In this lesson, you will not only learn about what inference means, but also how to apply it to determine what could or must be true within given facts and rules.

## Inference

When we make an inference, we are drawing conclusions that are related to facts and rules that are already known. Inferring involves taking one fact or rule and using it to determine another associated truth.

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Using inference, we are able to develop an informed opinion using evidence and reasoning, which is based on the information available to us. Inferring truth from facts allows us to draw logical conclusions that can be applied to things like math, science, and philosophy. If we use logic to arrive at our conclusions, we can be sure that they are sound.

## Examples of Inference

Suppose you are a juror in a difficult trial. To prove that the defendant is guilty or not guilty, both the defense and the prosecution will provide evidence to try and convince you that their side is the one telling the truth. It is up to you to determine what is true based on the evidence and facts they provide.

Say a witness comes forward claiming that he saw the defendant at the grocery store at the time of the murder. Because of this new alibi, you can infer that the defendant is not guilty because the facts prove it to be true.Another example of inference can be found by looking at the work that NASA does. NASA sends out telescopes, like the Hubble Space Telescope, in order to understand the universe around us.

The measurements taken by the telescope provide concrete facts that give us a better understanding of space. Based on these measurements, or facts, NASA can make inferences of what they think the facts mean. For example, while a black hole has never been directly observed, by examining gravitational interactions and magnetic fields detected by the telescope, scientists can infer that they exist.In both of these examples, we can see instances where specific facts lead us to be able to determine separate, but related truths.

## Inferring What Could be True and What Must be True

In order to determine what could be true versus what must be true, we need to look at the facts or rules of given statements. These facts or rules act like guidelines that let you know the parameters you have to work within to find the solution.

Consider what COULD be true based on the following statements.

• Most A are B.
• Some A are C.

What could we infer?It is true that A could be both B and C.As another example, take a look at the following statement.

• If it isn’t raining outside, Cindy will mow the law or walk the dog.

If we know that it is not raining outside, what could we infer based on this statement?A.) Cindy will mow the lawn, but not walk the dog.B.) Cindy will walk the dog, but not mow the lawn.

C.) Cindy will both mow the law and walk the dog.D.

) We could infer both A and B.The answer is that we could infer both A and B because, based on the statement, Cindy might mow the law or she might walk the dog. Either is a possibility but, because of the word ”or,” both are not possible.Now, let’s look at an example where what we infer MUST be true.

• Jesse likes oranges.

• Only men like oranges.

What must be inferred based on these two statements?Jesse is a man because only men like oranges. This statement must be true because of the rules that are provided in the previous statements.Let’s take a look at an additional example of inferring what must be true.

• Some professional ballet dancers also tap dance.
• No ballet dancers do ballroom dancing.
• Some of the ballroom dancers also tap dance.

Which of the following could be inferred from the statements above?A.) Most ballet dancers only dance ballet.B.

) Some of the ballroom dancers who tap dance also dance ballet.C.) There are some tap dancers who dance ballroom, but not ballet.D.) There are some tap dancers who dance neither ballroom, nor ballet.The answer is that there are some tap dancers who dance ballroom, but not ballet. We can infer this based on the statements, which tell us that there must be a subset of tap dancers who dance ballroom, but not ballet.

When an additional statement is added to the previous statement and conditions, it is considered an additional or substitute rule. An example would be:The soccer team needs 5 new players. Bob, Billy, John, Casey, Drew, Phil, Pete, and Carl are the options.

• Carl can’t be on the team if Bob is on the team.
• Casey can’t be on the team if Pete is on the team.
• Drew can’t be on the team if Carl is on the team.

Bob did not show up for tryouts, so who must be on the team?The answer is Carl, Casey, John, Phil, and Billy.

This would be the only possible option for the team selection. With the additional rule informing us that Bob did not show up for tryouts, we now have an new factor to incorporate when making our inference.What if the scenario was that the following team members must be on the team together?

• Carl and Bob
• Casey and Pete

The new team must be:Carl, Bob, Casey, Pete, and Billy.Because of the additional rule that tells us which team members must be on the team together, this new team is the only option.

## Lesson Summary

Inference occurs when we draw separate, but related, conclusions from the evidence, facts, or rules that are available to us. Using these facts, or guidelines, we are able to make determinations about other, related truths.

Furthermore, we are able to assess what could be true and what must be true based on the information we possess. While in some instances we can judge that something is a possibility, in others cases we can determine that something must be true. By inferring truth from facts and rules, we are using logic and reasoning to arrive at sound conclusions.

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