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Geometry: Inductive and Deductive Reasoning

Deductive Reasoning

Problems

Problems

Deductive reasoning, unlike inductive reasoning, is a valid form of proof. It is, in fact, the way in which geometric proofs are written. Deductive reasoning is the process by which a person makes conclusions based on previously known facts. An instance of deductive reasoning might go something like this: a person knows that all the men in a certain room are bakers, that all bakers get up early to bake bread in the morning, and that Jim is in that specific room. Knowing these statements to be true, a person could deductively reason that Jim gets up early in the morning. Such a method of reasoning is a step-by-step process of drawing conclusions based on previously known truths. Usually a general statement is made about an entire class of things, and then one specific example is given. If the example fits into the class of things previously mentioned, then deductive reasoning can be used. Deductive reasoning is the method by which conclusions are drawn in geometric proofs.

Deductive reasoning in geometry is much like the situation described above, except it relates to geometric terms. For example, given that a certain quadrilateral is a rectangle, and that all rectangles have equal diagonals, what can you deduce about the diagonals of this specific rectangle? They are equal, of course. An example of deductive reasoning in action.

Although deductive reasoning seems rather simple, it can go wrong in more than one way. When deductive reasoning leads to faulty conclusions, the reason is often that the premises were incorrect. In the example in the previous paragraph, it was logical that the diagonals of the given quadrilateral were equal. What if the quadrilateral wasn't a rectangle, though? Maybe it was actually a parallelogram, or a rhombus. In such a case, the process of deductive reasoning cannot be used. The fact that the diagonals of a rectangle are equal tells us nothing relevant about the diagonals of a parallelogram or a rhombus. The premises used in deductive reasoning are in many ways the most important part of the entire process of deductive reasoning. If they are incorrect, the foundation of the whole line of reasoning is faulty, and nothing can be reliably concluded. Even if just one conclusion is incorrect, every conclusion after that is unreliable, and may very well be incorrect, also.

Another instance in which deductive reasoning doesn't work is when it is not executed properly. Using the example in the first paragraph, let's add the premise that Bob is a baker. Can we deduce that Bob is in the room? We could only deduce this if we knew that everybody who was a baker was in the room. This was not one of the premises, though. When reading premises, it is very important not to assume anything more than exactly what is written. In Logic Statements we'll more carefully examine exactly what occurs when premises are misused and lead to false conclusions. For know, it is enough to know that deductive reasoning is perfectly effective when all of the premises are true, and each step in the process of deductive reasoning follows logically from the previous step.

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