2.3: Deductive Reasoning
Learning Objectives
- Apply basic rules of logic.
- Compare inductive reasoning and deductive reasoning.
Review Queue
- Write the converse, inverse, and contrapositive of the following statement: Football players wear shoulder pads.
- Are the converse, inverse or contrapositive of #1 true? If not, find a counterexample.
- An if-then statement is \begin{align*}p \rightarrow q\end{align*}.
- What is the inverse of \begin{align*}p \rightarrow q\end{align*}?
- What is the converse of the inverse of \begin{align*}p \rightarrow q\end{align*}?
Know What? In a fictitious far-away land, a peasant is awaiting his fate from the king. He is standing in a stadium, with two doors in front of him. Both doors have signs on them, which are below:
Door A | Door B |
---|---|
IN THIS ROOM THERE IS A LADY, AND IN THE OTHER ROOM THERE IS A TIGER. | IN ONE OF THESE ROOMS THERE IS A LADY, AND IN ONE OF THE OTHER ROOMS THERE IS A TIGER. |
The king states, “Only one of these statements is true. If you pick correctly, you will find the lady. If not, the tiger will be waiting for you.” Which door should the peasant pick?
Deductive Reasoning
Logic: The study of reasoning.
In the first section, you learned about inductive reasoning, making conclusions based upon patterns. Now, we will learn about deductive reasoning.
Deductive Reasoning: Drawing conclusion from facts. Conclusions are usually drawn from general statements about something more specific.
Example 1: Suppose Bea makes the following statements, which are known to be true.
If Central High School wins today, they will go to the regional tournament. Central High School won today.
What is the logical conclusion?
Solution: This is an example of deductive reasoning. These are true statements that we can take as facts. The conclusion is: Central High School will go to the regional tournament.
Example 2: Here are two true statements.
Every odd number is the sum of an even and an odd number.
5 is an odd number.
What can you conclude?
Solution: Based on only these two true statements, there is one conclusion: 5 is the sum of an even and an odd number. (This is true, \begin{align*}5 = 3 + 2\end{align*} or \begin{align*}4 + 1\end{align*}).
Law of Detachment
Let’s look at Example 2 and change it into symbolic form.
\begin{align*}p: \text{A number is odd} && q: \text{It is the sum of an even and odd number}\end{align*}
The first statement is \begin{align*}p \rightarrow q\end{align*}.
- The second statement in Example 2, “5 is an odd number,” is a specific example of \begin{align*}p\end{align*}. “A number” is 5.
- The conclusion is \begin{align*}q\end{align*}, “5 is the sum of an even and an odd number.”
The symbolic form of Example 2 is:
\begin{align*}& p \rightarrow q\\ & p\\ & \therefore q && \therefore \ \text{symbol for therefore}\end{align*}
All deductive arguments that follow this pattern have a special name, the Law of Detachment.
Law of Detachment: If \begin{align*}p \rightarrow q\end{align*} is true, and \begin{align*}p\end{align*} is true, then \begin{align*}q\end{align*} is true.
Example 3: Here are two true statements.
If \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are a linear pair, then \begin{align*}m \angle A + m \angle B = 180^\circ\end{align*}.
\begin{align*}\angle ABC\end{align*} and \begin{align*}\angle CBD\end{align*} are a linear pair.
What conclusion can you draw from this?
Solution: This is an example of the Law of Detachment, therefore:
\begin{align*}m\angle ABC + m \angle CBD = 180^\circ\end{align*}
Example 4: Here are two true statements. Be careful!
If \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are a linear pair, then \begin{align*}m \angle A + m \angle B = 180^\circ\end{align*}.
\begin{align*}m\angle 1 = 90^\circ\end{align*} and \begin{align*}m\angle 2 = 90^\circ\end{align*}.
What conclusion can you draw from these two statements?
Solution: Here there is NO conclusion. These statements are in the form:
\begin{align*}& p \rightarrow q\\ & q\end{align*}
We cannot conclude that \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are a linear pair.
Here are two counterexamples:
Law of Contrapositive
Example 5: The following two statements are true.
If a student is in Geometry, then he or she has passed Algebra I.
Daniel has not passed Algebra I.
What can you conclude from these two statements?
Solution: These statements are in the form:
\begin{align*}& p \rightarrow q\\ & \sim q\end{align*}
Not \begin{align*}q\end{align*} is the beginning of the contrapositive \begin{align*}(\sim q \rightarrow \sim p)\end{align*}, therefore the logical conclusion is not \begin{align*}p\end{align*}: Daniel is not in Geometry.
This example is called the Law of Contrapositive.
Law of Contrapositive: If \begin{align*}p \rightarrow q\end{align*} is true and \begin{align*}\sim q\end{align*} is true. Then, you can conclude \begin{align*}\sim p\end{align*}.
The Law of Contrapositive is a logical argument.
Example 6: Determine the conclusion from the true statements below.
Babies wear diapers.
My little brother does not wear diapers.
Solution: The second statement is the equivalent of \begin{align*}\sim q\end{align*}. Therefore, the conclusion is \begin{align*}\sim p\end{align*}, or: My little brother is not a baby.
Example 7: Determine the conclusion from the true statements below.
If you are not in Chicago, then you can’t be on the \begin{align*}L\end{align*}.
Bill is in Chicago.
Solution: If we were to rewrite this symbolically, it would look like:
\begin{align*}& \sim p \rightarrow \sim q\\ & p\end{align*} This is not in the form of the Law of Contrapositive or the Law of Detachment, so there is no logical conclusion.
Example 8: Determine the conclusion from the true statements below.
If you are not in Chicago, then you can’t be on the \begin{align*}L\end{align*}.
Sally is on the \begin{align*}L\end{align*}.
Solution: If we were to rewrite this symbolically, it would look like:
\begin{align*}& \sim p \rightarrow \sim q\\ & q\end{align*} Even though it looks a little different, this is an example of the Law of Contrapositive. Therefore, the logical conclusion is: Sally is in Chicago.
Law of Syllogism
Example 9: Determine the conclusion from the following true statements.
If Pete is late, Mark will be late.
If Mark is late, Karl will be late.
So, if Pete is late, what will happen?
Solution: If Pete is late, this starts a domino effect of lateness. Mark will be late and Karl will be late too. So, if Pete is late, then Karl will be late, is the logical conclusion.
Each “then” becomes the next “if” in a chain of statements. This is called the Law of Syllogism.
Law of Syllogism: If \begin{align*}p \rightarrow q\end{align*} and \begin{align*}q \rightarrow r\end{align*} are true, then \begin{align*}p \rightarrow r\end{align*} is true.
Inductive vs. Deductive Reasoning
Inductive Reasoning: Using Patterns
Deductive Reasoning: Using Facts
Example 10: Solving an equation for \begin{align*}x\end{align*} is an example of inductive or deductive reasoning?
Solution: Deductive reasoning. Solving an equation uses Properties of Equality (facts) to solve a problem for \begin{align*}x\end{align*}.
Example 11: 1, 10, 100, 1000, \begin{align*}\ldots\end{align*} is an example of inductive or deductive reasoning?
Solution: Inductive reasoning. This is a pattern.
Example 12: Doing an experiment and writing a hypothesis is an example of inductive or deductive reasoning?
Solution: Inductive reasoning. Making a hypothesis comes from the patterns found in the experiment. These are not facts.
Example 13: Proving the experiment from Example 12 is true is an example of inductive or deductive reasoning?
Solution: Deductive reasoning. Here you would have to use facts to prove what happened in the experiment is supposed to happen.
Know What? Revisited Analyze the two statements on the doors.
Door A: IN THIS ROOM THERE IS A LADY, AND IN THE OTHER ROOM THERE IS A TIGER.
Door B: IN ONE OF THESE ROOMS THERE IS A LADY, AND IN ONE OF THE OTHER ROOMS THERE IS A TIGER.
We know that one door is true, so the other one must be false. Read Door B carefully, it says “in one of these rooms,” which means the lady could be behind either door, which has to be true. So, because Door B is the true statement, Door A is false and the tiger is behind it. The peasant should pick Door B.
Review Questions
Determine the logical conclusion and state which law you used (Law of Detachment, Law of Contrapositive, or Law of Syllogism). If no conclusion can be drawn, write “no conclusion.”
- People who vote for Jane Wannabe are smart people. I voted for Jane Wannabe.
- If Rae is the driver today then Maria is the driver tomorrow. Ann is the driver today.
- All equiangular triangles are equilateral. \begin{align*}\triangle ABC\end{align*} is equiangular.
- If North wins, then West wins. If West wins, then East loses.
- If \begin{align*}z > 5\end{align*}, then \begin{align*}x > 3\end{align*}. If \begin{align*}x > 3\end{align*}, then \begin{align*}y > 7\end{align*}.
- If I am cold, then I wear a jacket. I am not wearing a jacket.
- If it is raining outside, then I need an umbrella. It is not raining outside.
- If a shape is a circle, then it never ends. If it never ends, then it never starts. If it never starts, then it doesn’t exist. If it doesn’t exist, then we don’t need to study it.
- If you text while driving, then you are unsafe. You are a safe driver.
- If you wear sunglasses, then it is sunny outside. You are wearing sunglasses.
- If you wear sunglasses, then it is sunny outside. It is cloudy.
- I will clean my room if my mom asks me to. I am not cleaning my room.
- Write the symbolic representation of #8. Include your conclusion. Does this argument make sense?
- Write the symbolic representation of #10. Include your conclusion.
- Write the symbolic representation of #11. Include your conclusion.
Determine if the problems below represent inductive or deductive reasoning. Briefly explain your answer.
- John is watching the weather. As the day goes on it gets more and more cloudy and cold. He concludes that it is going to rain.
- Beth’s 2-year-old sister only eats hot dogs, blueberries and yogurt. Beth decides to give her sister some yogurt because she is hungry.
- Nolan Ryan has the most strikeouts of any pitcher in Major League Baseball. Jeff debates that he is the best pitcher of all-time for this reason.
- Ocean currents and waves are dictated by the weather and the phase of the moon. Surfers use this information to determine when it is a good time to hit the water.
- As Rich is driving along the 405, he notices that as he gets closer to LAX the traffic slows down. As he passes it, it speeds back up. He concludes that anytime he drives past an airport, the traffic will slow down.
Determine if the following statements are true or false.
- The Law of Detachment uses an if-then statement and its hypothesis to draw a conclusion.
- There is a Law of Inverse.
- Sometimes arguments can be valid, but not make sense.
- The Law of Syllogism takes the conclusion from a statement and makes it the hypothesis of the next.
- Number patterns are an example of deductive reasoning.
Review Queue Answers
1. Converse: If you wear shoulder pads, then you are a football player.
Inverse: If you are not a football player, then you do not wear shoulder pads.
Contrapositive: If you do not wear shoulder pads, then you are not a football player.
2. The converse and inverse are both false. A counterexample for both could be a woman from the 80’s. They definitely wore shoulder pads!
3. (a) \begin{align*}\sim p \rightarrow \sim q\end{align*}
(b) The converse of the \begin{align*}\sim p \rightarrow \sim q\end{align*} is \begin{align*}\sim q \rightarrow \sim p\end{align*}, or the contrapositive.
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