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Estimated Learning Time = 6 hours

In Level 2: Logical Properties & Logical Relations we learned to construct truth-tables and how to use these logical devices to determine the logical properties of individual statements and logical relationships between statements. And in Level 3: Using Argument Forms To Test For Validity we were introduced to the concept of validity and how to use the form of an argument to test its validity. In this course we will bring these two topics together and learn how to use truth-tables to test any argument – no matter how unfamiliar its form – for validity.

Consider the following affirming the consequent argument:

(P

Q)

Q

 

 

P

 

 

Now, let’s set this argument out horizontally as a truth-table:

 

(P

Q)

Q

P

 
 

T

T

T

T

 

T

1

 

T

F

F

F

 

T

2

 

F

T

T

T

 

F

3

 

F

T

F

F

 

F

4

As we can see, there is one line (line 3) upon which both premises of the argument are true, but the conclusion of the argument is false. As there is a “TTF line” in the truth-table, it must be possible for the premises to be true at the same time that the conclusion false. Thus, the truth-table indicates that the argument is invalid.

But how do things look when we are dealing with a valid argument? Consider the following modus ponens argument:

(P

Q)

P

 

 

Q

   

Now, let’s set this argument out horizontally as a truth-table:

 

(P

Q)

P

Q

 
 

T

T

T

T

 

T

1

 

T

F

F

T

 

F

2

 

F

T

T

F

 

T

3

 

F

T

F

F

 

F

4

As we can see, there is only one line (Line 1) upon which both of our premises are true: and on that line, our conclusion is also true. Thus, there is no “TFF” line in our truth table. Hence, our truth-table clearly indicates that the argument form is valid.

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