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| Arguments and their Validity |
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| A compound proposition is a tautology if it is always true for all possible combination of the truth values of its components. |
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| If a compound proposition which is always false for all possible combination of the truth values, of its components then it is called a contradiction. |
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Consider two compound statements P and Q. If the conditional P Q is a tautology then we say that "Q logically follows from P or Q is a valid conclusion (consequence) of the premise P. (P;Q) is called the argument. |
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| In general, if P1, P2, P3 ….Pn are n compound statements and Q is a compound statement which follows from the other statements P1, P2, P3.........Pn, then we say the conclusion Q logically follows from the set of premises P1, P2, P3.......Pn. These sequences of premises ending with the conclusion is called an argument. |
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| (P1, P2, P3.......Pn;Q) is called an argument where P1, P2, P3.......Pn are premises and Q is a conclusion or consequence. Premises are also known as Hypotheses. |
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| The argument (P1, P2, P3.....Pn;Q) is said to be valid if Q is true whenever all P1, P2, P3.....Pn are true. |
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| We discuss two methods to determine the validity of the argument. |
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| Method 1: |
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| Construct the truth table of the conditional |
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P1  P2  P3  .... Pm  Q and find out whether it is a tautology or not. If it is a tautology then the argument is valid otherwise it is said to be invalid. |
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| Example: |
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| Test the validity of the following argument. |
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P1: p  q |
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P2: q  p, Q:p q |
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| To test the validity of the argument, we have to show |
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P1  P2  Q is a tautology. |
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| Since the truth value of the last column are not all T, |
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| Method 2: |
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| Steps to test the validity of an argument. |
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| Step 1: Construct the truth table for truth values of all the hypotheses and the conclusion. |
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| Step 2: Find the rows in which all the hypotheses have truth value T. Such a row is called critical row. |
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| Step 3: If in each critical row, the truth value of the conclusion is also true, then the argument is valid. |
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| If there is at least one critical row in which the conclusion is false, then the argument is invalid. |
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| Examine the validity of the following argument |
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| We select the critical rows and wherever the premises (hypotheses) have truth value T. |
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| Since the corresponding conclusion Q also has Truth value T in the critical rows, the given argument is valid. |
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