**Deductive reasoning
**

Much deductive reasoning is based on implications, which are statements of the form If . . . then . . . .

We often write the general statement as *If p then q,* where *p* and *q* are simple statements. For example, in the implication

*p* is the statement *If it's raining* and *q* is the statement *it's cloudy.* The statements are called *implications* because they can be rewritten as *p implies q.* For example, *rain implies cloudiness.* (We sometimes need to fudge a little with English to overcome awkwardness in stating implications.)

The pieces of implications are often negations of simple statements. The negation of a statement is another statement that is false when the original statement is true, and vice versa. For example, one negation of *The person is tall* is *The person is not tall.* Negations are not necessarily opposite; you wouldn't say that *The person is tall* and *The person is short* are negations of each other.

Proof by contradiction depends on negations.

To prove that a conjecture is false, you need to prove that its negation is true. If the conjecture is a claim that something is always true, finding one case in which its false is enough to prove that the entire conjecture is false. For example, to prove that the conjecture

If it's a trapezoid then it's a rectangleis false, you need only to show an example of a trapezoid that isn't a rectangle. Such an example is called a

**Disguises**

Implications are often disguised in forms other than *if then* and *implies.* For example, here are some other ways of saying *If it's raining then it's cloudy:*

It's cloudy if it's raining.

It's raining only if it's cloudy.

Clouds are necessary for rain.

Rain is sufficient to ensure clouds.

Implications *p implies q* and *q implies p* are called *converses* of each other. For example, *If a person is pregnant then the person is a woman* and *If a person is a woman then the person is pregnant* are converses of each other. Because not all women are pregnant, you can see that converses don't mean the same thing as each other.

On the other hand, implications *p implies q* and *not q implies not p* do mean the same thing logically. These are called *contrapositives* of each other. For example, *If a person is pregnant then the person is a woman* and *If a person is not a woman then the person is not pregnant* are contrapositives of each other and have the same logical meaning.

Proofs based on contrapositives are another form of indirect proof.

Underlying every direct proof are three kinds of reasoning:

From the two statementspandp implies qwe can conclude statementq.This fact goes by the namemodus ponens,Latin forpower method.From the two statements

pandnot q implies not pwe can conclude statementq.From the two statements

p implies qandq implies rwe can conclude statementp implies r.

*Modus ponens* is the basis for proof by induction.

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