Question

Show that each conditional statement is a tautology using the fact that a conditional statement is false exactly when the hypothesis is true and the conclusion is false. (Do not use truth tables.)

Answer 1

From Conditional Statements, there can be built a Tautology.

Explanation:

According to the question, when the Hypothesis is True and the Conclusion is False, the result will be False. And that's correct.

But when Conditional Statement can be a Tautology, given the fact that not always a Conditional Statement will return the logic value of truth? Look at this example below:

1)James is from Orlando and He lives in Florida. Therefore, James is from Orlando.

2)James is from Orlando and He doesn't live in Florida. Therefore, James is from Orlando.

3)James is not from Orlando e He lives in Florida. Therefore, James is from Orlando

4)James is not from Orlando e He doesn't live in Florida. Therefore, James is from Orlando

What we have here above is symbolic for:

p∧q→p

And this is tautological, since no matter, we have on line 2 for p∧q, q being false it does not change the value for false, in any way for the whole proposition p∧q→p . What gives us a Tautology

This, p∧q→p can be rewritten as p∧q⇒p