**Some Methods and Examples of Valid Reasoning**

**I. Euler diagram**

Draw pictures (diagrams). If any diagram that satisfies the hypotheses contradicts the conclusion, the argument is not valid.

Hypotheses:

All roses are red.

This flower is a rose.

Conclusion:

This flower is red.

**II. Direct reasoning**

If the statement "if p then q" is true and p is true, then q must be true.

Hypotheses:

If you stay up late, you will be tired in the morning.

I stayed up late last night.

Conclusion:

I'm tired this morning.

**III. Indirect reasoning**

If the statement “ if p the q” is true and q is false, then p must also be false.

Hypotheses:

If you stay up late, you will be tired in the morning.

I’m not tired this morning.

Conclusion:

I did not stay up late last night.

**IV. Chain rule **

If the statements “if p then q” and “if q then r” are both true, then if p is true, r must also be true.

Hypotheses:

If I work hard, I earn lots of $$$.

If I earn lots of $$$, I pay high taxes.

Conclusion:

If I work hard then I pay high taxes.