Aristotle was the first person to systematize the rules of logic. He thought of logic as “a study of proof.” His work on logic is found in his Organon. This is the title given to the logical works of Aristotle, implying that logic is an “organ” or “tool” used to acquire philosophical knowledge. In other words, logic is for practicing philosophy and is a kind of preparatory study for students of philosophy. There are two principle forms of logical inference: deduction, which yields certain conclusions, and induction, which yields probable conclusions.

Deductive Reasoning

Since Aristotle was chiefly concerned with the form of proof, he was most interested in syllogism, which he assumed provided certain knowledge concerning reality. For instance, in the syllogism “All men are mortal, Socrates is a man, therefore Socrates is mortal,” it is not merely that the conclusion is deduced according to the formal laws of logic; Aristotle assumes that the conclusion is verified in reality. With syllogisms, one reasons from the universal (e.g., a principle like “All men are mortal” or “All kindness is good”) to a particular instance (e.g., “Bill is mortal” or “Giving to the poor is good”).

These are called “categorical syllogisms,” since he is referring to categories like “men,” “mortal,” and a particular man, “Socrates.” If the placement of these terms is imperfect, then the form of the syllogism will not work. For instance, if one says, “All Men are Mortal, Socrates is Mortal, so Socrates is a Man,” the syllogism has bad form. For reasoning from the first two premises one can only conclude that Socrates is a man, not that he is mortal. Since the form of reasoning is in error, the syllogism commits a formal fallacy and is invalid, which means its conclusion does not “follow.” Aristotle observed that there are 256 different varieties of syllogism, but only 24 of them are valid, meaning that their conclusions follow with certainty.

Besides syllogistic deductive reasoning, there are three laws of thought that are also known with certainty according to Aristotle. The first is the principle of contradiction. This law, or principle, asserts that a statement cannot be true and false at the same time. For example, a person could not logically make the statement “It is raining and it is not raining.” Statements of this form are internally contradictory and can never be true.

The second law is the principle of the excluded middle. This law says that a statement must be true or false. “It is raining or it is not raining” is an example. It is called the principle of the “excluded middle” because it is either raining or not; there is no third or “middle” possibility.

The third law is the principle of identity. This principle asserts that everything is equal to itself. The statement “X is equal to X” is true without exception. As with the first two principles, one immediately apprehends this principle. That is, one has an intuitive grasp of its truth, quite apart from any knowledge about the facts of sense experience.

Inductive Reasoning

In the Analytics Aristotle considers not only deductive scientific proof or demonstration, but also induction. “Induction proceeds through an enumeration of all the cases,” he says. Induction, then, proceeds in the opposite direction, enabling one to reason from a particular instance, like “This swan is white” to a general conclusion like “All swans are white.” Proceeding from the particular to the general is just one way in which inductive arguments differ from deductive ones. They are also distinguished by being probable as opposed to being certain. But Aristotle's ideal for reasoning is deduction, or syllogistic, demonstration.

In everyday life, however, few arguments are deductive. There are times when the best one can do in arguing is to arrive at conclusions that are probable. So you might reason: “Whenever I have set my alarm for 6 A.M., it has sounded off at 6 A.M. Therefore, the next time I set it at 6 A.M., it will sound off at 6 A.M.” Now it may be true that your alarm will once again sound at 6 A.M., and you hope that it does. The probability that it will sound off is high — close to a probability of one, which is to say close to a certainty. But you cannot be certain that it will; it is a probability. So, inductive arguments are those whose conclusions follow with probability. Most of the conclusions about your everyday life would seem to be probable in a similar way.

You expect that it will take you thirty minutes to drive to your job from your home because it usually takes thirty minutes. In addition, since your car was running fine yesterday and the day before, you expect that it will run reliably today and tomorrow. Notice that the probability of your getting to work in thirty minutes depends on several factors that are also based on probabilities: the traffic, the weather, the condition of your car, road accidents, and so on. You could calculate the probability of getting to work in thirty minutes by multiplying the probabilities of each of these events.

Probabilities play out frequently on the field of sports. In years past, if you have seen a baseball player, say Boston's David Ortiz, hit for a batting average of .300. (equivalent to 30 hits for every 100 at bats), you might expect him to hit .300 this year. In all reasoning of this kind, your conclusions are established by the evidence with a degree of probability (with probabilities landing on a spectrum somewhere between 0 and 1) and so are inductive. One of the characteristics of such inductive arguments is that you expect that the future will resemble the past.