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Estimated Learning Time = 6 hours

What is the indirect method, and why should I learn it?

If we only have two simple sentences to deal with, our truth-table will only be four lines deep. However, with each additional simple sentence the number of lines doubles. This means that if we have four simple sentences our truth-table will be thirty-two lines deep! And with thirty two lines to worry about, there is a significant chance that a mere mortal such as you or I will make a mistake. This is where The Indirect Method comes in handy.

In this course, you will learn to use the indirect method to test the validity of sequents and to test for logical relationships within pairs of logical formulae. Here’s an example in which the indirect method is used to test the validity of a sequent.

(~

Q

x∨

P)

P

Q

The first step in applying the indirect method in determining entailment is to assign a T to each of the two premises and an F to the conclusion as follows (the application of this step amounts to our assumption that our premises are true and our conclusion false), then carry any assigned simple-sentence truth-value over to instances of the same simple-sentence:

(~

Q

x∨

P)

P

Q

T

T

F

The next step is to carry the truth-values of the simple sentences across to other instances of the same sentence as follows.

(~

Q

x∨

P)

P

Q

F

T

T

T

F

The final step is to fill in the truth-value for the negation sign as follows:

(~

Q

x∨

P)

P

Q

T

F

T

T

T

F

As P is true and ~Q is true then both disjuncts of (~Q x∨ P) will be true. And, an exclusive disjunction with two true disjuncts should be false, rather than true. Thus, we have a contradiction in our first premise. As the assumption that our two premises are true and our conclusion is false leads unavoidably into contradiction, we can conclude that it is impossible for our premises to be true at the same time that our conclusion is false, and thus that our sequent is valid.

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