Table 162

Valid and invalid inferences for the conditional

Valid

Premises

If he is a cowboy, then he is a chair He is a cowboy.

Conclusion

Therefore, he is a chair.

Therefore, Q

Premises

If it is raining, then Alicia gets wet Alicia does not get wet.

Conclusion

Therefore, it is not raining.

Therefore, not P

Modus ponens

Therefore, Q

Modus tollens If P then Q

not Q

Therefore, not P Invalid

Affirmation of the consequent If P then Q

Therefore, P

Denial of the antecedent If P then Q, not-P

Therefore, not-Q_

Modus ponens and modus tollens are the two valid inferences that can be drawn from simple conditional arguments. Two other inferences can be drawn but they are invalid (even though people often think them plausible). They are called the "affirmation of the consequent" and the "denial of the antecedent". In the affirmation of the consequent, if P then Q is true and Q is true, for instance: Invalid: Affirmation of the consequent

Premises

If it is raining, then Alicia gets wet If P then Q,

Alicia is wet. Q,

Conclusion

Therefore, it is raining. Therefore, P

One can see where this form gets its name, because the consequent of the conditional premise (i.e., Q) has been affirmed. But why is it considered to be invalid? If you examine the truth table for the lines where IfP then Q is true and Q is true, you can see that there are two lines that meet this description. On one of these lines, P is true and on the other P is false. This means that logically speaking the most we can conclude is that "no conclusion can be made". So, a conclusion which asserts that P is true is considered invalid. A similar explanation can be made for the other invalid form, the denial of the antecedent for example: Invalid: Denial of the antecedent

Premises

If it is raining, then Alicia gets wet If P then Q,

It is not raining. not P,

Conclusion

Therefore, Alicia does not get wet. Therefore, not Q

Here one has denied the antecedent of the conditional (i.e., the P). There are two lines in the truth table where If P then Q is true and P is false. In one of these lines, Q is false and in the other Q is true. Therefore, again, no firm conclusion can be made. So, to conclude that not-Q is the case is invalid (these forms are summarised in Table 16.2). If you think that these two invalid forms yield plausible conclusions, you are not alone.

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