Deductive Reasoning
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.
