Geometry/Chapter 2/Lesson 5
Introduction
[edit | edit source]Chapter 2, Lesson 5 will introduce you to the concept of deductive reasoning. We will also review these two laws: The Law of Detachment and the Law of Syllogism.
Deductive Reasoning
[edit | edit source]Deductive reasoning is different from inductive reasoning---Deductive Reasoning is reaching logical and sensible conclusions by the means of facts, rules, definitions, and properties. Deductive reasoning is what doctors use to reach a conclusion on how much medicine a patient must take. Doctors don't use inductive reasoning (examples) to reach conclusions!
An example of deductive reasoning is the father of this concept, Aristotle. He concluded Socrates as a mortal from these two facts:
- Humans are mortals
- Socrates is a human
This is known as the Law of Syllogism, which we will get to in section 1.1.2.
Law of Detachment
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The Law of Detachment is a form of deductive reasoning that is used to draw conclusions. This law has a specific format that distinguishes itself as a law.
- Refer back to Lesson 3 if you do not remember logic symbols
- p → q
- p
- q
If the order is switched, such as q being first instead of p, it is no longer known as the Law of Detachment. Instead, it is invalid. In order for it to be the Law of Detachment, it must follow the format described above. Here, we will look at this with an example:
- If the measure of an angle is greater than 90, then it is obtuse.
- m ∠T is greater than 90.
- ∠T is obtuse.
- If the measure of an angle is greater than 90, then it is obtuse.
This example exactly follows the format of the Law of Detachment:
- p → greater than 90 || q → it is obtuse
- p → greater than 90
- q → is obtuse
But on the other hand, this example does not follow the Law of Detachment:
- If Pedro is taking history, then he will study about WWII.
- Pedro will study about WWII.
- Pedro is taking history.
- If Pedro is taking history, then he will study about WWII.
- Closer evaluation
- p → taking history || q → will study about WWII
- q → will study about WWII
- p → taking history
As you can see, this statement switched the orders of the p and q. Thus, making this statement invalid and not Law of Detachment.
Law of Syllogism
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The Law of Syllogism is another law of logic and is similar to the Transitive Property. This law, like the Law of Detachment, has a specific format to it.
- p → q
- q → r
- p → r
If any part of the format is changed, then it is no longer the Law of Syllogism, and thus, is invalid. Here, we will look at an example here that follows the Law of Syllogism correctly:
- Statement 1: If it continues to rain (p), then the soccer field will become wet and muddy (q). This becomes if p, then q.
- Statement 2: If the soccer field becomes wet and muddy (q), then the game will be canceled (r). This becomes if q, then r.
- Statement 3: If it continues to rain (p), then the game will be canceled (r). This final statement is the conclusion and becomes p, then r.
- Statement 3: If it continues to rain (p), then the game will be canceled (r). This final statement is the conclusion and becomes p, then r.
(Source of problem is on the back)
As you can see, these statements follow the p to q, q to r, and finally p to r format, and thus, follows the Law of Syllogism. Now, we will look at an example that does not follow the Law of Syllogism, and thus, is invalid.
- Statement 1: If I take this test, then I will get a better grade than 0.
- Statement 2: If I get a better grade than 0, then I will make my parents happy.
- Statement 3: If I take this test, then I will get a better grade than 0.
This does not follow the Law of Syllogism because it follows the p to q, q to r, and p to q format--and is invalid.