Logical Reasoning:

In the perfect world of Geometry, logical reasoning is basically a bunch of different kinds of statements and what order you put them in.

Conditionals:

The force of logic comes from the way information is structured.

You are probably wondering what this means. Here is an easier way to think of it. When you tell someone a story, you try to make it make sense, right? The person listening is trying to get some logic from it. But when you mix it up, they could get totally different logic from it. And maybe not get any at all. In other words, logic comes from the way the information is structured or formed or worded, etc.

Drawing Conclusions from Conditionals:

Drawing a conclusion from a conditional isn't that hard. Like, all Corvettes are Chevrolets, this can be shown by a Venn diagram. (A Venn Diagram: a diagram represented by using rectangles and circles. Used for solving problems in logic. Created by John Venn in 1894.) Here is a Venn diagram showing this fact:



Looking at this Venn diagram, we can esily see that the following statement is true:

If a car is a Corvette, then it is a Chevrolet.

Statements like this one are called if-then or conditional statements. In logical reasoning, conditional statements are usually written like this:

If P then Q

In conditional statements the part follows the "if" and is the hypothesis. The part follows the "then" and is the conclusion.

If a car is a Corvette, then it is a Chevrolet.

hypothesis, conclusion

Now look at the following statement: Kay's car is a Corvette.
By putting Kay's Corvette into a Venn diagram, it will show that it is a Chevrolet. By adding this we can use what is called deductive reasoning or deduction. Which looks something like this:

#1: If a car is a Corvette, then it is a Chevrolet.
#2: Kay's car is a Corvette.
#3: Therefore, Kay's car is a Chevrolet.

"True" Conditionals:
A true conditional is a conditional statement that leads to a true conclusion. In other words, if the conditional statement is true then the conclusion of the statement must be true to be a true conditional. Now, if you were to have something that disagrees with that statement, then that is a counterexample. For example, if you had the statement:

If someone plays jazz music, then they hate classical.

Plus, it just so happens you have a friend who plays jazz but absolutely loves classical, then that is a counterexample. If you were to try to put that on a Venn diagram, then there would be no place for your friend. They would not be either inside or outside of the diagram. You can see how it doesn't fit in this Venn diagram:



Reverse Conditionals:
A reverse conditional is when the "if" and the "then" parts of a conditional are switched to form a new statement, which also is a conditional statement, and can be called the converse of the original statement. Here is an example of one:

Statement: If a car is a Corvette, then it is a Chevrolet.
Converse: If a car is a Chevrolet, then it is a Corvette.

Now, you can see that the converse is just the statement parts reversed. This converse happens to be false because there are Chevrolets that are not Corvettes. If the converse is true, then the statement is a definition. Such as:

Statement: If a shape has 3 sides, then it is a triangle.
Converse: If a shape is a triangle, then it has 3 sides.
This converse is true, this statement is a definition.

Logical Chains:
Conditional statements can be linked together. The linked statements are what is called a logical chain. In this next example, there are 3 conditionals linked together to form a chain. Plus, it doesn't matter whether the statements are true or not.

#1: If it is summer, then the days are longer.
#2: If the days are longer, then kids can play more.
#3: If kids can play more, then they are happy.

Here it is shown in a zigzag pattern:



In the end you can conclude that:
If it is summer, then kids are happy.

By concluding this, you have just used a property without even mentioning it! You're probably asking yourself what that property is and here it is:



Definitions:



If you were to look at card #1, you would see some creatures that are called eeps. You will notice that all three of the eeps have 2 things in common. They all have 3 legs and 1 eye. The definition for eeps would be:

An eep is a figure with 3 legs and 1 eye.



Now, if you were to look at card #2, you would see some creatures that are not eeps. Now that you know what an eep looks like, look at card #3. It shows you four creatures. Can you tell which ones are eeps or not? If you guessed 1, then your right! There is only one eep on that card.

Now we can put our definition into a conditional statement:

If a figure is an eep, then it has 3 legs and 1 eye.

Plus, you could write a converse of it:

If a figure has 3 legs and 1 eye, then it is an eep.

Since both statements are true, and this is the case with all definitions, then they can be combined to form one sentence by joining the hypothesis and the conclusion with an "if and only if" phrase. Here is the technical formula:

P if and only if Q

If we took our definition and put into one of these statements, it would look like this:

A figure is an eep if and only if it has 3 legs and 1 eye.

Venn diagrams can be used to represent the 2 parts of our definition