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## Implication

Definition 2.3.13   Let and be two propositions. Then the truth table of is as follows:   T T T T F F F T T F F T

This means that by a correct argument, when the premice is true, then the conclusion is true. When the premice is false, you can get any conclusion, even with a correct argument. For example the sentence Jerusalem is Pakistan's capital,thus an apple has wings'' is logically true.

Proposition 2.3.14   Let , and are three propositions, then implies .

Proof. As usual, we use a truth table.        T T T T T T T T T T F T F F F T T F T F T F T T T F F F T F F T F T T T T T T T F T F T F F T T F F T T T T T T F F F T T T T T
The last column shows that we have here a tautology. Proposition 2.3.15   Let and be two propositions. Then is logically equivalent to the statement .

The proof has to be done with a truth table, which we leave to the reader. This proposition will reveal useful for finding the negation of an implication. For example:
• Consider the sentence If I swim in the sea, then I'm weat''. Its negation is I swim in the sea and I'm not weat''. Check which one is true.
• Take the sentence: In order to be a policeman, one has to succeed in an examination and to have a good physical condition''. This means that when you see a policeman, you know that his exam was successful and that he was in good physical condition. If somebody tells you that he failed to enter the police, you know that either he did not succeed at the exam or he was in poor physical condition (or both, eventually). Pay attention to the fact that here, finding the negation involved the negation of two connectors : implication used as in Proposition and De Morgans law (see Proposition ).

Example 2.3.16   In Calculus, we learn the definition of a continous function at a point . A function , defined on a neighborhood of a real is continuous at if the following holds: The negation of this, i.e. the fact that is discontinuous at is described by the following statement:     Next: Equivalence Up: Logical connectors Previous: Tautologies and contradictions   Contents
root 2002-06-10