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Estimated Learning Time = 7 hours

Truth-tables are powerful tools with a variety of applications in logical analysis. In the first part of this course you will learn to construct truth-tables for logical formulae. Next, you will learn to use these truth-tables 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 truth-table. The first step is to analyse the sentence. Here’s what the entire analysed sentence looks like with the truth-values 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 truth-table:

(P

~

P)

 T

 T

 F

 T

 F

 T

 T

 F

Now, focus on the truth-values under the operator with the broadest scope, “∨”. The truth-value under the broadest operator on a given row is the truth-value of the entire formula on that row. And as we can see, the truth-value under the main operator “∨” is T on both rows. This means that (P∨~P) is always true, no matter what the truth-value 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 truth-table. Again, the first step is to analyse the sentence. Here’s what the entire formula looks like with the truth-values 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 truth-table:

(P

&

~

P)

 T

 F

 F

 T

 F

 F

 T

 F

Now, focus again on the truth-values under the operator with the broadest scope, which this time around is “&”. The truth-value under the broadest operator on a given row is the truth-value of the entire formula on that row, and as we can see, the truth-value 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 truth-value of P.

Once we have mastered the use of truth-tables for identifying logical properties, we will learn to use truth-tables to test for relations of consistency, contradiction, equivalence and entailment within pairs of formulae. In Level 4: Using Truth-Tables To Test For Validity, we will once again return to the topic of truth-tables, this time using them to test argument forms or sequents for validity.

On completion of this course you will have the ability to:

(1) Construct truth-tables for logical formulae.

(2) Use truth-tables to determine the logical properties (necessary truth, necessary falsehood, contingency) of logical formulae.

(3) Use truth-tables to test for logical relations (consistency, contradiction, entailment, equivalence) within pairs of logical formulae.

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Level 2.1 is available as a free sample!

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