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Estimated Learning Time = 7 hours
Truthtables are powerful tools with a variety of applications in logical analysis. In the first part of this course you will learn to construct truthtables for logical formulae. Next, you will learn to use these truthtables to identify the logical properties of statements. Here’s an example:
–“It is currently raining in Paris or it is not currently raining in Paris.”
–Intuitively, we know that this sentence must be true regardless of the state of the world. Further, we can prove that this statement must be true using a truthtable. The first step is to analyse the sentence. Here’s what the entire analysed sentence looks like with the truthvalues filled out under the simple sentences:
–P = “It is raining in Paris”
–
(P 
∨ 
~ 
P) 
T 
T 

F 
F 
–
And, here’s how it looks when we fill out the entire truthtable:
–
(P 
∨ 
~ 
P) 
T 
T 
F 
T 
F 
T 
T 
F 
–
Now, focus on the truthvalues under the operator with the broadest scope, “∨”. The truthvalue under the broadest operator on a given row is the truthvalue of the entire formula on that row. And as we can see, the truthvalue under the main operator “∨” is T on both rows. This means that (P∨~P) is always true, no matter what the truthvalue P happens to be. Thus, (P ∨ ~P) is a necessary truth.
–Let’s look at another example:
–“It is currently raining in Paris and it is not currently raining in Paris.”
–Intuitively, we know that this sentence must be false, and we can prove that this using a truthtable. Again, the first step is to analyse the sentence. Here’s what the entire formula looks like with the truthvalues filled out under the simple sentences:
–P = “It is raining in Paris”
–
(P 
& 
~ 
P) 
T 
T 

F 
F 
–
And, here’s how it looks when we fill out the entire truthtable:
–
(P 
& 
~ 
P) 
T 
F 
F 
T 
F 
F 
T 
F 
–
Now, focus again on the truthvalues under the operator with the broadest scope, which this time around is “&”. The truthvalue under the broadest operator on a given row is the truthvalue of the entire formula on that row, and as we can see, the truthvalue under the main operator “∨” is F on both rows. This means that (P & ~P) is a necessary falsehood. It is false no matter what the truthvalue of P.
–Once we have mastered the use of truthtables for identifying logical properties, we will learn to use truthtables to test for relations of consistency, contradiction, equivalence and entailment within pairs of formulae. In Level 4: Using TruthTables To Test For Validity, we will once again return to the topic of truthtables, this time using them to test argument forms or sequents for validity.
Upon completion of this course you will have the ability to:
(1) Construct truthtables for logical formulae.
(2) Use truthtables to determine the logical properties (necessary truth, necessary falsehood, contingency) of logical formulae.
(3) Use truthtables to test for logical relations (consistency, contradiction, entailment, equivalence) within pairs of logical formulae.
Level 2.1 is available as a free sample!
