Logic

Now the study of formal inference, as already explained, is essentially a study of the implications of propositions. For this purpose it is necessary to ascertain the main types of proposition, their several meanings and implications. This part of logic is known as the study of immediate inference. But under certain circumstances, which will be explained in due course, two or more propositions may, between them, imply something over and above the implications which each of them has separately, some thing that is not even the mere sum of their several implications. The study of these implications is known as the study of mediate inferences. These are the two principal divisions of formal inferences, and each of them contains a number of varieties, which we shall presently describe and explain. But before doing so it will be necessary to deal first with the ultimate assumptions or postulates of logic. These are really ultimate assumptions of reason, without which there would be no such thing as reason ing, or at least no distinction between valid and invalid reason ing. It is the business of logic to formulate them explicitly.

Ultimate Postulates of Logic.

In all reasoning (as distin guished from dogmatic assertion, e.g.) some proposition is as serted to be justified by some other proposition or propositions, called the evidence. Dismissing those cases in which one verbal expression is substituted for another with which it is synonymous, and which do not constitute real reasoning, one proposition can only be justified or implied by another or by others in so far as the contents of the propositions are connected in some definite way—in other words, in so far as the facts to which the propo sitions in question refer stand in certain connections. It is the belief in, or the knowledge of, such objective connections among natural events or objects that constitutes in the last resort the basis of all reasoning or inference. Such objective connections cannot, strictly speaking, be proved, although they are commonly assumed to exist as a matter of course. But in a study like logic it is necessary to state explicitly every assumption that is neces sary for its very existence. Accordingly, we may formulate as

the first postulate of logic the assumption that there are connec tions among the events and objects of the real world. If the world were not, to some extent at least, an orderly cosmos, if things just happened by chance or anyhow, there could be no genuine reasoning, and consequently no logic. This assumption may be described briefly as the postulate of cosmic connectedness.

Another postulate is what is known as the principle of the uni formity of nature, which postulates that the connections among the events and objects of the world are uniform in character. In other words, phenomena of the same kind exhibit the same kind of objective connections. From the point of view of logic this postulate is of importance as the basis of what the writer has called the principle of the uniformity of reasons, namely, whatever is regarded as a sufficient reason in any one case must be regarded as a sufficient reason in all cases of the same type, or expressed negatively, nothing can be regarded as a sufficient reason in any one case unless it can also be regarded as a sufficient reason in all cases of that kind. This principle is of special importance in connection with inductive inference, inasmuch as it formulates the logical basis of generalization. But it is also of importance in connection with the study of formal inference, for it is the basis of our derivation of the general rules of formal inference from an inspection of the form of inference in particular cases.

Two other postulates required for formal inference are known as the law of contradiction and the law of excluded middle. The law of contradiction is to the effect that the same predicate cannot be both affirmed and denied of the same subject—S cannot both be P and not be P. The law of excluded middle postulates that a given predicate must be either affirmed or denied of a given subject—S must either be P or not be P. Since "not to be P" is the same as "to be non-P" (that is, something other than P), the two laws may also be expressed respectively in the following formulae—_S cannot be both P and P, and S must be either P or P (when = non-P)