Forms of Reasoning
Inductive Reasoning:
This type of reasoning does not use a form of logical syllogism. Rather the inductive conclusion is determined by a combination of gathering specific information with previous knowledge and learned experience. One of the more famous users of inductive reasoning (mistakenly refered to as 'deductive') is the character, Sherlock Holmes.

Here's an example of Inductive Reasoning:  
From "The Adventure of the Speckled Band" by Sir Arthur Conan Doyle

  "It is not cold which makes me shiver," said the woman in a low voice, changing her seat as requested.
  "What, then?"
  "It is fear, Mr. Holmes. It is terror." She raised her veil as she spoke, and we could see that she was indeed in a pitiable
state of agitation, her face all drawn and gray, with restless frightened eyes, like those of some hunted animal. Her features
and figure were those of a woman of thirty, but her hair was shot with premature gray, and her expression was weary and haggard.
Sherlock Holmes ran her over with one of his quick, all-comprehensive glances.
  "You must not fear," said he soothingly, bending forward and patting her forearm. "We shall soon set matters right, I have
no doubt. You have come in by train this morning, I see."
  "You know me, then?"
  "No, but I observe the second half of a return ticket in the palm of your left glove. You must have started early, and yet
you had a good drive in a dog-cart, along heavy roads, before you reached the station."
  The lady gave a violent start and stared in bewilderment at my companion.
  "There is no mystery, my dear madam," said he, smiling." The left arm of your jacket is spattered with mud in no less 
than seven places. The marks are perfectly fresh. There is no vehicle save a dog-cart which throws up mud in that way, 
and then only when you sit on the left-hand side of the driver."
  "Whatever your reasons may be, you are perfectly correct," said she.
http://www.citsoft.com/holmes/adventures/speckled.band.txt


Other Forms of Inductive Reasoning

Generalization: A general statement is made based upon observations of specific members of a particular group. When the observer makes specific observations as the basis of a general conclusion, an inductive leap, is made.

Analogy: A conclusion is drawn by arguing that there are distinct similarities between two different events.




Yet Another Form of Inductive Reasoning
The Inference
Inference: When conclusions are drawn from known or assumed facts or statements.

Statistical Inference: Whenever conclusions are made that something is true of a population as a whole because it is true of a certain portion of the population (ie., polling samples).



Deductive:

Where inductive reasoning is generally refered to as a "bottom up" thought process deductive reasoning is known as a "top-down" thought process. The first (major) premise is a statement of general truth. The second (minor) premise contains a statement of specific/particular truth. The conclusion is a synthesis of the realtionship between major and minor premises. And as long as the first two premises are true, the conclusion will be sound and true.

A standard example:
1) All men are mortal
2) Michael Collins was a man
Therefore, Michael Collins was mortal

In the movie, A Few Good Men, Lt. Daniel Caffee challenges the assertions made by the commander of Guantanamo Bay, Col. Nathan Jessup, about the death of PFC William Santiago. One of the logical arguments made is this...

1) Orders at Gitmo are followed to the letter
2) Col. Jessup gave a standing order that PFC Santiago should not be harmed
Therefore, PFC Santiago was safe from harm and did not need to be transferred

Through a series of statements and questions, Lt. Caffee argues that Santiago's death, at the hands of two marines, proves that Jessup's syllogism contains a false premise. Indeed, Caffee's questioning brings about Jessup's confession that he'd actually ordered a 'Code Red' for PFC Santiago.


Abduction:
This form of reasoning is similar to induction. As a form of critical thinking, abductive reasoning looks for patterns in a phenomena that suggest a possible hypothesis. Generally speaking, abduction is based on observed phenomena where some elements of the situation are missing.

Ex.
Margaret is walking down the street. As she passes a set of apartment buildings, she notices that a cloud of smoke is coming out of a third floor window. Without a second thought she reaches for her cellphone and calls for the fire trucks. After the trucks arrive a grease fire is put out.

What Maggie sees is smoke coming out of a window rather than a chimney. What she guestimates from the location and the amount of smoke is that there is an unwanted fire in the apartment.

see: Charles Sanders Pierce <http://carbon.cudenver.edu/~mryder/mem19.html>