Deductive reasoning is the process of reaching a conclusion that is guaranteed to follow, if the evidence provided is true and the reasoning used to reach the conclusion is correct. The conclusion also must be based only on the evidence previously provided; it cannot contain new information about the subject matter. This is the kind of reasoning that is done in high school geometry classes using formal proofs. Deductive reasoning was first described by Greek philosophers/mathematicians.

     Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the conclusion of an argument is very likely to be true, but not certain, given the premises. It is to ascribe properties or relations to types based on limited observations of particular tokens, or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is used, for example, in using specific propositions such as:
     The ice is cold.
     A billiard ball moves when struck with a cue.
to infer general propositions such as:
     All ice is cold.
     For every action, there is an equal and opposite re-action.

     Inductive reasoning (scientific reasoning, probabilistic inquiry, experimental inquiry, solving problems under uncertainty) has many similarities with the kind of reasoning used by Sherlock Holmes in the works by A. Conan Doyle. This kind of reasoning involves the claim, not that reasons give conclusive evidence for the truth of a conclusion, but that they provide some support for.

     Both types of reasoning are routinely employed, but inductive reasoning is the more intuitive of the two while deductive reasoning must be learned. One difference between them is that in deductive reasoning, the evidence provided must be a set about which everything is known before the conclusion can be drawn. Since it is difficult to know everything before drawing a conclusion, deductive reasoning has limited use in the real world. This is where inductive reasoning steps in. Given a set of evidence, however incomplete the knowledge is, the conclusion is likely to follow, but one gives up the guarantee that the conclusion follows, and, one gives up the guarantee that if a conclusion follows that it is correct. Induction does provide the ability to learn new things that are not obvious from the evidence.

Reasoning, Dr Watson style!
Taking a well-earned break from the detective business, Sherlock Holmes and Watson were on a camping/hiking trip. They had gone to bed early and awoke during the night, lying there looking up at the sky. Holmes said, "Watson, look up. What do you see?" "Well, I see thousands of stars." "And what does that mean to you?"
     "Well, I suppose it means that of all the planets and suns and moons in the universe, that we are truly the one most blessed with the reason to deduce theorems to make our way in this world of criminal enterprises and blind greed. It means that we are truly small in the eyes of God but struggle each day to be worthy of the senses and spirit we have been blessed with. And, I suppose, at the very least, in the meteorological sense, it means that it is most likely that we will have another nice day tomorrow. What does it mean to you, Holmes?" "To me, it means someone has stolen our tent."
     Many incorrectly teach that deductive reasoning goes from the general to the specific and that inductive reasoning travels in the opposite direction. Deductive reasoning is fundamentally in the form of an assertion of idea to materialization, while inductive reasoning is from empirical evidence to formulate the generalized knowledge of the observation thereof. It is not unusual therefore for science, in its beginning form, to be induction based. However, since the discovery of quantum physics, it is realized that higher science via deduction post greater possibility in resolving higher theoretical scientific problems.

Deductive Reasoning:  Deductive reasoning is when you move from things you know or assume to be true - called 'premises' - to conclusions that must follow from them. The most famous example of deduction is:
    Socrates is a man.
    All men are mortal.
    Therefore, Socrates is mortal.
The first two statements are premises, and the third statement is a conclusion. By the rules of deduction, if the first two statements are true, the conclusion must be true. Note that this is the case even if the premises appear to be nonsense:
    All ducks play golf.
    No one who plays golf is a dentist.
    Therefore, no ducks are dentists.
The conclusion follows deductively, or is 'deduced', from the premises. IF the premises are true, THEN the conclusion must be true. If the premises happen to be false, then of course all bets are off. The conclusion might still be true, or it might not. Lewis Carroll was fond of using examples like these in his books on symbolic logic, partly because they prevent you from applying your 'common sense'. After all, the whole point of logic is to make up for errors in common sense!

     Inductive Reasoning: Inductive reasoning is when you move from a set of examples to a theory that you think explains all the examples, as well as examples that will appear in the future. The simplest kind of induction looks like this: The sun came up this morning. The sun came up the day before that. The sun came up the day before that. . . Therefore, the sun comes up every day, and will come up tomorrow too. Note that while a conclusion deduced by deduction must be true if the premises are true, the conclusions induced by induction may be true, or they may not be true. For example, people who visit Seattle for short periods often find that it rains every day of the visit. They could induce (or infer, or draw the conclusion) that it rains every day in Seattle - but this wouldn't be true.

     In mathematics, inductive reasoning is often used to make a guess at a property, and then deductive reasoning is used to prove that the property must hold for all cases, or for some delimited set of cases. For example, by playing around, you might notice that every time you inscribe a triangle in a circle so that one leg lies along a diameter of the circle, it seems to be a right triangle. You might then guess that this is always the case. That's induction. You would then set out to use the axioms of geometry to prove that this must be the case. That's deduction.