogs and continued selling hot dogs through the third period. When the game ended, Bob sold half the remaining grilled hot dogs. If given the information that 10 grilled hot dogs didn’t sell, how many hot dogs did Bob grill before the game started?

Make a plan to solve the problem backward using critical thinking and logic. In the concession stand example, you know that Bob had 10 unsold grilled hot dogs when the game ended.

Working backward, start with the known quantity of 10 unsold, grilled hot dogs. You were also told that Bob sold half the remaining hot dogs when the game ended. Therefore, the second half of unsold hot dogs totals 10. Multiply 10 by 2 = 20 hot dogs. Earlier, Bob had sold an additional 10 hot dogs to equal a running total of 30 hot dogs. Continuing to work backward, you recall that Bob sold one-third of his hot dogs in the first period, meaning that two-thirds remained, which equals 30. Now that you’ve determined that two-thirds equal 30 hot dogs, you can surmise that one-third equals 15. Add 15 + 30 = 45. Your final calculation reveals that Bob grilled 45 hot dogs before the game started.

To check the accuracy of your work, do the problem in reverse using logical reasoning. Start with your final answer – 45 hot dogs grilled before the game started. This time, however, work forward. Bob sold one-third of his hot dogs during the first period of the hockey game. Make the calculation. Divide 45 by three, which equals 15. When you subtract 15 from 45, the answer is 30. Because Bob sold 10 more hot dogs during the second period, subtract 10 from 30, which is 20. Half of 20 is 10, which is the number of hot dogs remaining. Arriving at this solution confirms your logical reasoning abilities.

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  • Education.com: The Strategies of Working Backward and Using Logical Reasoning in Word Problems: Study Guide

About the Author

Nicholas Smith has written political articles for SmithonPolitics.com, "The Daily Californian" and other publications since 2004. He is a former commissioner with the city of Berkeley, Calif. He holds a Bachelor of Arts in political science from the University of California-Berkeley and a Juris Doctor from St. John's University School of Law.

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problem solving using logical reasoning

How logical reasoning works

logical reasoning cognitive skill

What is logical reasoning?

Logical reasoning is the process of using rational and systematic series of steps to come to a conclusion for a given statement. The situations that ask for logical reasoning require structure, a relationship between given facts and chains of reasoning that are sensible. Because you have to study a problem objectively with logical reasoning, analysing is an important factor within the process. Logical reasoning starts with a proposition or statement. This statement can be both true or false.  

Why is logical reasoning important?

Logical reasoning, in combination with other cognitive skills, is an important skill you use during all kinds of daily situations. It helps you make important decisions, discern the truth, solve problems, come up with new ideas and set achievable goals. Logical reasoning is also an important aspect of measuring intelligence during an IQ-test.  

The three types of logical reasoning

Logical reasoning can be divided into deductive-, inductive- and abductive reasoning. While inductive reasoning starts with a specific instance and moves into a generalized conclusion, deductive reasoning goes from a generalized principle that is known to be true to a specific conclusion that is true. And abductive reasoning is making a probable conclusion from what you know.  

logical reasoning types

We’ll explain each type of logical reasoning further:

Inductive reasoning

With inductive reasoning, a number of specific observations lead to a general rule. With this method, the premises are viewed as supplying some evidence for the truth of a conclusion. With inductive reasoning, there is an element of probability. In other words, forming a generalization based on what is known or observed.   While this sounds like the theory you will use during a debate or discussion, this is something you do every day in much simpler situations as well. We’ll explain this type of logical reasoning with an example: There are 28 balls within a basket, which are either red or white. To estimate the amount of red and white balls, you take a sample of four balls. The sample you took, exists out of three red and one white ball. Using good inductive generalization would be that there are 21 red and 7 white balls in the basket. As already explained, the conclusion drawn from his type of reasoning isn’t certain but is probable based on the evidence given (the sample of balls you took). Questions which require to perform inductive reasoning are a part of IQ-tests. An example of a little more complex question like just explained with the balls is the one of the image below. To come to a conclusion to solve this problem, both inductive reasoning and pattern recognition skills are required. Looking at the sequence of tiles with different patterns of dots, which tile should be on the place of the question mark? A, B, C, D, E or F?  

inductive reasoning example question

Deductive reasoning

With deductive reasoning, factual statements are used to come to a logical conclusion. If all the premises (factual statements) are true, the terms are clear and all the rules of deductive logic are followed to come to a conclusion, then the conclusion will also be true. In this case, the conclusion isn’t probable, but certain. Deductive reasoning is also known as “top-down” logic, because it (in most cases) starts with a general statement and will end with a specific conclusion.

We’ll explain deductive reasoning with an example, with 2 given premises:

It’s dangerous to drive while it’s freezing (premise 1)

It is currently freezing outside (premise 2)

So, we now know that it is dangerous to drive when it is freezing, and it is currently freezing outside. Using deductive reasoning, these two premises can help us form necessarily true conclusion, which is:

It is currently dangerous to drive outside (conclusion)

Situations in which you use deductive reasoning can come in many forms, such as mathematics. Whether you are designing your own garden or managing your time, you use deductive reasoning while doing math daily. An example is solving the following math problem:

All corners of a rectangle are always 180° (premise 1)

The following rectangle has one right angle, which is always 90° (premise 2)

The second angle is 60° (premise 3)  

deductive example math

How much degrees is the third angle (X)? To answer this question, you can use the three premises to come to the conclusion how much degrees the third hook is. The conclusion should be 180° (premise 1) -90° (premise 2) - 60° (premise 3) = 30° (conclusion)

Abductive reasoning

With abductive reasoning, the major premise is evident but the minor premise(s) is probable. Therefore, defining a conclusion would also make this conclusion probable. You start with an observation, followed by finding the most likely explanation for the observations. In other words, it is a type of logical reasoning you use when you form a conclusion with the (little) information that is known. An example of using abductive reasoning to come to a conclusion is a decision made by a jury. In this case, a group of people have to come to a solution based on the available evidence and witness testimonies. Based on this possibly incomplete information, they form a conclusion. A more common example is when you wake up in the morning, and you head downstairs. In the kitchen, you find a plate on the table, a half-eaten sandwich and half a glass of milk. From the premises that are available, you will come up with the most likely explanation for this. Which could be that your partner woke up before you and left in a hurry, without finishing his or her breakfast.  

inductive deductive abductive reasoning example

How does logical thinking relate to problem-solving?

As previously mentioned, the different types of logical reasoning (inductive, deductive and abductive) help you to form conclusions based on the current situation and known facts. This very closely correlates to problem-solving, as finding the most probable solution to resolve a problem is a similar conclusion. Logical thinking, and thereby problem solving, goes through the following five steps to draw a conclusion and/or find a solution:

Collecting information about the current situation. Determining what the current problem is, and what premises apply. Let’s say you want to go out for a drive, but it’s freezing outside.

Analyzing this information. What information is relevant to the situation, and what isn’t. In this case, the fact that it’s freezing is relevant for your safety on the road. The fact that you might get cold isn’t, as you’d be in your car.

Forming a conclusion. What can you conclude from this information? The roads might be more dangerous because it’s freezing.

Support your conclusion. You might look at traffic information to see that there have been more accidents today, in which case, that supports the conclusion that driving is more dangerous today.

  • Defend your conclusion. Is this conclusion correct for your case? If you don’t have winter tires it would be more accurate than when you do.  

problem solving steps

How to improve logical thinking and problem-solving skills?

Because there are so many different situations in which you use logical thinking and problem-solving, this isn’t a cognitive skill you can train specifically. Luckily, there are many methods that might help you to improve your logical thinking skills. These include methods to keep your general cognitive abilities healthy as well as methods to train your logical thinking skills. These are:

Learning something new

Social interaction

Healthy nutrition

Ensure enough sleep

Avoid stress

Preferably no alcohol

Spend time on creative hobbies

Practice questioning

Try to anticipate the outcome of your decisions

Brain training to challenge your logical reasoning skills

improve logical thinking skill

Logical Reasoning

I. definition.

Logical reasoning (or just “logic” for short) is one of the fundamental skills of effective thinking. It works by raising questions like:

  • If this is true, what else must be true?
  • If this is true, what else is probably true?
  • If this isn’t true, what else can’t be true?

These are all inferences : they’re connections between a given sentence (the “premise”) and some other sentence (the “conclusion”). Inferences are the basic building blocks of logical reasoning, and there are strict rules governing what counts as a valid inference and what doesn’t — it’s a lot like math, but applied to sentences rather than numbers.

If there is someone at the door, the dog will bark.

Assuming this sentence holds true, there are some other sentences that must also be true.

  • If the dog didn’t bark, there is no one at the door.
  • Just because the dog barked doesn’t mean there’s someone at the door.

There are also a few sentences that are probably true, such as:

  • The dog can sense (hear or smell) when someone is at the door.
  • The dog belongs to the people who live in the house where the door is located.

II. Types of Logical Reasoning

There are two basic types of logic, each defined by its own type of inference. They correspond to the two categories in the example from section 1.

  • Deduction is when the conclusion, based on the premises, must be true. For example, if it’s true that the dog always barks when someone is at the door and it’s true that there’s someone at the door, then it must be true that the dog will bark. Of course, the real world is messy and doesn’t always conform to the strictures of deductive reasoning (there are probably no actual dogs who always bark when someone’s at the door), but deductive reasoning is still important in fields like law, engineering, and science, where strict truths still hold. All math is deductive.
  • Induction is when the conclusion, based on the premises, is probably The answers are less definitive than they are in deductive reasoning, but they are often more useful. Induction is our only way of predicting what will happen in the future: we look at the way things are, and the way they have been in the past, and we make an educated guess about what will probably happen. But all predictions are based on probability, not certainty: for example, it’s extremely probable that the sun will rise tomorrow morning. But it’s not certain , since there are all sorts of catastrophes that could happen in between now and then.

III. Logical Reasoning vs. Critical Thinking

Logic is one of the main pillars of critical thinking . And there’s no question that critical thinking would be impossible without some understanding of logical reasoning. However, there are many other skills involved in critical thinking, such as:

  • Empathy , or the ability to imagine what someone else is feeling or experiencing. This is a crucial skill for critical thinking, since it allows you to broaden your perspective and reflect on your actions and beliefs. Empathy also makes you a better student of philosophy because it enables you to put yourself in the author’s shoes and understand the argument from within.
  • Analogy , or noticing similarities and thinking them through. Analogies allow us to draw conclusions about, for example, the similarity between our own time and some moment in history, and thus try to make better decisions in the future. This skill is closely related to inductive logic.
  • Creativity . Critical thinking is all about innovative problem-solving and coming up with new ideas, so it’s heavily dependent on creativity. Just like a creative art, critical thinking depends on assembling old parts in new ways, working inventively within constraints, and matching moments of inspiration with hours of rigorous craft.

III. Quotes About Logical Reasoning

“I am convinced that the act of thinking logically cannot possibly be natural to the human mind. If it were, then mathematics would be everybody’s easiest course at school and our species would not have taken several millennia to figure out the scientific method.” (Neil Degrasse Tyson)

Neil Degrasse Tyson is an astrophysicist and TV personality who passionately advocates for science and critical thinking. In this quote, he suggests that science and logical reasoning are inherently difficult tasks for the human mind, an organ that evolved to perform a very different set of tasks under very different conditions from the ones we live in today.

“Logic takes care of itself; all we have to do is to look and see how it does it.” (Ludwig Wittgenstein)

Wittgenstein was probably the most influential philosopher of the 20th century, but his views changed dramatically over the course of his life, leading to some controversy as to what he actually thought. This quote is a good example. Early on, Wittgenstein believed that logical reasoning was autonomous — that logical truth was an objective truth, out there in the world for anyone to see if they knew how to look. Later on, though, Wittgenstein started to believe that culture and nature influence the way we see logic, and that logic is therefore not perfectly objective. It’s a tricky question, whether logical reasoning is universal or cultural — it must be tricky if a genius like Wittgenstein couldn’t make up his mind on it!

IV. The History and Importance of Logical Reasoning

Logic is a universal part of the human experience — agriculture would be impossible without inductive reasoning about weather and sunlight, and construction would be impossible without mathematics and deductive reasoning about what makes a structure sturdy.

Formalized logic has appeared in several places with more or less similar results. The Greek philosopher Aristotle is credited with being the first to develop a formal system of logical reasoning, but there were already people in India and China working on formal logic long before Aristotle was born. The Indian, Chinese, and Greek systems were all remarkably similar in their rules, which suggests that there may have been some mutual influence despite the distance. Traders and travelling scholars may have brought ideas about logical reasoning with them all over the world, allowing for rapid development of new ideas.

Logic may seem like a stuffy, abstract discipline used only by philosophers and lawyers, but it has had a profound influence on the history of science and technology as well. Alan Turing, the inventor of the modern computer, was a logician rather than a tinkerer or engineer, and his famous “Turing Machine” was a product of his rigorous training in formal logical reasoning.

V. Logical reasoning in Popular Culture

“Vulcanians do not speculate. I speak from pure logic.” (Spock, Star Trek )

Mr. Spock was raised on Vulcan and trained to be perfectly rational, ignoring all emotion and concentrating on logical reasoning instead. This represents a widespread trope in popular culture — that logic and the emotions are at odds with each other (the head pulling one way and the heart pulling in another). But there’s no reason why logic and the emotions have to be enemies. Uncontrolled emotion certainly clouds logical reasoning — it’s difficult to think rationally if you’re in a rage, for example — but many traditions argue that logic and the emotions should be partners rather than rivals, each providing its own sort of insight in harmony with the other.

On Sherlock , the great detective Sherlock Holmes has a website called “The Art of Deduction,” in which he explains his methods for solving crimes. However, the website has the wrong name — nearly all of Sherlock’s inferences are inductive rather than deductive. That is, they bring together bits and pieces of evidence to develop a theory about what probably happened in a particular crime. They’re not based on the kind of logical certainty that we saw in section 1, but rather on reasoning about likelihoods and probabilities. It’s always logically possible that Sherlock could have it wrong, even though that rarely seems to happen.

a. Reduction and induction

b. Deduction and induction

c. Greek logic and Chinese logic

d. Formal logic and informal logic

d. All of the above

a. Deduction

b. Inference

c. Induction

b. Creativity

Maths with David

Problem solving: logically reason.

Here are some guidelines that can assist with logical reasoning:

  • Read and understand the problem and consider any assumptions being made;
  • Work out what is mathematically true and what could not possibly be true;
  • Think about things that might be true and use mathematical reasoning to find out whether they are true or not;
  • Break a larger problem into smaller steps to see if they can lead to a solution;
  • Keep a clear record of your work, so that you can use it to justify your solution.

Logical reasoning is particularly helpful where we know about one particular case and need to think about another case. Although to some extent we use logical reasoning with almost all problems.

Worked Examples

First we will read all three examples and have a quick think about them and then we will look at how logical reasoning can help us with each one:

Biology Example

The following diagram shows a magnified cell. Four students have calculated the actual length of the longest side of the cell, but some of them have got the wrong answer. Which of the answers is correct? (a) 5.6µm, (b) 56µm, (c)560µm, (d) 5600µm

problem solving using logical reasoning

Cube Example

The following six pictures all show the same cube. Which symbol is opposite the red cross?

problem solving using logical reasoning

Circle in Square Example

Below is a diagram of a red square with a purple circle drawn through its vertices and a blue square that touches the purple circle on all sides.

What is the relationship between the area of the red square and the blue square and how do you know?

problem solving using logical reasoning

Worked Solutions to Examples

The following diagram shows a magnified cell. Four students have calculated the actual length of the longest side of the cell, but some of them have got the wrong answer.

Effectively we can use rounding, by noticing that the length of the cell is around three times the marked length, so around 60µm. This leads us to conclude that only one of the answers given is realistically possible.

By looking at the different cubes and taking a little information from each of them we can draw out the net of the cube

problem solving using logical reasoning

So, what do we know at first?

Clearly the blue square is bigger than the red square, but not massively bigger, e.g. we could say it is less than four times bigger. This helps us to establish initial boundaries for our answer.

We don’t know any measurements, so perhaps we will have to name one of our sides of a square x, or would it be better to name the radius of the circle x?

Are there extra lines we could add to the diagram that would help us?

Here is one way that we could solve this problem.

If we ignore the circle and add the following lines onto the diagram, it becomes easier to see the relationship between the two square:

problem solving using logical reasoning

Here it becomes more immediately apparent that the red square has half the area of the blue square.

19 Questions of Increasing Difficulty

problem solving using logical reasoning

Solutions to Question Set

problem solving using logical reasoning

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Solving Propositional Logic Word Problem

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Propositional logic is a formal language that treats propositions as atomic units. A typical propositional logic word problem is as follows:

A, B, C, D are quarreling quadruplets. If A goes to the party, then B will not go. If C goes to the party, then B will not go. What is the largest possible number that will go to the party?

Logic is the study of valid reasoning. It is applied not only in studies but also in our day-to-day lives. Using simple logical reasoning and deduction, we are able to obtain information from a certain premise. Likewise, we can also verify or disprove statements.

We will be discussing ways of identifying common mistakes and how to avoid them. We will also talk about different proof techniques, such as using Venn diagrams and analogies so that you have a toolkit for solving logic word problems.

What's wrong with this proof?

Proof by venn diagram, proof by analogy.

In this section, we will use familiar notations used in propositional logic. You might want to familiarize yourself with Propositional Logic first.

Like solving any other questions, we should always ask ourselves what we can and can't do when writing out our reasoning. The first step to learn how to solve propositional logic problems is to list out what can't be done or what is not a possibility so we can narrow down what the possible scenarios are. Remember that it is very easy to fall into an erroneous conclusion based on faulty reasoning. Take the statements below as an example, if the first statement is true, is the second statement also true?

\[\begin{array} &\text{"If it's raining, then I can't play soccer".} &\text{"If I can't play soccer, then it's raining."} \end{array}\]

It is pretty clear that the issue here is: there can be other reasons why I can't play soccer, which doesn't necessarily depend on the weather. If we make such simple errors in reasoning when the context is very clear, just imagine what will happen when you are less certain about statements that are more vague. In the next paragraph, we will be introduced to these errors.

Converse and Inverse Errors

As a beginner, the most common mistake you can make is to assume that the converse and/or inverse of the original statement is also true. Take a look at the two sections below:

Introduction to Converse Error with erroneous reasoning:

Premise : If it's raining, then I can't play soccer. Conclusion : If I can't play soccer, then it's raining. Explanation : From the first statement, we are given a condition and a result: "raining" as a condition and "I can't play soccer" as a result. The entire premise is phrased in such a way that if the condition is fulfilled, then the result will occur. However, the conclusion shows that if the result is fulfilled, then the condition will occur. This does not make sense because it is not necessary for the condition to take place if the result occurs first. This is known as a converse error. In a general form, the argument for a converse error is as follows: If P occurs, then Q occurs. Q occurs. Therefore, P also occurs.

Introduction to Inverse Error with erroneous reasoning:

Premise : If it's raining, then I can't play soccer. Conclusion : If it's not raining, then I can play soccer. Explanation : From the first statement, we are given a condition and a result: "raining" as a condition and "I can't play soccer" as a result. The entire premise is phrased in such a way that if the condition is fulfilled, then the result will occur. However, the conclusion shows that if the condition does not occur, then the result does not occur either. This does not make sense because there can be other reasons/factors such that the result does occur. This is known as an inverse error. In a general form, the argument for an inverse error is as follows: If P occurs, then Q occurs. P does not occur. Therefore, Q also does not occur.

It may now be abundantly clear that it is easy to identify we've made an erroneous reasoning. However, what if the statements given appear more vague? This is the reason why we introduce the two errors above (converse error and inverse error) to show that not all wrong statements are easily identifiable. Simply put, the relationship between two events do not necessarily imply that one causes the other. In short, we are pointing out the common fact that "correlation does not imply causation".

Now that we have seen these mistakes first hand, let's do another example to remind ourselves that they are mistakes and we can hopefully avoid them in the future. Keep in mind that some of the converse/inverse statements can appear ridiculous but some do not.

We are given the following statement: If today is Sunday, then the weather is sunny. \[\] \(\qquad \text{ (i)}\) Write the inverse and converse of this statement. \(\qquad \text{(ii)}\) Identify which of these statements you have made is not logical and explain why. \(\text{(i)}\) Inverse and Converse Inverse: If today is not Sunday, then the weather is not sunny. Converse: If the weather is sunny, then today is Sunday. \(\text{(ii)}\) Logical or Illogical Though they are the inverse and converse of the original statement, we must keep in mind that they might not necessarily be an error. However, there is no harm in checking whether they are correct or not. The inverse statement implies that the day has a direct relation on the weather being sunny or not, which is ludicrous because there can also be non-sunny days which do not fall on a Sunday. The converse statement implies that only if the weather is sunny then the day is Sunday, which is also ludicrous because they can also have a sunny weather on days not falling on a Sunday. \(_\square\)

Pinpoint the exact error

Now that we can identify how the errors occur, let's take a step further and apply these techniques so that we can pinpoint exactly where the error occurs. Note that the easiest way to identify where the error arose is to convert logical statements into symbolic forms (like P implies Q). Let's try the following example.

Taking the long view on your education, you go to the Prestige Corporation and ask what you should do in college to be hired when you graduate. The personnel director replies that you will be hired only if you major in mathematics or computer science, get a \(\text{B}^\text{+}\) average or better, and take accounting. You do, in fact, become a math major, get a \(\text{B}\) average and take accounting. You return to Prestige Corporation, make a formal application, and are turned down. Did the personnel lie to you? Let's list down the requirements to be hired: \(\begin{array}{r r l} & \text{(i)} & \text{Major in mathematics or computer science}\\ & \text{(ii)} & \text{Get a } \text{B}^\text{+} \text{ average or better}\\ & \text{(iii)} & \text{Take accounting}\\ \end{array}\) Since you became a math major, criteria \(\text{(i)}\) is satisfied. Since you got a \(\text{B}\) average instead of a \(\text{B}^\text{+}\) average, criteria \(\text{(ii)}\) is not satisfied. Since you took accounting, criteria \(\text{(iii)}\) is satisfied. Since you did not satisfy all the criteria and were turned down, the personnel didn't lie to you. \(_\square\)

Now that you're familiar with writing out these statements and identifying possible errors, let's try another example that uses such a property!

A store has been raided by looter/s, who drove away in a car. Three well-known criminals Satvik, Krishna and Sharky are brought to the police station for questioning. Inspector Aditya of the police extracts the following facts:

\((1)\) None other than Satvik, Krishna and Sharky was involved in the robbery. \((2)\) Sharky never does a job without using Satvik (and possibly others) as accomplices. \((3)\) Krishna doesn't know how to drive.

Find the person who, in any case, is guilty.

This problem is the part of my set "Is This What You Call Logic?"

Formal terminologies.

In the previous sections, we have learned the two most common errors that students will make when solving a logical reasoning problem. However, we did not formally touch on the terminologies for those terms: converse error and inverse error. Let's begin!

Contrapositive : A statement is logically equivalent to its contrapositive. The contrapositive negates both terms in an implication and switches their positions. For example, the contrapositive of "P implies Q" is the negation of Q implies the negation of P.

Converse : The converse switches the positions of the terms. The converse of "P implies Q" is "Q implies P".

"If and only if", sometimes written as iff and known as equivalence, is implication that works in both directions. "P if and only if Q" means that both "P implies Q" and "Q implies P".

Let's try a few examples that cover this area!

\(\text{ (i)}\) Write down the contrapositive statement for \[\text{"If you are human, then you have DNA."}\] \(\text{(ii)}\) Write down the two if-then statements for \[\text{"A polygon is a quadrilateral if and only if the polygon has 4 sides."}\] \(\text{ (i)}\) contrapositive \(\qquad\) If you do not have DNA, then you are not human. \(\text{(ii)}\) if-then statements \(\qquad\) If a polygon is a quadrilateral, then it has 4 sides. \(\qquad\) If a polygon has 4 sides, then it is a quadrilateral. \(_\square\)

Simple, isn't it? Let's try some problems that apply the techniques we have learned above.

If Jeff spends 5 hours playing video games, then he cannot finish his math homework.

If Jeff finishes his math homework, then he will do well on his next math test.

Based on this information, which of the following is logically correct?

Selena, Jennifer and Miley wear a blue dress, yellow dress, and green dress in an unknown order. It is known that:

1) If Selena wears blue, then Jennifer wears green. 2) If Selena wears yellow, then Miley wears green. 3) If Jennifer does not wear yellow, then Miley wears blue.

What is the color of the dress Selena is wearing?

Now that we have mastered these techniques, let's move on to the following section for other cool proof techniques!

In this section, we will be applying some basic rules of set notations. You might want to familiarize yourself with sets and Venn diagram first.

In the previous section, we have learned the most common ways in identifying and pinpointing errors. In this section, we will apply the use of Venn Diagram as an alternative proof in solving logical reasoning problems. But what's the benefit of this? Well, it's simple: We do not need to verbalize these statements and we can use the visual aids to guide us to solve these problems.

Recap of set notations and Venn diagram \[\]

Let's do a brief recap for the application of Venn Diagrams by taking the following as an explicit example:

Consider \(W,X,Y,Z\) as sets, each with their own elements in them. Then by interpreting the Venn Diagram, we can obtain information like:

  • All elements in set \(W\) is in set \(Y\).
  • All elements in set \(X\) is in set \(Z\).
  • Not all elements in set \(Y\) is in set \(W\).

How to use Venn diagrams to solve a logical word problem \[\]

Let us consider the following statements and deduce whether the conclusion is true or false by Venn diagram:

True or false? \[\] It is given that all birds have wings. All chickens are birds. Therefore, all chickens have wings. Explanation : By Venn diagram, the statement "A chicken is a bird." implies that the set "all chickens" is a subset of "all birds." Thus we can say that all chickens have the same characteristics as a bird. Because it is given that all birds have wings (a characteristic), all chickens have wings too. Thus the conclusion is correct. Note : We should keep in mind that this only works if the premise is true. For example, if we replace the word "wings" by "forearms" in the first statement, then the conclusion of "All chickens have forearms." will inevitably be true despite its ridiculous claim.

Food for thought : If all phones have batteries and I have a phone, does it mean that my phone has a battery?

Careful! There are other ways of drawing out Venn diagrams!

Though it may appear very simple to set up a Venn diagram, the setup may not necessarily be unique. Let us consider a revised version of the statements above and deduce whether the conclusion is true or false by Venn diagram.

True or false? \[\] It is given that all birds have wings. All chickens are birds. Therefore, all birds are chickens. Explanation : By Venn diagram, the statement "A chicken is a bird" implies that the set "all chickens" is a subset of "all birds." Thus we can say that all chickens have the same characteristics as a bird. However, it is not necessarily true that all birds share the same characteristics of a chicken. (Sounds familiar? It's converse error.) So the claim "All birds are chicken" must be false. Note : To fix the conclusion, you should say " Some birds are chickens" instead of " All birds are chickens."

Now that we know the fundamental applications of proof by Venn diagram, let's apply these knowledge we learned on the following examples:

True or False?

\(\quad\) All pangs are pings. \(\quad\) Some pings are pongs. \(\quad\) Therefore, some pangs are pongs.

Johannes has several written publications on his bookshelf. Albert notices that all of the comics are paperback books, and that some of the paperback books are manga. Are all comics manga?

Image Credit : Wikimedia Johannes Jansson

Isn't the proof by Venn diagram fun? You don't need to use actual words to formalize these statements. Looks very unusual, right? But it works. Speaking of unusual, is it possible to solve these logical statements if we were to spice things up by dramatizing out the statements? Yes, we can! Proof by analogy is another proof technique to solve logical problems. See the following section:

How do we solve texts that are seemingly hard to decipher?

All pangs are pings. Some pings are pongs. Therefore some pangs are pongs.

Consider the logical statements given above. Since we can't relate to or identify what pangs, pings, or pongs are, it will appear that these terms are vague or all too similar. How are we supposed to solve problems like this if we have little to no clue to what is going on? Well, proof by analogy will be useful here: this is when we dramatize or caricaturize the terms used.

For example, we may call pangs as humans, pings as apes, and pongs as gorillas. With these new terms, we are able to visualize what they are. Rewriting them into the original 3 statements shows that

\[\text{All humans are apes. Some apes are gorillas. Therefore some humans are gorillas.}\]

So the given conclusion is wrong because of the ridiculousness of the conclusion "Some humans are gorillas."

However, an important question to ask is why this works. This is too good to be true, right? Or, are we running into some wrong argument? Why does this work?

Explanation of how this works

The reason why proof by analogy works is because we make an inference that if the objects have multiple similar characteristics, and it is given that you know one of them have an extra characteristics (call it X), then it is not a bad inference to conclude that the other object shares that same characteristic X.

To put it short, the generalized/structured form for proof by analogy is:

P and Q has similar properties \(x_1, x_2, x_3, \ldots, x_n\). We know that P has a further property \(y\). Therefore, Q probably has property \(y\) too.

Now let's try a modified version of the ping-pang-pong question from earlier!

True or false? \[\] \(\qquad\) All yangs are yengs and yings. \(\qquad\) Some yengs are yings. \(\qquad\) Then, all yengs are yangs. This is false. Let "yangs" be defined as "pets", "yengs" as "tigers", and "yings" as "cats". So it is true (or at least still reasonable) that all pets are cats and tigers and that some tigers are cats. But it is not true that all tigers are pets. \(_\square\) Note: The reason why proof by analogy works best here is because we couldn't label or identify any characteristics for yangs, yengs, and yings. Therefore, a sensible approach is to prove by analogy.

Now that you're ready to solve logical problems by analogy, let's try to solve the following problem again, but this time by analogy!

Suitability of analogy

Notice from the previous section that we've mentioned that "Q probably has property \(y\) too." instead of "Q definitely has a property \(y\) too." This is because the argument may provide what appears to be the right evidence, but the conclusion does not always follow. This subsection explains why this proof (arguemnt) might not always work.

Though it is true that we highlight or amplify parallel characteristics, the differences between things can often overwhelm their similiarities. One might note that it is always possible to extend an analogy to the point of absurdity. For illustration, take the following famous "Information Argument":

DNA is a code. A code requires an intelligence. Therefore, DNA comes from an intelligence.

Yes, this will sound completely logical if we apply the implications approach. That is, it's P implies Q and Q implies R, so P implies R. However, the argument here is not valid because the statement "DNA is a code." is purely an analogy and thus it is not an entirely accurate statement to begin with. Thus, we have started with a wrong premise. So the merits of analogy do not hold. This is further explained in Analyzing arguments from analogy .

We can see that proof by analogy is very useful and can also be used to make incorrect conclusions. Thus, one must be careful in labeling certain characteristics when using this method. Let's see the following examples to see how the proof by analogy backfires:

True or false? \[\] All squares and rectangless are convex, have four sides and form right angles at their vertices. All squares have sides of the same length. Therefore, all rectangles have sides of the same length. This is obviously false because by definition, all rectangles do not have sides of the same length, but only squares have sides of the same length. We make the wrong conclusion that rectangles also have this characteristic because it is known previously that both share a number of characteristics. \(_\square\)

True or false?

It is given that Amy, Bernadette and Penny are good friends of Sheldon and Leonard.

Leslie is a good friend of Leonard.

Therefore, Leslie is a friend of Sheldon as well.

Image Credit: Wikimedia TBBT. No copyright infringement intended.

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10.2: Reasoning and Logic

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Logic is a tool to solve problems. In mathematical logic, a PROPOSITION (or STATEMENT) is a sentence that is either true or false. It cannot be both. One way to think about it is that in order for it to be true, it must be always be true. There can be no qualifiers.

The following are NOT statements in mathematical logic:

  • He has black hair. (This depends on who “he” is, so we can’t tell if it is true or false.)
  • x + 3y = 6 (This depends on the values of x and y.)
  • 7 + 16 (This is not even a sentence.)
  • Go to the store. (This has nothing to do with being true or false.)
  • Are you a woman? (This is a question.)
  • Napoleon was short. (This depends on how someone defines short.)

How to learn logical reasoning for coding and beyond

problem solving using logical reasoning

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From the binary digits that computers use to represent data, to the most sophisticated AI programs, logic is key to the development and implementation of the technologies we rely on. Logic isn’t just a tool for programmers, though. We all use logical reasoning on a daily basis to think through problems and make decisions.

There are a lot of advantages to burnishing your logic skills. If you’re learning to code , practicing logical reasoning will help you understand how programming languages work and how programmers use them to solve problems. If you’re not at that stage yet, that’s all right: learning logical reasoning will make you a more effective and self-aware thinker , and that has applications far beyond programming.

Today, we’re going to discuss logical reasoning: what it is, why it’s important in the context of programming, and how you can start learning it.

We’ll cover :

Logical reasoning defined

2 types of logical reasoning, how programmers use logical reasoning, strategies for practicing logical reasoning, next steps for learning logical reasoning.

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Narrowly defined, logical reasoning is the practice of applying induction, deduction, or another logical method to a problem. More broadly, using logical reasoning means analyzing the relationships among the constituent parts of an argument or process . It involves thinking about how elements interact to bring about a certain result.

Logical reasoning has tons of practical applications. You can use it to construct strong arguments and analyze the arguments of others. You can use it to develop step-based processes that are efficient, effective, and internally coherent. This is why logical reasoning is so important to so many disciplines. Philosophers, scientists, mathematicians, and computer programmers all use logic in their work.

There are many types of logical reasoning. We’ll discuss two of the most important before exploring the role of logical reasoning in programming, specifically.

Inductive reasoning involves evaluating a body of information to derive a general conclusion . Whenever you engage in research, you’re using inductive reasoning insofar as you’re gathering evidence and using it to better understand the topic at hand. It’s important to remember, however, that the conclusions you’ll arrive at through induction will often be built on evidence that is incomplete. This means that the outcome of inductive reasoning tends to be probable rather than certain.

Deductive reasoning is the process of deriving conclusions from general statements or principles called premises . The syllogism is one of the most basic forms of deductive reasoning. A syllogism includes a major premise, a minor premise, and a conclusion. Here’s an example:

“All bears have fur. A polar bear is a type of bear. Therefore, polar bears have fur.”

In this case, the premises are true, and they lead logically to the conclusion. A deductive argument can easily go wrong if the conclusion doesn’t follow logically from the premises or if one or more of the premises are false. The following syllogism is flawed:

“All bears live in forests. A polar bear is a type of bear. Therefore, polar bears live in forests.”

The major premise is incorrect, so the conclusion is also incorrect.

In summary, while induction involves deriving general principles from specific cases, deduction involves applying general principles to specific cases.

Logical reasoning plays an important role in programming. Let’s discuss some of the ways that programmers use logic in their work.

Problem-solving

Programmers use logical reasoning for problem-solving . Before you start coding a program, you’ll have to wrestle with questions about what the program is trying to accomplish, what features it will need to have, what programming language you’ll write it in, and so on. Inductive and deductive reasoning are useful tools for answering process questions like these.

Writing code

Coding is the process of writing instructions in a language a computer can read. These instructions direct the computer to perform a set of operations . Writing functioning code requires applying logic. Imperative programming , for instance, is a paradigm based on issuing commands for a program to follow on a step-by-step basis. Writing these commands is a logical process. It requires thinking about the relationships between parts. Do the commands interact with each other to bring about the desired outcome? If not, the code isn’t going to work.

Writing functioning code is also dependent on understanding the logic of the programming language you’re using. For example, the logical operators and, or, and not in Python are derived from Boolean logic , and they’re an important part of conditional statements and other logical expressions.

Creating algorithms

Algorithms used in machine learning and artificial intelligence also rely on logic. An algorithm is a set of instructions that tells a program what to do on the basis of certain inputs. A simple decision tree algorithm, for instance, will consist of a series of branching decisions. The logic of each branch is fairly simple: “If a certain condition is met, do A . If not, do B .” These branching decisions are repeated through later stages of the program to specify the appropriate end result based on the inputs.

The best way to practice logical reasoning is by consciously implementing it in your daily life. Here are a few exercises that might be helpful.

Find a news article or thinkpiece in a reputable publication. Read it and try to pick out the author’s primary claim . Then, try to identify the pieces of evidence that the author uses to support that claim. Does the author’s claim follow logically from the evidence? Or does the claim go beyond what the evidence actually supports?

Analyzing how authors use evidence to support their arguments is a great way of practicing inductive reasoning.

Next time you’re planning how to tackle a new project, break your process down into steps and then write them down on paper. Does the end result follow logically from the steps that precede it? Or does the process contain steps that are redundant, unnecessary, or counterproductive?

Remember that logic is all about coherence among the constituent elements of an argument or process. Thinking logically means thinking holistically about how relationships and interactions lead to a certain result.

Find an issue being presented for discussion at the next meeting of your local City Council or zoning board. Formulate a position on it, and use logical reasoning to construct an argument that articulates and defends that position .

As we discussed earlier, logical reasoning is a great tool for analyzing others’ arguments, but you can also use it to construct sound arguments of your own. The best way to practice this is by taking a position on an issue and trying to convince others that you’re right.

There’s a good chance you already use the forms of logical reasoning we’ve described in this article. We arrive at judgments and decisions by evaluating evidence (induction) and applying principles that we believe to be true (deduction). Even so, studying logical reasoning and consciously applying it can make you a more rigorous and self-aware thinker. It will help you to pick out flaws in your logic and others’ , and it will help you to solve problems and build more efficient processes – in programming work and beyond.

If you’re interested in learning more about how programmers think, consider exploring Educative’s learn to code courses.

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They provide an introduction to programming and how programmers solve problems. It also offers some insight into the logical structure of programming languages, explaining, for instance, how conditional statements work to ensure that programs perform the right operations in response to inputs.

Whether you’re learning to code or just trying to strengthen your problem-solving skills, you’ll find that the rigorous, step-by-step thinking that logical reasoning requires is a major asset.

Happy learning!

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How can I practice my logic?

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What Is Logical Reasoning?

Logical reasoning is a type of problem-solving that involves working through a set of rules that govern a scenario. This set of rules or steps is referred to as an algorithm. Logical reasoning involves testing different sets of steps - or algorithms - to determine which sequence of rules leads to the correct solution.

In practice, it involves using given data to determine or to deduce other facts. To reason logically, you’ll need to draw accurate conclusions based on identified premises.

Common forms of logical reasoning include transitive inference (exploring the relationship between two given premises to draw a conclusion) and conditional reasoning (involving an ‘if….then…’ proposition). These structures are encountered across the different types of logical reasoning.

The three types of logical reasoning

logical reasoning

Logical reasoning is an umbrella term that encompasses several different types of reasoning: deductive, inductive, and abductive.

Deductive reasoning starts by presenting premises and relations, which can be followed to reach a solid conclusion. There is a guaranteed certainty involved in deductive reasoning. It takes a general rule (or rules) and uses them to arrive at a specific conclusion that is always true.

Example: All dogs are four-legged. Poppy is a dog. Poppy has four legs.

Inductive reasoning has a different starting point. Observations are used to reach conclusions about the premises and relations at play, which can then be used to draw conclusions. It involves beginning with a specific observation and arriving at a general conclusion. This conclusion has less certainty than in deductive reasoning.

Example: Poppy is a dog. Poppy is four-legged. All dogs have four legs.

Abductive reasoning involves an incomplete observation (or set of observations) that are used to determine the best prediction. This may be true, but ultimately it is a best guess and is open to inaccuracy.

Example: All dogs have four legs. Poppy is four-legged. Poppy is a dog.

When might you use logical reasoning?

Our brains employ logical reasoning skills throughout daily life, making deductions and visualising outcomes so the best course of action can be selected or the appropriate decision made. This rapid reasoning process is unconscious but governs much of our sense-making and behaviour.

The chances are you probably consciously use logical reasoning skills a little less often. If you enjoy completing puzzles or playing games you will be relying upon logical reasoning skills. These may be word problems, spatial problems, or involve numbers (such as Sudoku puzzles).

More formal logical reasoning is used as a form of assessment within the educational system and in recruitment.

Logical reasoning tests are a common form of aptitude test encountered during the recruitment process. These tests are often set alongside verbal reasoning , numerical reasoning , situational judgement , and personality tests , to get a fuller picture of candidate ability and suitability for a job role.

If you’re applying for a position, it is likely you’ll encounter a logical reasoning test during the early screening stage of the hiring process. Thankfully, logical reasoning, like other reasoning skills, can be improved and honed with practice.

What are common types of logical reasoning tests?

There are several different styles of logical reasoning assessment used within the recruitment process. Logical reasoning can be both verbal and non-verbal. This means that some tests feature paragraphs of text, whilst others involve diagrams.

Deductive reasoning tests

Deductive reasoning tests present candidates with a series of word problems. A set of premises will be outlined, and you’ll need to use the information contained in these to come to the correct conclusion.

These premises may be contained within a paragraph of text, or laid out as a series of statements. You’ll use the evidence provided to select the answer that is true from the list of multiple-choice options.

A simpler level of deductive reasoning is also included within the more general verbal reasoning tests, in questions where the candidate is given a statement and must indicate whether it is ‘true’, ‘false’, or if they ‘cannot say’ based upon the written information given.

Inductive reasoning tests

Inductive reasoning tests - commonly known as diagrammatic reasoning tests , abstract reasoning tests or simply as logical reasoning tests - involve making general inferences from non-verbal information presented in diagrams involving shapes and symbols.

You’ll need to analyse the sequence of shapes given, identify patterns and the rules governing them, and then use these to determine the next figure in the sequence or the missing part of the diagram.

Inductive reasoning tests are multiple choice, so you’ll have a list of answer options to select from.

How to improve your logical reasoning

logical reasoning

Incorporate more conscious logical reasoning into your life

Completing puzzles and games that involve the direct application of logical thinking will get you used to actively working through and deducing the logical steps needed to reach a conclusion.

Incorporating learning with fun means you can hone your logical reasoning skills in an enjoyable way, and it has also been shown to aid the acquisition and retention of skills.

Take stock of all available information

Effective logical reasoning involves rapidly understanding and analysing data - whether in the form of words or diagrams. You’ll need to be able to quickly identify which information is salient and which is superfluous to the problem at hand.

Improving your ability to determine the factors that are influencing situations will assist in identifying the information relevant to logical reasoning questions.

Make sure you understand the different types of logical reasoning

Ensure you know the difference between deductive, inductive, and abductive logic in practice, as this will help you to understand the processes involved in reaching a conclusion in each reasoning type. Working through a scenario from each point of view - similar to in the examples given, will ensure you have a firm grasp on the differences in the ordering of the logic.

Take practice logical reasoning tests

Practicing online logical reasoning tests is the best way to improve your logical reasoning skills. The more practice tests you sit, the more familiar you’ll get with the format and structure of the test.

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7 Module 7: Thinking, Reasoning, and Problem-Solving

This module is about how a solid working knowledge of psychological principles can help you to think more effectively, so you can succeed in school and life. You might be inclined to believe that—because you have been thinking for as long as you can remember, because you are able to figure out the solution to many problems, because you feel capable of using logic to argue a point, because you can evaluate whether the things you read and hear make sense—you do not need any special training in thinking. But this, of course, is one of the key barriers to helping people think better. If you do not believe that there is anything wrong, why try to fix it?

The human brain is indeed a remarkable thinking machine, capable of amazing, complex, creative, logical thoughts. Why, then, are we telling you that you need to learn how to think? Mainly because one major lesson from cognitive psychology is that these capabilities of the human brain are relatively infrequently realized. Many psychologists believe that people are essentially “cognitive misers.” It is not that we are lazy, but that we have a tendency to expend the least amount of mental effort necessary. Although you may not realize it, it actually takes a great deal of energy to think. Careful, deliberative reasoning and critical thinking are very difficult. Because we seem to be successful without going to the trouble of using these skills well, it feels unnecessary to develop them. As you shall see, however, there are many pitfalls in the cognitive processes described in this module. When people do not devote extra effort to learning and improving reasoning, problem solving, and critical thinking skills, they make many errors.

As is true for memory, if you develop the cognitive skills presented in this module, you will be more successful in school. It is important that you realize, however, that these skills will help you far beyond school, even more so than a good memory will. Although it is somewhat useful to have a good memory, ten years from now no potential employer will care how many questions you got right on multiple choice exams during college. All of them will, however, recognize whether you are a logical, analytical, critical thinker. With these thinking skills, you will be an effective, persuasive communicator and an excellent problem solver.

The module begins by describing different kinds of thought and knowledge, especially conceptual knowledge and critical thinking. An understanding of these differences will be valuable as you progress through school and encounter different assignments that require you to tap into different kinds of knowledge. The second section covers deductive and inductive reasoning, which are processes we use to construct and evaluate strong arguments. They are essential skills to have whenever you are trying to persuade someone (including yourself) of some point, or to respond to someone’s efforts to persuade you. The module ends with a section about problem solving. A solid understanding of the key processes involved in problem solving will help you to handle many daily challenges.

7.1. Different kinds of thought

7.2. Reasoning and Judgment

7.3. Problem Solving

READING WITH PURPOSE

Remember and understand.

By reading and studying Module 7, you should be able to remember and describe:

  • Concepts and inferences (7.1)
  • Procedural knowledge (7.1)
  • Metacognition (7.1)
  • Characteristics of critical thinking:  skepticism; identify biases, distortions, omissions, and assumptions; reasoning and problem solving skills  (7.1)
  • Reasoning:  deductive reasoning, deductively valid argument, inductive reasoning, inductively strong argument, availability heuristic, representativeness heuristic  (7.2)
  • Fixation:  functional fixedness, mental set  (7.3)
  • Algorithms, heuristics, and the role of confirmation bias (7.3)
  • Effective problem solving sequence (7.3)

By reading and thinking about how the concepts in Module 6 apply to real life, you should be able to:

  • Identify which type of knowledge a piece of information is (7.1)
  • Recognize examples of deductive and inductive reasoning (7.2)
  • Recognize judgments that have probably been influenced by the availability heuristic (7.2)
  • Recognize examples of problem solving heuristics and algorithms (7.3)

Analyze, Evaluate, and Create

By reading and thinking about Module 6, participating in classroom activities, and completing out-of-class assignments, you should be able to:

  • Use the principles of critical thinking to evaluate information (7.1)
  • Explain whether examples of reasoning arguments are deductively valid or inductively strong (7.2)
  • Outline how you could try to solve a problem from your life using the effective problem solving sequence (7.3)

7.1. Different kinds of thought and knowledge

  • Take a few minutes to write down everything that you know about dogs.
  • Do you believe that:
  • Psychic ability exists?
  • Hypnosis is an altered state of consciousness?
  • Magnet therapy is effective for relieving pain?
  • Aerobic exercise is an effective treatment for depression?
  • UFO’s from outer space have visited earth?

On what do you base your belief or disbelief for the questions above?

Of course, we all know what is meant by the words  think  and  knowledge . You probably also realize that they are not unitary concepts; there are different kinds of thought and knowledge. In this section, let us look at some of these differences. If you are familiar with these different kinds of thought and pay attention to them in your classes, it will help you to focus on the right goals, learn more effectively, and succeed in school. Different assignments and requirements in school call on you to use different kinds of knowledge or thought, so it will be very helpful for you to learn to recognize them (Anderson, et al. 2001).

Factual and conceptual knowledge

Module 5 introduced the idea of declarative memory, which is composed of facts and episodes. If you have ever played a trivia game or watched Jeopardy on TV, you realize that the human brain is able to hold an extraordinary number of facts. Likewise, you realize that each of us has an enormous store of episodes, essentially facts about events that happened in our own lives. It may be difficult to keep that in mind when we are struggling to retrieve one of those facts while taking an exam, however. Part of the problem is that, in contradiction to the advice from Module 5, many students continue to try to memorize course material as a series of unrelated facts (picture a history student simply trying to memorize history as a set of unrelated dates without any coherent story tying them together). Facts in the real world are not random and unorganized, however. It is the way that they are organized that constitutes a second key kind of knowledge, conceptual.

Concepts are nothing more than our mental representations of categories of things in the world. For example, think about dogs. When you do this, you might remember specific facts about dogs, such as they have fur and they bark. You may also recall dogs that you have encountered and picture them in your mind. All of this information (and more) makes up your concept of dog. You can have concepts of simple categories (e.g., triangle), complex categories (e.g., small dogs that sleep all day, eat out of the garbage, and bark at leaves), kinds of people (e.g., psychology professors), events (e.g., birthday parties), and abstract ideas (e.g., justice). Gregory Murphy (2002) refers to concepts as the “glue that holds our mental life together” (p. 1). Very simply, summarizing the world by using concepts is one of the most important cognitive tasks that we do. Our conceptual knowledge  is  our knowledge about the world. Individual concepts are related to each other to form a rich interconnected network of knowledge. For example, think about how the following concepts might be related to each other: dog, pet, play, Frisbee, chew toy, shoe. Or, of more obvious use to you now, how these concepts are related: working memory, long-term memory, declarative memory, procedural memory, and rehearsal? Because our minds have a natural tendency to organize information conceptually, when students try to remember course material as isolated facts, they are working against their strengths.

One last important point about concepts is that they allow you to instantly know a great deal of information about something. For example, if someone hands you a small red object and says, “here is an apple,” they do not have to tell you, “it is something you can eat.” You already know that you can eat it because it is true by virtue of the fact that the object is an apple; this is called drawing an  inference , assuming that something is true on the basis of your previous knowledge (for example, of category membership or of how the world works) or logical reasoning.

Procedural knowledge

Physical skills, such as tying your shoes, doing a cartwheel, and driving a car (or doing all three at the same time, but don’t try this at home) are certainly a kind of knowledge. They are procedural knowledge, the same idea as procedural memory that you saw in Module 5. Mental skills, such as reading, debating, and planning a psychology experiment, are procedural knowledge, as well. In short, procedural knowledge is the knowledge how to do something (Cohen & Eichenbaum, 1993).

Metacognitive knowledge

Floyd used to think that he had a great memory. Now, he has a better memory. Why? Because he finally realized that his memory was not as great as he once thought it was. Because Floyd eventually learned that he often forgets where he put things, he finally developed the habit of putting things in the same place. (Unfortunately, he did not learn this lesson before losing at least 5 watches and a wedding ring.) Because he finally realized that he often forgets to do things, he finally started using the To Do list app on his phone. And so on. Floyd’s insights about the real limitations of his memory have allowed him to remember things that he used to forget.

All of us have knowledge about the way our own minds work. You may know that you have a good memory for people’s names and a poor memory for math formulas. Someone else might realize that they have difficulty remembering to do things, like stopping at the store on the way home. Others still know that they tend to overlook details. This knowledge about our own thinking is actually quite important; it is called metacognitive knowledge, or  metacognition . Like other kinds of thinking skills, it is subject to error. For example, in unpublished research, one of the authors surveyed about 120 General Psychology students on the first day of the term. Among other questions, the students were asked them to predict their grade in the class and report their current Grade Point Average. Two-thirds of the students predicted that their grade in the course would be higher than their GPA. (The reality is that at our college, students tend to earn lower grades in psychology than their overall GPA.) Another example: Students routinely report that they thought they had done well on an exam, only to discover, to their dismay, that they were wrong (more on that important problem in a moment). Both errors reveal a breakdown in metacognition.

The Dunning-Kruger Effect

In general, most college students probably do not study enough. For example, using data from the National Survey of Student Engagement, Fosnacht, McCormack, and Lerma (2018) reported that first-year students at 4-year colleges in the U.S. averaged less than 14 hours per week preparing for classes. The typical suggestion is that you should spend two hours outside of class for every hour in class, or 24 – 30 hours per week for a full-time student. Clearly, students in general are nowhere near that recommended mark. Many observers, including some faculty, believe that this shortfall is a result of students being too busy or lazy. Now, it may be true that many students are too busy, with work and family obligations, for example. Others, are not particularly motivated in school, and therefore might correctly be labeled lazy. A third possible explanation, however, is that some students might not think they need to spend this much time. And this is a matter of metacognition. Consider the scenario that we mentioned above, students thinking they had done well on an exam only to discover that they did not. Justin Kruger and David Dunning examined scenarios very much like this in 1999. Kruger and Dunning gave research participants tests measuring humor, logic, and grammar. Then, they asked the participants to assess their own abilities and test performance in these areas. They found that participants in general tended to overestimate their abilities, already a problem with metacognition. Importantly, the participants who scored the lowest overestimated their abilities the most. Specifically, students who scored in the bottom quarter (averaging in the 12th percentile) thought they had scored in the 62nd percentile. This has become known as the  Dunning-Kruger effect . Many individual faculty members have replicated these results with their own student on their course exams, including the authors of this book. Think about it. Some students who just took an exam and performed poorly believe that they did well before seeing their score. It seems very likely that these are the very same students who stopped studying the night before because they thought they were “done.” Quite simply, it is not just that they did not know the material. They did not know that they did not know the material. That is poor metacognition.

In order to develop good metacognitive skills, you should continually monitor your thinking and seek frequent feedback on the accuracy of your thinking (Medina, Castleberry, & Persky 2017). For example, in classes get in the habit of predicting your exam grades. As soon as possible after taking an exam, try to find out which questions you missed and try to figure out why. If you do this soon enough, you may be able to recall the way it felt when you originally answered the question. Did you feel confident that you had answered the question correctly? Then you have just discovered an opportunity to improve your metacognition. Be on the lookout for that feeling and respond with caution.

concept :  a mental representation of a category of things in the world

Dunning-Kruger effect : individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

inference : an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

metacognition :  knowledge about one’s own cognitive processes; thinking about your thinking

Critical thinking

One particular kind of knowledge or thinking skill that is related to metacognition is  critical thinking (Chew, 2020). You may have noticed that critical thinking is an objective in many college courses, and thus it could be a legitimate topic to cover in nearly any college course. It is particularly appropriate in psychology, however. As the science of (behavior and) mental processes, psychology is obviously well suited to be the discipline through which you should be introduced to this important way of thinking.

More importantly, there is a particular need to use critical thinking in psychology. We are all, in a way, experts in human behavior and mental processes, having engaged in them literally since birth. Thus, perhaps more than in any other class, students typically approach psychology with very clear ideas and opinions about its subject matter. That is, students already “know” a lot about psychology. The problem is, “it ain’t so much the things we don’t know that get us into trouble. It’s the things we know that just ain’t so” (Ward, quoted in Gilovich 1991). Indeed, many of students’ preconceptions about psychology are just plain wrong. Randolph Smith (2002) wrote a book about critical thinking in psychology called  Challenging Your Preconceptions,  highlighting this fact. On the other hand, many of students’ preconceptions about psychology are just plain right! But wait, how do you know which of your preconceptions are right and which are wrong? And when you come across a research finding or theory in this class that contradicts your preconceptions, what will you do? Will you stick to your original idea, discounting the information from the class? Will you immediately change your mind? Critical thinking can help us sort through this confusing mess.

But what is critical thinking? The goal of critical thinking is simple to state (but extraordinarily difficult to achieve): it is to be right, to draw the correct conclusions, to believe in things that are true and to disbelieve things that are false. We will provide two definitions of critical thinking (or, if you like, one large definition with two distinct parts). First, a more conceptual one: Critical thinking is thinking like a scientist in your everyday life (Schmaltz, Jansen, & Wenckowski, 2017).  Our second definition is more operational; it is simply a list of skills that are essential to be a critical thinker. Critical thinking entails solid reasoning and problem solving skills; skepticism; and an ability to identify biases, distortions, omissions, and assumptions. Excellent deductive and inductive reasoning, and problem solving skills contribute to critical thinking. So, you can consider the subject matter of sections 7.2 and 7.3 to be part of critical thinking. Because we will be devoting considerable time to these concepts in the rest of the module, let us begin with a discussion about the other aspects of critical thinking.

Let’s address that first part of the definition. Scientists form hypotheses, or predictions about some possible future observations. Then, they collect data, or information (think of this as making those future observations). They do their best to make unbiased observations using reliable techniques that have been verified by others. Then, and only then, they draw a conclusion about what those observations mean. Oh, and do not forget the most important part. “Conclusion” is probably not the most appropriate word because this conclusion is only tentative. A scientist is always prepared that someone else might come along and produce new observations that would require a new conclusion be drawn. Wow! If you like to be right, you could do a lot worse than using a process like this.

A Critical Thinker’s Toolkit 

Now for the second part of the definition. Good critical thinkers (and scientists) rely on a variety of tools to evaluate information. Perhaps the most recognizable tool for critical thinking is  skepticism (and this term provides the clearest link to the thinking like a scientist definition, as you are about to see). Some people intend it as an insult when they call someone a skeptic. But if someone calls you a skeptic, if they are using the term correctly, you should consider it a great compliment. Simply put, skepticism is a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided. People from Missouri should recognize this principle, as Missouri is known as the Show-Me State. As a skeptic, you are not inclined to believe something just because someone said so, because someone else believes it, or because it sounds reasonable. You must be persuaded by high quality evidence.

Of course, if that evidence is produced, you have a responsibility as a skeptic to change your belief. Failure to change a belief in the face of good evidence is not skepticism; skepticism has open mindedness at its core. M. Neil Browne and Stuart Keeley (2018) use the term weak sense critical thinking to describe critical thinking behaviors that are used only to strengthen a prior belief. Strong sense critical thinking, on the other hand, has as its goal reaching the best conclusion. Sometimes that means strengthening your prior belief, but sometimes it means changing your belief to accommodate the better evidence.

Many times, a failure to think critically or weak sense critical thinking is related to a  bias , an inclination, tendency, leaning, or prejudice. Everybody has biases, but many people are unaware of them. Awareness of your own biases gives you the opportunity to control or counteract them. Unfortunately, however, many people are happy to let their biases creep into their attempts to persuade others; indeed, it is a key part of their persuasive strategy. To see how these biases influence messages, just look at the different descriptions and explanations of the same events given by people of different ages or income brackets, or conservative versus liberal commentators, or by commentators from different parts of the world. Of course, to be successful, these people who are consciously using their biases must disguise them. Even undisguised biases can be difficult to identify, so disguised ones can be nearly impossible.

Here are some common sources of biases:

  • Personal values and beliefs.  Some people believe that human beings are basically driven to seek power and that they are typically in competition with one another over scarce resources. These beliefs are similar to the world-view that political scientists call “realism.” Other people believe that human beings prefer to cooperate and that, given the chance, they will do so. These beliefs are similar to the world-view known as “idealism.” For many people, these deeply held beliefs can influence, or bias, their interpretations of such wide ranging situations as the behavior of nations and their leaders or the behavior of the driver in the car ahead of you. For example, if your worldview is that people are typically in competition and someone cuts you off on the highway, you may assume that the driver did it purposely to get ahead of you. Other types of beliefs about the way the world is or the way the world should be, for example, political beliefs, can similarly become a significant source of bias.
  • Racism, sexism, ageism and other forms of prejudice and bigotry.  These are, sadly, a common source of bias in many people. They are essentially a special kind of “belief about the way the world is.” These beliefs—for example, that women do not make effective leaders—lead people to ignore contradictory evidence (examples of effective women leaders, or research that disputes the belief) and to interpret ambiguous evidence in a way consistent with the belief.
  • Self-interest.  When particular people benefit from things turning out a certain way, they can sometimes be very susceptible to letting that interest bias them. For example, a company that will earn a profit if they sell their product may have a bias in the way that they give information about their product. A union that will benefit if its members get a generous contract might have a bias in the way it presents information about salaries at competing organizations. (Note that our inclusion of examples describing both companies and unions is an explicit attempt to control for our own personal biases). Home buyers are often dismayed to discover that they purchased their dream house from someone whose self-interest led them to lie about flooding problems in the basement or back yard. This principle, the biasing power of self-interest, is likely what led to the famous phrase  Caveat Emptor  (let the buyer beware) .  

Knowing that these types of biases exist will help you evaluate evidence more critically. Do not forget, though, that people are not always keen to let you discover the sources of biases in their arguments. For example, companies or political organizations can sometimes disguise their support of a research study by contracting with a university professor, who comes complete with a seemingly unbiased institutional affiliation, to conduct the study.

People’s biases, conscious or unconscious, can lead them to make omissions, distortions, and assumptions that undermine our ability to correctly evaluate evidence. It is essential that you look for these elements. Always ask, what is missing, what is not as it appears, and what is being assumed here? For example, consider this (fictional) chart from an ad reporting customer satisfaction at 4 local health clubs.

problem solving using logical reasoning

Clearly, from the results of the chart, one would be tempted to give Club C a try, as customer satisfaction is much higher than for the other 3 clubs.

There are so many distortions and omissions in this chart, however, that it is actually quite meaningless. First, how was satisfaction measured? Do the bars represent responses to a survey? If so, how were the questions asked? Most importantly, where is the missing scale for the chart? Although the differences look quite large, are they really?

Well, here is the same chart, with a different scale, this time labeled:

problem solving using logical reasoning

Club C is not so impressive any more, is it? In fact, all of the health clubs have customer satisfaction ratings (whatever that means) between 85% and 88%. In the first chart, the entire scale of the graph included only the percentages between 83 and 89. This “judicious” choice of scale—some would call it a distortion—and omission of that scale from the chart make the tiny differences among the clubs seem important, however.

Also, in order to be a critical thinker, you need to learn to pay attention to the assumptions that underlie a message. Let us briefly illustrate the role of assumptions by touching on some people’s beliefs about the criminal justice system in the US. Some believe that a major problem with our judicial system is that many criminals go free because of legal technicalities. Others believe that a major problem is that many innocent people are convicted of crimes. The simple fact is, both types of errors occur. A person’s conclusion about which flaw in our judicial system is the greater tragedy is based on an assumption about which of these is the more serious error (letting the guilty go free or convicting the innocent). This type of assumption is called a value assumption (Browne and Keeley, 2018). It reflects the differences in values that people develop, differences that may lead us to disregard valid evidence that does not fit in with our particular values.

Oh, by the way, some students probably noticed this, but the seven tips for evaluating information that we shared in Module 1 are related to this. Actually, they are part of this section. The tips are, to a very large degree, set of ideas you can use to help you identify biases, distortions, omissions, and assumptions. If you do not remember this section, we strongly recommend you take a few minutes to review it.

skepticism :  a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

bias : an inclination, tendency, leaning, or prejudice

  • Which of your beliefs (or disbeliefs) from the Activate exercise for this section were derived from a process of critical thinking? If some of your beliefs were not based on critical thinking, are you willing to reassess these beliefs? If the answer is no, why do you think that is? If the answer is yes, what concrete steps will you take?

7.2 Reasoning and Judgment

  • What percentage of kidnappings are committed by strangers?
  • Which area of the house is riskiest: kitchen, bathroom, or stairs?
  • What is the most common cancer in the US?
  • What percentage of workplace homicides are committed by co-workers?

An essential set of procedural thinking skills is  reasoning , the ability to generate and evaluate solid conclusions from a set of statements or evidence. You should note that these conclusions (when they are generated instead of being evaluated) are one key type of inference that we described in Section 7.1. There are two main types of reasoning, deductive and inductive.

Deductive reasoning

Suppose your teacher tells you that if you get an A on the final exam in a course, you will get an A for the whole course. Then, you get an A on the final exam. What will your final course grade be? Most people can see instantly that you can conclude with certainty that you will get an A for the course. This is a type of reasoning called  deductive reasoning , which is defined as reasoning in which a conclusion is guaranteed to be true as long as the statements leading to it are true. The three statements can be listed as an  argument , with two beginning statements and a conclusion:

Statement 1: If you get an A on the final exam, you will get an A for the course

Statement 2: You get an A on the final exam

Conclusion: You will get an A for the course

This particular arrangement, in which true beginning statements lead to a guaranteed true conclusion, is known as a  deductively valid argument . Although deductive reasoning is often the subject of abstract, brain-teasing, puzzle-like word problems, it is actually an extremely important type of everyday reasoning. It is just hard to recognize sometimes. For example, imagine that you are looking for your car keys and you realize that they are either in the kitchen drawer or in your book bag. After looking in the kitchen drawer, you instantly know that they must be in your book bag. That conclusion results from a simple deductive reasoning argument. In addition, solid deductive reasoning skills are necessary for you to succeed in the sciences, philosophy, math, computer programming, and any endeavor involving the use of logic to persuade others to your point of view or to evaluate others’ arguments.

Cognitive psychologists, and before them philosophers, have been quite interested in deductive reasoning, not so much for its practical applications, but for the insights it can offer them about the ways that human beings think. One of the early ideas to emerge from the examination of deductive reasoning is that people learn (or develop) mental versions of rules that allow them to solve these types of reasoning problems (Braine, 1978; Braine, Reiser, & Rumain, 1984). The best way to see this point of view is to realize that there are different possible rules, and some of them are very simple. For example, consider this rule of logic:

therefore q

Logical rules are often presented abstractly, as letters, in order to imply that they can be used in very many specific situations. Here is a concrete version of the of the same rule:

I’ll either have pizza or a hamburger for dinner tonight (p or q)

I won’t have pizza (not p)

Therefore, I’ll have a hamburger (therefore q)

This kind of reasoning seems so natural, so easy, that it is quite plausible that we would use a version of this rule in our daily lives. At least, it seems more plausible than some of the alternative possibilities—for example, that we need to have experience with the specific situation (pizza or hamburger, in this case) in order to solve this type of problem easily. So perhaps there is a form of natural logic (Rips, 1990) that contains very simple versions of logical rules. When we are faced with a reasoning problem that maps onto one of these rules, we use the rule.

But be very careful; things are not always as easy as they seem. Even these simple rules are not so simple. For example, consider the following rule. Many people fail to realize that this rule is just as valid as the pizza or hamburger rule above.

if p, then q

therefore, not p

Concrete version:

If I eat dinner, then I will have dessert

I did not have dessert

Therefore, I did not eat dinner

The simple fact is, it can be very difficult for people to apply rules of deductive logic correctly; as a result, they make many errors when trying to do so. Is this a deductively valid argument or not?

Students who like school study a lot

Students who study a lot get good grades

Jane does not like school

Therefore, Jane does not get good grades

Many people are surprised to discover that this is not a logically valid argument; the conclusion is not guaranteed to be true from the beginning statements. Although the first statement says that students who like school study a lot, it does NOT say that students who do not like school do not study a lot. In other words, it may very well be possible to study a lot without liking school. Even people who sometimes get problems like this right might not be using the rules of deductive reasoning. Instead, they might just be making judgments for examples they know, in this case, remembering instances of people who get good grades despite not liking school.

Making deductive reasoning even more difficult is the fact that there are two important properties that an argument may have. One, it can be valid or invalid (meaning that the conclusion does or does not follow logically from the statements leading up to it). Two, an argument (or more correctly, its conclusion) can be true or false. Here is an example of an argument that is logically valid, but has a false conclusion (at least we think it is false).

Either you are eleven feet tall or the Grand Canyon was created by a spaceship crashing into the earth.

You are not eleven feet tall

Therefore the Grand Canyon was created by a spaceship crashing into the earth

This argument has the exact same form as the pizza or hamburger argument above, making it is deductively valid. The conclusion is so false, however, that it is absurd (of course, the reason the conclusion is false is that the first statement is false). When people are judging arguments, they tend to not observe the difference between deductive validity and the empirical truth of statements or conclusions. If the elements of an argument happen to be true, people are likely to judge the argument logically valid; if the elements are false, they will very likely judge it invalid (Markovits & Bouffard-Bouchard, 1992; Moshman & Franks, 1986). Thus, it seems a stretch to say that people are using these logical rules to judge the validity of arguments. Many psychologists believe that most people actually have very limited deductive reasoning skills (Johnson-Laird, 1999). They argue that when faced with a problem for which deductive logic is required, people resort to some simpler technique, such as matching terms that appear in the statements and the conclusion (Evans, 1982). This might not seem like a problem, but what if reasoners believe that the elements are true and they happen to be wrong; they will would believe that they are using a form of reasoning that guarantees they are correct and yet be wrong.

deductive reasoning :  a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

argument :  a set of statements in which the beginning statements lead to a conclusion

deductively valid argument :  an argument for which true beginning statements guarantee that the conclusion is true

Inductive reasoning and judgment

Every day, you make many judgments about the likelihood of one thing or another. Whether you realize it or not, you are practicing  inductive reasoning   on a daily basis. In inductive reasoning arguments, a conclusion is likely whenever the statements preceding it are true. The first thing to notice about inductive reasoning is that, by definition, you can never be sure about your conclusion; you can only estimate how likely the conclusion is. Inductive reasoning may lead you to focus on Memory Encoding and Recoding when you study for the exam, but it is possible the instructor will ask more questions about Memory Retrieval instead. Unlike deductive reasoning, the conclusions you reach through inductive reasoning are only probable, not certain. That is why scientists consider inductive reasoning weaker than deductive reasoning. But imagine how hard it would be for us to function if we could not act unless we were certain about the outcome.

Inductive reasoning can be represented as logical arguments consisting of statements and a conclusion, just as deductive reasoning can be. In an inductive argument, you are given some statements and a conclusion (or you are given some statements and must draw a conclusion). An argument is  inductively strong   if the conclusion would be very probable whenever the statements are true. So, for example, here is an inductively strong argument:

  • Statement #1: The forecaster on Channel 2 said it is going to rain today.
  • Statement #2: The forecaster on Channel 5 said it is going to rain today.
  • Statement #3: It is very cloudy and humid.
  • Statement #4: You just heard thunder.
  • Conclusion (or judgment): It is going to rain today.

Think of the statements as evidence, on the basis of which you will draw a conclusion. So, based on the evidence presented in the four statements, it is very likely that it will rain today. Will it definitely rain today? Certainly not. We can all think of times that the weather forecaster was wrong.

A true story: Some years ago psychology student was watching a baseball playoff game between the St. Louis Cardinals and the Los Angeles Dodgers. A graphic on the screen had just informed the audience that the Cardinal at bat, (Hall of Fame shortstop) Ozzie Smith, a switch hitter batting left-handed for this plate appearance, had never, in nearly 3000 career at-bats, hit a home run left-handed. The student, who had just learned about inductive reasoning in his psychology class, turned to his companion (a Cardinals fan) and smugly said, “It is an inductively strong argument that Ozzie Smith will not hit a home run.” He turned back to face the television just in time to watch the ball sail over the right field fence for a home run. Although the student felt foolish at the time, he was not wrong. It was an inductively strong argument; 3000 at-bats is an awful lot of evidence suggesting that the Wizard of Ozz (as he was known) would not be hitting one out of the park (think of each at-bat without a home run as a statement in an inductive argument). Sadly (for the die-hard Cubs fan and Cardinals-hating student), despite the strength of the argument, the conclusion was wrong.

Given the possibility that we might draw an incorrect conclusion even with an inductively strong argument, we really want to be sure that we do, in fact, make inductively strong arguments. If we judge something probable, it had better be probable. If we judge something nearly impossible, it had better not happen. Think of inductive reasoning, then, as making reasonably accurate judgments of the probability of some conclusion given a set of evidence.

We base many decisions in our lives on inductive reasoning. For example:

Statement #1: Psychology is not my best subject

Statement #2: My psychology instructor has a reputation for giving difficult exams

Statement #3: My first psychology exam was much harder than I expected

Judgment: The next exam will probably be very difficult.

Decision: I will study tonight instead of watching Netflix.

Some other examples of judgments that people commonly make in a school context include judgments of the likelihood that:

  • A particular class will be interesting/useful/difficult
  • You will be able to finish writing a paper by next week if you go out tonight
  • Your laptop’s battery will last through the next trip to the library
  • You will not miss anything important if you skip class tomorrow
  • Your instructor will not notice if you skip class tomorrow
  • You will be able to find a book that you will need for a paper
  • There will be an essay question about Memory Encoding on the next exam

Tversky and Kahneman (1983) recognized that there are two general ways that we might make these judgments; they termed them extensional (i.e., following the laws of probability) and intuitive (i.e., using shortcuts or heuristics, see below). We will use a similar distinction between Type 1 and Type 2 thinking, as described by Keith Stanovich and his colleagues (Evans and Stanovich, 2013; Stanovich and West, 2000). Type 1 thinking is fast, automatic, effortful, and emotional. In fact, it is hardly fair to call it reasoning at all, as judgments just seem to pop into one’s head. Type 2 thinking , on the other hand, is slow, effortful, and logical. So obviously, it is more likely to lead to a correct judgment, or an optimal decision. The problem is, we tend to over-rely on Type 1. Now, we are not saying that Type 2 is the right way to go for every decision or judgment we make. It seems a bit much, for example, to engage in a step-by-step logical reasoning procedure to decide whether we will have chicken or fish for dinner tonight.

Many bad decisions in some very important contexts, however, can be traced back to poor judgments of the likelihood of certain risks or outcomes that result from the use of Type 1 when a more logical reasoning process would have been more appropriate. For example:

Statement #1: It is late at night.

Statement #2: Albert has been drinking beer for the past five hours at a party.

Statement #3: Albert is not exactly sure where he is or how far away home is.

Judgment: Albert will have no difficulty walking home.

Decision: He walks home alone.

As you can see in this example, the three statements backing up the judgment do not really support it. In other words, this argument is not inductively strong because it is based on judgments that ignore the laws of probability. What are the chances that someone facing these conditions will be able to walk home alone easily? And one need not be drunk to make poor decisions based on judgments that just pop into our heads.

The truth is that many of our probability judgments do not come very close to what the laws of probability say they should be. Think about it. In order for us to reason in accordance with these laws, we would need to know the laws of probability, which would allow us to calculate the relationship between particular pieces of evidence and the probability of some outcome (i.e., how much likelihood should change given a piece of evidence), and we would have to do these heavy math calculations in our heads. After all, that is what Type 2 requires. Needless to say, even if we were motivated, we often do not even know how to apply Type 2 reasoning in many cases.

So what do we do when we don’t have the knowledge, skills, or time required to make the correct mathematical judgment? Do we hold off and wait until we can get better evidence? Do we read up on probability and fire up our calculator app so we can compute the correct probability? Of course not. We rely on Type 1 thinking. We “wing it.” That is, we come up with a likelihood estimate using some means at our disposal. Psychologists use the term heuristic to describe the type of “winging it” we are talking about. A  heuristic   is a shortcut strategy that we use to make some judgment or solve some problem (see Section 7.3). Heuristics are easy and quick, think of them as the basic procedures that are characteristic of Type 1.  They can absolutely lead to reasonably good judgments and decisions in some situations (like choosing between chicken and fish for dinner). They are, however, far from foolproof. There are, in fact, quite a lot of situations in which heuristics can lead us to make incorrect judgments, and in many cases the decisions based on those judgments can have serious consequences.

Let us return to the activity that begins this section. You were asked to judge the likelihood (or frequency) of certain events and risks. You were free to come up with your own evidence (or statements) to make these judgments. This is where a heuristic crops up. As a judgment shortcut, we tend to generate specific examples of those very events to help us decide their likelihood or frequency. For example, if we are asked to judge how common, frequent, or likely a particular type of cancer is, many of our statements would be examples of specific cancer cases:

Statement #1: Andy Kaufman (comedian) had lung cancer.

Statement #2: Colin Powell (US Secretary of State) had prostate cancer.

Statement #3: Bob Marley (musician) had skin and brain cancer

Statement #4: Sandra Day O’Connor (Supreme Court Justice) had breast cancer.

Statement #5: Fred Rogers (children’s entertainer) had stomach cancer.

Statement #6: Robin Roberts (news anchor) had breast cancer.

Statement #7: Bette Davis (actress) had breast cancer.

Judgment: Breast cancer is the most common type.

Your own experience or memory may also tell you that breast cancer is the most common type. But it is not (although it is common). Actually, skin cancer is the most common type in the US. We make the same types of misjudgments all the time because we do not generate the examples or evidence according to their actual frequencies or probabilities. Instead, we have a tendency (or bias) to search for the examples in memory; if they are easy to retrieve, we assume that they are common. To rephrase this in the language of the heuristic, events seem more likely to the extent that they are available to memory. This bias has been termed the  availability heuristic   (Kahneman and Tversky, 1974).

The fact that we use the availability heuristic does not automatically mean that our judgment is wrong. The reason we use heuristics in the first place is that they work fairly well in many cases (and, of course that they are easy to use). So, the easiest examples to think of sometimes are the most common ones. Is it more likely that a member of the U.S. Senate is a man or a woman? Most people have a much easier time generating examples of male senators. And as it turns out, the U.S. Senate has many more men than women (74 to 26 in 2020). In this case, then, the availability heuristic would lead you to make the correct judgment; it is far more likely that a senator would be a man.

In many other cases, however, the availability heuristic will lead us astray. This is because events can be memorable for many reasons other than their frequency. Section 5.2, Encoding Meaning, suggested that one good way to encode the meaning of some information is to form a mental image of it. Thus, information that has been pictured mentally will be more available to memory. Indeed, an event that is vivid and easily pictured will trick many people into supposing that type of event is more common than it actually is. Repetition of information will also make it more memorable. So, if the same event is described to you in a magazine, on the evening news, on a podcast that you listen to, and in your Facebook feed; it will be very available to memory. Again, the availability heuristic will cause you to misperceive the frequency of these types of events.

Most interestingly, information that is unusual is more memorable. Suppose we give you the following list of words to remember: box, flower, letter, platypus, oven, boat, newspaper, purse, drum, car. Very likely, the easiest word to remember would be platypus, the unusual one. The same thing occurs with memories of events. An event may be available to memory because it is unusual, yet the availability heuristic leads us to judge that the event is common. Did you catch that? In these cases, the availability heuristic makes us think the exact opposite of the true frequency. We end up thinking something is common because it is unusual (and therefore memorable). Yikes.

The misapplication of the availability heuristic sometimes has unfortunate results. For example, if you went to K-12 school in the US over the past 10 years, it is extremely likely that you have participated in lockdown and active shooter drills. Of course, everyone is trying to prevent the tragedy of another school shooting. And believe us, we are not trying to minimize how terrible the tragedy is. But the truth of the matter is, school shootings are extremely rare. Because the federal government does not keep a database of school shootings, the Washington Post has maintained their own running tally. Between 1999 and January 2020 (the date of the most recent school shooting with a death in the US at of the time this paragraph was written), the Post reported a total of 254 people died in school shootings in the US. Not 254 per year, 254 total. That is an average of 12 per year. Of course, that is 254 people who should not have died (particularly because many were children), but in a country with approximately 60,000,000 students and teachers, this is a very small risk.

But many students and teachers are terrified that they will be victims of school shootings because of the availability heuristic. It is so easy to think of examples (they are very available to memory) that people believe the event is very common. It is not. And there is a downside to this. We happen to believe that there is an enormous gun violence problem in the United States. According the the Centers for Disease Control and Prevention, there were 39,773 firearm deaths in the US in 2017. Fifteen of those deaths were in school shootings, according to the Post. 60% of those deaths were suicides. When people pay attention to the school shooting risk (low), they often fail to notice the much larger risk.

And examples like this are by no means unique. The authors of this book have been teaching psychology since the 1990’s. We have been able to make the exact same arguments about the misapplication of the availability heuristics and keep them current by simply swapping out for the “fear of the day.” In the 1990’s it was children being kidnapped by strangers (it was known as “stranger danger”) despite the facts that kidnappings accounted for only 2% of the violent crimes committed against children, and only 24% of kidnappings are committed by strangers (US Department of Justice, 2007). This fear overlapped with the fear of terrorism that gripped the country after the 2001 terrorist attacks on the World Trade Center and US Pentagon and still plagues the population of the US somewhat in 2020. After a well-publicized, sensational act of violence, people are extremely likely to increase their estimates of the chances that they, too, will be victims of terror. Think about the reality, however. In October of 2001, a terrorist mailed anthrax spores to members of the US government and a number of media companies. A total of five people died as a result of this attack. The nation was nearly paralyzed by the fear of dying from the attack; in reality the probability of an individual person dying was 0.00000002.

The availability heuristic can lead you to make incorrect judgments in a school setting as well. For example, suppose you are trying to decide if you should take a class from a particular math professor. You might try to make a judgment of how good a teacher she is by recalling instances of friends and acquaintances making comments about her teaching skill. You may have some examples that suggest that she is a poor teacher very available to memory, so on the basis of the availability heuristic you judge her a poor teacher and decide to take the class from someone else. What if, however, the instances you recalled were all from the same person, and this person happens to be a very colorful storyteller? The subsequent ease of remembering the instances might not indicate that the professor is a poor teacher after all.

Although the availability heuristic is obviously important, it is not the only judgment heuristic we use. Amos Tversky and Daniel Kahneman examined the role of heuristics in inductive reasoning in a long series of studies. Kahneman received a Nobel Prize in Economics for this research in 2002, and Tversky would have certainly received one as well if he had not died of melanoma at age 59 in 1996 (Nobel Prizes are not awarded posthumously). Kahneman and Tversky demonstrated repeatedly that people do not reason in ways that are consistent with the laws of probability. They identified several heuristic strategies that people use instead to make judgments about likelihood. The importance of this work for economics (and the reason that Kahneman was awarded the Nobel Prize) is that earlier economic theories had assumed that people do make judgments rationally, that is, in agreement with the laws of probability.

Another common heuristic that people use for making judgments is the  representativeness heuristic (Kahneman & Tversky 1973). Suppose we describe a person to you. He is quiet and shy, has an unassuming personality, and likes to work with numbers. Is this person more likely to be an accountant or an attorney? If you said accountant, you were probably using the representativeness heuristic. Our imaginary person is judged likely to be an accountant because he resembles, or is representative of the concept of, an accountant. When research participants are asked to make judgments such as these, the only thing that seems to matter is the representativeness of the description. For example, if told that the person described is in a room that contains 70 attorneys and 30 accountants, participants will still assume that he is an accountant.

inductive reasoning :  a type of reasoning in which we make judgments about likelihood from sets of evidence

inductively strong argument :  an inductive argument in which the beginning statements lead to a conclusion that is probably true

heuristic :  a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

availability heuristic :  judging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

representativeness heuristic:   judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

Type 1 thinking : fast, automatic, and emotional thinking.

Type 2 thinking : slow, effortful, and logical thinking.

  • What percentage of workplace homicides are co-worker violence?

Many people get these questions wrong. The answers are 10%; stairs; skin; 6%. How close were your answers? Explain how the availability heuristic might have led you to make the incorrect judgments.

  • Can you think of some other judgments that you have made (or beliefs that you have) that might have been influenced by the availability heuristic?

7.3 Problem Solving

  • Please take a few minutes to list a number of problems that you are facing right now.
  • Now write about a problem that you recently solved.
  • What is your definition of a problem?

Mary has a problem. Her daughter, ordinarily quite eager to please, appears to delight in being the last person to do anything. Whether getting ready for school, going to piano lessons or karate class, or even going out with her friends, she seems unwilling or unable to get ready on time. Other people have different kinds of problems. For example, many students work at jobs, have numerous family commitments, and are facing a course schedule full of difficult exams, assignments, papers, and speeches. How can they find enough time to devote to their studies and still fulfill their other obligations? Speaking of students and their problems: Show that a ball thrown vertically upward with initial velocity v0 takes twice as much time to return as to reach the highest point (from Spiegel, 1981).

These are three very different situations, but we have called them all problems. What makes them all the same, despite the differences? A psychologist might define a  problem   as a situation with an initial state, a goal state, and a set of possible intermediate states. Somewhat more meaningfully, we might consider a problem a situation in which you are in here one state (e.g., daughter is always late), you want to be there in another state (e.g., daughter is not always late), and with no obvious way to get from here to there. Defined this way, each of the three situations we outlined can now be seen as an example of the same general concept, a problem. At this point, you might begin to wonder what is not a problem, given such a general definition. It seems that nearly every non-routine task we engage in could qualify as a problem. As long as you realize that problems are not necessarily bad (it can be quite fun and satisfying to rise to the challenge and solve a problem), this may be a useful way to think about it.

Can we identify a set of problem-solving skills that would apply to these very different kinds of situations? That task, in a nutshell, is a major goal of this section. Let us try to begin to make sense of the wide variety of ways that problems can be solved with an important observation: the process of solving problems can be divided into two key parts. First, people have to notice, comprehend, and represent the problem properly in their minds (called  problem representation ). Second, they have to apply some kind of solution strategy to the problem. Psychologists have studied both of these key parts of the process in detail.

When you first think about the problem-solving process, you might guess that most of our difficulties would occur because we are failing in the second step, the application of strategies. Although this can be a significant difficulty much of the time, the more important source of difficulty is probably problem representation. In short, we often fail to solve a problem because we are looking at it, or thinking about it, the wrong way.

problem :  a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

problem representation :  noticing, comprehending and forming a mental conception of a problem

Defining and Mentally Representing Problems in Order to Solve Them

So, the main obstacle to solving a problem is that we do not clearly understand exactly what the problem is. Recall the problem with Mary’s daughter always being late. One way to represent, or to think about, this problem is that she is being defiant. She refuses to get ready in time. This type of representation or definition suggests a particular type of solution. Another way to think about the problem, however, is to consider the possibility that she is simply being sidetracked by interesting diversions. This different conception of what the problem is (i.e., different representation) suggests a very different solution strategy. For example, if Mary defines the problem as defiance, she may be tempted to solve the problem using some kind of coercive tactics, that is, to assert her authority as her mother and force her to listen. On the other hand, if Mary defines the problem as distraction, she may try to solve it by simply removing the distracting objects.

As you might guess, when a problem is represented one way, the solution may seem very difficult, or even impossible. Seen another way, the solution might be very easy. For example, consider the following problem (from Nasar, 1998):

Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 miles per hour. At the same time, a fly that travels at a steady 15 miles per hour starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner until he is crushed between the two front wheels. Question: what total distance did the fly cover?

Please take a few minutes to try to solve this problem.

Most people represent this problem as a question about a fly because, well, that is how the question is asked. The solution, using this representation, is to figure out how far the fly travels on the first leg of its journey, then add this total to how far it travels on the second leg of its journey (when it turns around and returns to the first bicycle), then continue to add the smaller distance from each leg of the journey until you converge on the correct answer. You would have to be quite skilled at math to solve this problem, and you would probably need some time and pencil and paper to do it.

If you consider a different representation, however, you can solve this problem in your head. Instead of thinking about it as a question about a fly, think about it as a question about the bicycles. They are 20 miles apart, and each is traveling 10 miles per hour. How long will it take for the bicycles to reach each other? Right, one hour. The fly is traveling 15 miles per hour; therefore, it will travel a total of 15 miles back and forth in the hour before the bicycles meet. Represented one way (as a problem about a fly), the problem is quite difficult. Represented another way (as a problem about two bicycles), it is easy. Changing your representation of a problem is sometimes the best—sometimes the only—way to solve it.

Unfortunately, however, changing a problem’s representation is not the easiest thing in the world to do. Often, problem solvers get stuck looking at a problem one way. This is called  fixation . Most people who represent the preceding problem as a problem about a fly probably do not pause to reconsider, and consequently change, their representation. A parent who thinks her daughter is being defiant is unlikely to consider the possibility that her behavior is far less purposeful.

Problem-solving fixation was examined by a group of German psychologists called Gestalt psychologists during the 1930’s and 1940’s. Karl Dunker, for example, discovered an important type of failure to take a different perspective called  functional fixedness . Imagine being a participant in one of his experiments. You are asked to figure out how to mount two candles on a door and are given an assortment of odds and ends, including a small empty cardboard box and some thumbtacks. Perhaps you have already figured out a solution: tack the box to the door so it forms a platform, then put the candles on top of the box. Most people are able to arrive at this solution. Imagine a slight variation of the procedure, however. What if, instead of being empty, the box had matches in it? Most people given this version of the problem do not arrive at the solution given above. Why? Because it seems to people that when the box contains matches, it already has a function; it is a matchbox. People are unlikely to consider a new function for an object that already has a function. This is functional fixedness.

Mental set is a type of fixation in which the problem solver gets stuck using the same solution strategy that has been successful in the past, even though the solution may no longer be useful. It is commonly seen when students do math problems for homework. Often, several problems in a row require the reapplication of the same solution strategy. Then, without warning, the next problem in the set requires a new strategy. Many students attempt to apply the formerly successful strategy on the new problem and therefore cannot come up with a correct answer.

The thing to remember is that you cannot solve a problem unless you correctly identify what it is to begin with (initial state) and what you want the end result to be (goal state). That may mean looking at the problem from a different angle and representing it in a new way. The correct representation does not guarantee a successful solution, but it certainly puts you on the right track.

A bit more optimistically, the Gestalt psychologists discovered what may be considered the opposite of fixation, namely  insight . Sometimes the solution to a problem just seems to pop into your head. Wolfgang Kohler examined insight by posing many different problems to chimpanzees, principally problems pertaining to their acquisition of out-of-reach food. In one version, a banana was placed outside of a chimpanzee’s cage and a short stick inside the cage. The stick was too short to retrieve the banana, but was long enough to retrieve a longer stick also located outside of the cage. This second stick was long enough to retrieve the banana. After trying, and failing, to reach the banana with the shorter stick, the chimpanzee would try a couple of random-seeming attempts, react with some apparent frustration or anger, then suddenly rush to the longer stick, the correct solution fully realized at this point. This sudden appearance of the solution, observed many times with many different problems, was termed insight by Kohler.

Lest you think it pertains to chimpanzees only, Karl Dunker demonstrated that children also solve problems through insight in the 1930s. More importantly, you have probably experienced insight yourself. Think back to a time when you were trying to solve a difficult problem. After struggling for a while, you gave up. Hours later, the solution just popped into your head, perhaps when you were taking a walk, eating dinner, or lying in bed.

fixation :  when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

functional fixedness :  a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

mental set :  a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

insight :  a sudden realization of a solution to a problem

Solving Problems by Trial and Error

Correctly identifying the problem and your goal for a solution is a good start, but recall the psychologist’s definition of a problem: it includes a set of possible intermediate states. Viewed this way, a problem can be solved satisfactorily only if one can find a path through some of these intermediate states to the goal. Imagine a fairly routine problem, finding a new route to school when your ordinary route is blocked (by road construction, for example). At each intersection, you may turn left, turn right, or go straight. A satisfactory solution to the problem (of getting to school) is a sequence of selections at each intersection that allows you to wind up at school.

If you had all the time in the world to get to school, you might try choosing intermediate states randomly. At one corner you turn left, the next you go straight, then you go left again, then right, then right, then straight. Unfortunately, trial and error will not necessarily get you where you want to go, and even if it does, it is not the fastest way to get there. For example, when a friend of ours was in college, he got lost on the way to a concert and attempted to find the venue by choosing streets to turn onto randomly (this was long before the use of GPS). Amazingly enough, the strategy worked, although he did end up missing two out of the three bands who played that night.

Trial and error is not all bad, however. B.F. Skinner, a prominent behaviorist psychologist, suggested that people often behave randomly in order to see what effect the behavior has on the environment and what subsequent effect this environmental change has on them. This seems particularly true for the very young person. Picture a child filling a household’s fish tank with toilet paper, for example. To a child trying to develop a repertoire of creative problem-solving strategies, an odd and random behavior might be just the ticket. Eventually, the exasperated parent hopes, the child will discover that many of these random behaviors do not successfully solve problems; in fact, in many cases they create problems. Thus, one would expect a decrease in this random behavior as a child matures. You should realize, however, that the opposite extreme is equally counterproductive. If the children become too rigid, never trying something unexpected and new, their problem solving skills can become too limited.

Effective problem solving seems to call for a happy medium that strikes a balance between using well-founded old strategies and trying new ground and territory. The individual who recognizes a situation in which an old problem-solving strategy would work best, and who can also recognize a situation in which a new untested strategy is necessary is halfway to success.

Solving Problems with Algorithms and Heuristics

For many problems there is a possible strategy available that will guarantee a correct solution. For example, think about math problems. Math lessons often consist of step-by-step procedures that can be used to solve the problems. If you apply the strategy without error, you are guaranteed to arrive at the correct solution to the problem. This approach is called using an  algorithm , a term that denotes the step-by-step procedure that guarantees a correct solution. Because algorithms are sometimes available and come with a guarantee, you might think that most people use them frequently. Unfortunately, however, they do not. As the experience of many students who have struggled through math classes can attest, algorithms can be extremely difficult to use, even when the problem solver knows which algorithm is supposed to work in solving the problem. In problems outside of math class, we often do not even know if an algorithm is available. It is probably fair to say, then, that algorithms are rarely used when people try to solve problems.

Because algorithms are so difficult to use, people often pass up the opportunity to guarantee a correct solution in favor of a strategy that is much easier to use and yields a reasonable chance of coming up with a correct solution. These strategies are called  problem solving heuristics . Similar to what you saw in section 6.2 with reasoning heuristics, a problem solving heuristic is a shortcut strategy that people use when trying to solve problems. It usually works pretty well, but does not guarantee a correct solution to the problem. For example, one problem solving heuristic might be “always move toward the goal” (so when trying to get to school when your regular route is blocked, you would always turn in the direction you think the school is). A heuristic that people might use when doing math homework is “use the same solution strategy that you just used for the previous problem.”

By the way, we hope these last two paragraphs feel familiar to you. They seem to parallel a distinction that you recently learned. Indeed, algorithms and problem-solving heuristics are another example of the distinction between Type 1 thinking and Type 2 thinking.

Although it is probably not worth describing a large number of specific heuristics, two observations about heuristics are worth mentioning. First, heuristics can be very general or they can be very specific, pertaining to a particular type of problem only. For example, “always move toward the goal” is a general strategy that you can apply to countless problem situations. On the other hand, “when you are lost without a functioning gps, pick the most expensive car you can see and follow it” is specific to the problem of being lost. Second, all heuristics are not equally useful. One heuristic that many students know is “when in doubt, choose c for a question on a multiple-choice exam.” This is a dreadful strategy because many instructors intentionally randomize the order of answer choices. Another test-taking heuristic, somewhat more useful, is “look for the answer to one question somewhere else on the exam.”

You really should pay attention to the application of heuristics to test taking. Imagine that while reviewing your answers for a multiple-choice exam before turning it in, you come across a question for which you originally thought the answer was c. Upon reflection, you now think that the answer might be b. Should you change the answer to b, or should you stick with your first impression? Most people will apply the heuristic strategy to “stick with your first impression.” What they do not realize, of course, is that this is a very poor strategy (Lilienfeld et al, 2009). Most of the errors on exams come on questions that were answered wrong originally and were not changed (so they remain wrong). There are many fewer errors where we change a correct answer to an incorrect answer. And, of course, sometimes we change an incorrect answer to a correct answer. In fact, research has shown that it is more common to change a wrong answer to a right answer than vice versa (Bruno, 2001).

The belief in this poor test-taking strategy (stick with your first impression) is based on the  confirmation bias   (Nickerson, 1998; Wason, 1960). You first saw the confirmation bias in Module 1, but because it is so important, we will repeat the information here. People have a bias, or tendency, to notice information that confirms what they already believe. Somebody at one time told you to stick with your first impression, so when you look at the results of an exam you have taken, you will tend to notice the cases that are consistent with that belief. That is, you will notice the cases in which you originally had an answer correct and changed it to the wrong answer. You tend not to notice the other two important (and more common) cases, changing an answer from wrong to right, and leaving a wrong answer unchanged.

Because heuristics by definition do not guarantee a correct solution to a problem, mistakes are bound to occur when we employ them. A poor choice of a specific heuristic will lead to an even higher likelihood of making an error.

algorithm :  a step-by-step procedure that guarantees a correct solution to a problem

problem solving heuristic :  a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

confirmation bias :  people’s tendency to notice information that confirms what they already believe

An Effective Problem-Solving Sequence

You may be left with a big question: If algorithms are hard to use and heuristics often don’t work, how am I supposed to solve problems? Robert Sternberg (1996), as part of his theory of what makes people successfully intelligent (Module 8) described a problem-solving sequence that has been shown to work rather well:

  • Identify the existence of a problem.  In school, problem identification is often easy; problems that you encounter in math classes, for example, are conveniently labeled as problems for you. Outside of school, however, realizing that you have a problem is a key difficulty that you must get past in order to begin solving it. You must be very sensitive to the symptoms that indicate a problem.
  • Define the problem.  Suppose you realize that you have been having many headaches recently. Very likely, you would identify this as a problem. If you define the problem as “headaches,” the solution would probably be to take aspirin or ibuprofen or some other anti-inflammatory medication. If the headaches keep returning, however, you have not really solved the problem—likely because you have mistaken a symptom for the problem itself. Instead, you must find the root cause of the headaches. Stress might be the real problem. For you to successfully solve many problems it may be necessary for you to overcome your fixations and represent the problems differently. One specific strategy that you might find useful is to try to define the problem from someone else’s perspective. How would your parents, spouse, significant other, doctor, etc. define the problem? Somewhere in these different perspectives may lurk the key definition that will allow you to find an easier and permanent solution.
  • Formulate strategy.  Now it is time to begin planning exactly how the problem will be solved. Is there an algorithm or heuristic available for you to use? Remember, heuristics by their very nature guarantee that occasionally you will not be able to solve the problem. One point to keep in mind is that you should look for long-range solutions, which are more likely to address the root cause of a problem than short-range solutions.
  • Represent and organize information.  Similar to the way that the problem itself can be defined, or represented in multiple ways, information within the problem is open to different interpretations. Suppose you are studying for a big exam. You have chapters from a textbook and from a supplemental reader, along with lecture notes that all need to be studied. How should you (represent and) organize these materials? Should you separate them by type of material (text versus reader versus lecture notes), or should you separate them by topic? To solve problems effectively, you must learn to find the most useful representation and organization of information.
  • Allocate resources.  This is perhaps the simplest principle of the problem solving sequence, but it is extremely difficult for many people. First, you must decide whether time, money, skills, effort, goodwill, or some other resource would help to solve the problem Then, you must make the hard choice of deciding which resources to use, realizing that you cannot devote maximum resources to every problem. Very often, the solution to problem is simply to change how resources are allocated (for example, spending more time studying in order to improve grades).
  • Monitor and evaluate solutions.  Pay attention to the solution strategy while you are applying it. If it is not working, you may be able to select another strategy. Another fact you should realize about problem solving is that it never does end. Solving one problem frequently brings up new ones. Good monitoring and evaluation of your problem solutions can help you to anticipate and get a jump on solving the inevitable new problems that will arise.

Please note that this as  an  effective problem-solving sequence, not  the  effective problem solving sequence. Just as you can become fixated and end up representing the problem incorrectly or trying an inefficient solution, you can become stuck applying the problem-solving sequence in an inflexible way. Clearly there are problem situations that can be solved without using these skills in this order.

Additionally, many real-world problems may require that you go back and redefine a problem several times as the situation changes (Sternberg et al. 2000). For example, consider the problem with Mary’s daughter one last time. At first, Mary did represent the problem as one of defiance. When her early strategy of pleading and threatening punishment was unsuccessful, Mary began to observe her daughter more carefully. She noticed that, indeed, her daughter’s attention would be drawn by an irresistible distraction or book. Fresh with a re-representation of the problem, she began a new solution strategy. She began to remind her daughter every few minutes to stay on task and remind her that if she is ready before it is time to leave, she may return to the book or other distracting object at that time. Fortunately, this strategy was successful, so Mary did not have to go back and redefine the problem again.

Pick one or two of the problems that you listed when you first started studying this section and try to work out the steps of Sternberg’s problem solving sequence for each one.

a mental representation of a category of things in the world

an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

knowledge about one’s own cognitive processes; thinking about your thinking

individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

Thinking like a scientist in your everyday life for the purpose of drawing correct conclusions. It entails skepticism; an ability to identify biases, distortions, omissions, and assumptions; and excellent deductive and inductive reasoning, and problem solving skills.

a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

an inclination, tendency, leaning, or prejudice

a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

a set of statements in which the beginning statements lead to a conclusion

an argument for which true beginning statements guarantee that the conclusion is true

a type of reasoning in which we make judgments about likelihood from sets of evidence

an inductive argument in which the beginning statements lead to a conclusion that is probably true

fast, automatic, and emotional thinking

slow, effortful, and logical thinking

a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

udging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

noticing, comprehending and forming a mental conception of a problem

when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

a sudden realization of a solution to a problem

a step-by-step procedure that guarantees a correct solution to a problem

The tendency to notice and pay attention to information that confirms your prior beliefs and to ignore information that disconfirms them.

a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

Introduction to Psychology Copyright © 2020 by Ken Gray; Elizabeth Arnott-Hill; and Or'Shaundra Benson is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Chapter 11, Lesson 4: Problem-Solving Investigation: Use Logical Reasoning

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Using Folktales to Teach Logical Reasoning

Having young students write a pourquoi—an origin folktale—is a great way to cultivate problem-solving and divergent thinking.

A teacher helping elementary students with a reading and writing assignment in class

Logical reasoning is more than interpreting patterns, number sequences, and the relationships between shapes in math—it can also be successfully integrated into language arts. The underlying skill of divergent thinking can be developed in young students through the process of writing a creative pourquoi—a folktale that explains how or why something came to be.

There’s a strong need for educators to take a multidisciplinary approach to teaching logical reasoning skills to students starting at a young age. American students fall significantly behind their peers in other developed nations when it comes to their ability to problem-solve and use divergent thinking skills: On the creative problem-solving portion of the Programme for International Student Assessment in 2012, nearly 10 percent of students in Singapore and Korea performed well enough to be “highly skilled problem solvers,” while only 2.7 percent of American students earned that ranking.

Using Pourquoi Tales to Foster Logical Reasoning

A pourquoi (French for “why”) is a terrific way to cultivate problem-solving and divergent thinking—core skills of logical reasoning—in the student learning framework. Logical reasoning is associated with the capacity to generate alternative ideas, and pourquois involve creating alternative ideas to explain a natural phenomenon in the world (e.g., why stars shine at night or how zebras got stripes).

Pourquois are further a fun and playful way for students to apply inductive and deductive reasoning skills—they can identify conditional reasoning premises like “if P, then Q” in these tales (e.g., if the lion fires an arrow at the raincloud, rain will fall down).

Students can also examine how a single cause leads to multiple far-reaching effects (e.g., banging a drum causes the earth to quake, which in turn causes animals to scamper).

To introduce a pourquoi unit, create a “Wonderings” anchor chart upon which students can brainstorm natural phenomena they wonder about, activating their prior knowledge. This empowers students by embracing their curiosity, and it also enables them to make powerful real-world connections.

Next, explain that pourquois fall into the folktale genre and have distinguishing elements, including nameless characters, a subtle moral or lesson, use of personification, and more.

There are outstanding multicultural pourquoi tales you can read aloud to students, as well as incorporate into guided reading groups, literacy centers, or written assignments. Such tales can foster open-mindedness and inclusiveness toward other cultures, helping to shape students who are world citizens and globally minded.

Read-aloud pourquoi tales:

  • Why The Sky Is Far Away by Mary-Joan Gerson and Carla Golembe
  • Why Mosquitos Buzz in People’s Ears by Verna Aardema and Leo and Diane Dillon
  • How Snake Got His Hiss by Marguerite Davol and Mercedes McDonald
  • How the Elephant Got Its Trunk by Jean Richards and Norman Gorbaty
  • How Tiger Got His Stripes by Rob Cleveland and Baird Hoffmire

Formative and Summative Assessment

There are a variety of formative assessments targeting higher-order thinking skills that you can use in a pourquoi unit. Have students map out the “if P, then Q” premises in a pourquoi onto a cause and effect graphic organizer. This teaches students how to make connections between conditional syllogisms, which is paramount to logical reasoning.

You can also have students compare and contrast elements in two or more pourquois. A third assessment is to have students make text-to-self connections using a pourquoi (e.g., “I am similar to the lion because my anger caused me to act rashly”).

As a summative assessment, students can plan, draft, and publish original pourquois. First, students should identify what natural phenomenon in the world they wish to explain. Next, they should identify their characters, setting, a problem, and a solution. Their tales should contain a beginning, middle, and ending. For enrichment or extension opportunities, have students research and select a culture upon which to base their tale.

Their writing will be enhanced if you routinely share student samples that demonstrate rich writing skills, including use of similes, metaphors, dialogue, and descriptive language.

As a best practice, incorporate one-on-one conferencing and peer editing into the writing process. End the unit with a publishing party to celebrate the authors and their process of arriving at an alternative explanation through use of logical reasoning skills.

Through designing and implementing a comprehensive, structured pourquoi unit, teachers can effectively integrate logical reasoning into the language arts discipline.

XDA Developers

Gemini 1.5 has massive improvements, but it's not available for everyone yet

  • Gemini 1.5 Pro increases the potential context window from 32K tokens to 1 million, allowing it to process larger amounts of data and answer questions more effectively.
  • The introduction of Mixture of Experts (MoE) in Gemini 1.5 Pro improves computational efficiency during training and offers potential for faster inference times.
  • Gemini 1.5 Pro demonstrates significant improvements in logic, understanding multimodal prompts, and logical reasoning, making it more capable in problem-solving tasks.

Google has been hard at work with Gemini since its rebrand, releasing Gemini Advanced powered by Gemini 1.0 Ultra. However, the underlying Gemini Pro model that powered the base version of Gemini was available in Bard for quite a while, but now Google is upgrading it. Gemini 1.5 is here, and it's coming to the Pro model in the free version soon. There are some pretty big architectural changes that Google say will put it on par in terms of performance with 1.0 Ultra.

Google Gemini: What is it, and how does it work?

Gemini 1.5 is a massive step forward, it beats gpt-4 turbo in some key areas.

First and foremost, one of the biggest steps that Gemini 1.5 Pro has taken is that it increases the context window from 32K tokens to 1 million tokens. The context window is essentially how much the LLM can "see" at any given time, which is a huge increase. As well, GPT-4 Turbo only has a 128K context window, and Google says that Gemini 1.5 Pro can go up to 10 million in research contexts.

This expanded context window has implications for how much data it can take in at any given time. For example, Google demonstrated how Gemini 1.5 Pro can take in the Apollo 11 Air-to-Ground voice transcript, which is 402 pages, and answer questions about it with ease.

Gemini 1.5 Pro also has another major improvement dubbed Mixture of Experts (MoE), which we've already seen with Mixtral 8x7B. Mixtral employs a MoE architecture to process incoming tokens, directing them to specialized neural networks within the system based on their relevance. The Mixtral 8x7B model features eight such experts. Notably, it's possible to structure these experts in a hierarchical manner, where an expert itself may be another MoE. Upon receiving a prompt, Mixtral 8x7B utilizes a routing network to determine the most suitable expert for each token. In this setup, each token is evaluated by two experts, and the final response is a blend of their outputs.

The MoE approach offers several benefits, particularly in terms of computational efficiency during the initial training phase, although it can be prone to overfitting during the fine-tuning stage. Overfitting occurs when a model becomes too familiar with its training data, leading to a tendency to reproduce it exactly in its responses.

What is Mixtral 8x7B? The open LLM giving GPT-3.5 a run for its money

Another advantage of MoEs is their potential for faster inference times, as they activate only a subset of experts for each query. However, accommodating a model like Mixtral, with its 47 billion parameters, requires substantial RAM. The model's overall parameter count is 47 billion rather than 56 billion because it shares many parameters across all experts and does not simply multiply the 7 billion parameters of each expert by eight.

While the above explains how MoE works for Mixtral, the same architectural improvements of an MoE will be present in Gemini too, undoubtedly with some other changes brought in by Google, too. Google didn't reveal how many parameters are powering Gemini 1.5 Pro, but we expected that a MoE will still make it significantly more efficient to run.

Reasoning and problem solving improvements

Gemini should be better at logic and understanding.

Google demonstrated in its reveal of Gemini 1.5 Pro that it's capable of significantly improved multimodal prompt understanding and logical reasoning. When given a 44-minute silent film, as per the above clip, it can identify plot points and events.

Those improvements aren't just for multi-modal prompts, though. Google says that when given a prompt of over 100,000 lines of code, Gemini can still provide modifications, solutions, and other changes based on prompts given by the user.

Gemini 1.5 Pro will be available soon

It's rolling out to developers first.

If you're a developer that's been using Google's AI Studio or Vertex AI, a limited preview will be available for you to try Gemini 1.5 Pro out now. Google says that it can outperform Gemini 1.0 Pro in 87% of its benchmarks and "performs at a broadly similar level" when compared to Gemini 1.0 Ultra. Interestingly as well, Google says that the model can learn in-conversation, without needing any fine-tuning. Its context window being so large is a first of its kind, which enables it to do significantly more than was previously possible with an LLM.

When it reaches its wider release, Google says that Gemini 1.5 Pro will be available with a 128K context window. It also says that it plans to introduce "pricing tiers" starting at the standard 128K context window and going up to 1 million tokens. Early testers are advised to expect higher latency with the higher context window limit, but that there is no cost for it during its testing stage.

For now, normal users who want to give it a try will have to wait. Google isn't saying when it will roll out, aside from "soon," which doesn't really say a whole lot. Given that developers are already able to try it out now, it seems likely that it will be available in the coming weeks or months for most people.

Gemini 1.5 has massive improvements, but it's not available for everyone yet

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  22. Chapter 11, Lesson 4: Problem-Solving Investigation: Use Logical Reasoning

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  23. PDF USE LOGICAL REASONING

    USE LOGICAL REASONING This strategy is really used for all problem solving. In many ways students use this strategy category as a catch-all because these strategies are not usually used on their own but in combination with other strategies mentioned above.

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  27. Gemini 1.5 has massive improvements, but it's not available for

    Gemini 1.5 Pro demonstrates significant improvements in logic, understanding multimodal prompts, and logical reasoning, making it more capable in problem-solving tasks.