LOGIC ([Greek: logikê], sc. [Greek: technê], the art of reasoning), the name given to one of the four main departments of philosophy, though its sphere is very variously delimited. The present article is divided into 1. _The Problems of Logic_, II. _History_. I. _The Problems of Logic._ _Introduction._--Logic is the science of the processes of inference, what, then, is inference? It is that mental operation which proceeds by combining two premises so as to cause a consequent conclusion. Some suppose that we may infer from one premise by a so-called "immediate inference." But one premise can only reproduce itself in another form, e.g. all men are some animals; therefore some animals are men. It requires the combination of at least two premises to infer a conclusion different from both. There are as many kinds of inference as there are different ways of combining premises, and in the main three types:-- 1. _Analogical Inference_, from particular to particular: e.g. border-war between Thebes and Phocis is evil; border-war between Thebes and Athens is similar to that between Thebes and Phocis; therefore, border-war between Thebes and Athens is evil. 2. _Inductive Inference_, from particular to universal: e.g. border-war between Thebes and Phocis is evil; all border-war is like that between Thebes and Phocis; therefore, all border-war is evil. 3. _Deductive or Syllogistic Inference_, from universal to particular, e.g. all border-war is evil; border-war between Thebes and Athens is border-war; therefore border-war between Thebes and Athens is evil. In each of these kinds of inference there are three mental judgments capable of being expressed as above in three linguistic propositions; and the two first are the premises which are combined, while the third is the conclusion which is consequent on their combination. Each proposition consists of two terms, the subject and its predicate, united by the copula. Each inference contains three terms. In syllogistic inference the subject of the conclusion is the minor term, and its predicate the major term, while between these two extremes the term common to the two premises is the middle term, and the premise containing the middle and major terms is the major premise, the premise containing the middle and minor terms the minor premise. Thus in the example of syllogism given above, "border-war between Thebes and Athens" is the minor term, "evil" the major term, and "border-war" the middle term. Using S for minor, P for major and M for middle, and preserving these signs for corresponding terms in analogical and inductive inferences, we obtain the following formula of the three inferences:--