Reasoning is either Inductive or Deductive. The latter is again either (1) Immediate, or (2) Mediate, according as the conclusion follows from one premiss or from more than one. A Mediate Deductive Reasoning is called a Syllogism, when it conforms to the axiom called Dictum de omni et nullo,-"Whatever is affirmed or denied of a class distributively, may be affirmed or denied of any thing belonging to that class," or to some similar axiom or axioms. It may be called Mathematical, when it conforms to some one or other of the axioms in mathematics, such as (1) that things which are equal to the same thing are equal to one another, (2) that the sums of equals are equal, (3) the principle or axiom called Argumentum a fortiori, that 'a thing which is greater than a second, which is greater than a third, is greater than the third.' The subdivisions of the other main division cannot be discussed in this book. A Reasoning, regarded objectively, is the inference of a relation from one or more given relations among things and attributes. When a general or universal relation is inferred from one, a few, or many particular relations, the reasoning or inference is Inductive. When the relation inferred is not more general than the given relation or relations, and is, in fact, contained in, or implied by, the latter, the reasoning or inference is called Deductive. It is Immediate when the inference is drawn from one given relation or premiss, and Mediate when drawn from more than one. The word inference, it should be noted, has, at least, three meanings:(1) the process of reasoning, (2) the product of reasoning consisting of the premisses and the conclusion, and (3) the conclusion only. We have here used the word in the second sense, but it is frequently used in the first, and more frequently in the third. A reasoning, expressed in language, is called an Argument. There are thus as many kinds or varieties of the latter as there are of the former. The simplest form of argument corresponding to the simplest form of reasoning, namely, Immediate, consists of two propositions, the premiss and the conclusion. A Mediate deductive reasoning gives rise to an argument consisting of more than two propositions, namely, the premisses and the conclusion. An Inductive reasoning gives rise to an argument consisting of many propositions, namely, the particular instances constituting the data, and the general conclusion justified by them. The word 'argument' also denotes a series of reasonings advanced to establish a certain conclusion. It should be carefully noted that so far as Logic is concerned with reasoning, it treats of the principles of correct reasoning, and lays down the conditions to which reasoning must conform in order that it may be valid. It is no part of Logic to give an account of the various processes according to which men do or may reason, but of those according to which they ought to reason, and must reason if their reasonings are to be valid. The former is the business of the science of Psychology, the latter only is the business of Logic1. Examples of Different Kinds of Reasoning or Inference. 1 No attempt is made here to give an exhaustive account of all the processes of reasoning either from the psychological or from the logical point of view. In this chapter, the subject is treated for the purposes of this work. There is great diversity of view among Logicians (1) as to the nature of reasoning or inference,-as to what is and what is not inference, and (2) as to its fundamental kinds and varieties. The theory of Reasoning and Inference, like the theory of Predication and of the Import of Propositions, is a most important subject in Logic and Psychology, and would demand a thorough treatment in a complete treatise on Logic, No man is perfect, All philosophers are men; No philosopher is perfect. All metals are elements, Gold is a metal; Gold is an element. B.-Non-Syllogistic. e.g., Mathematical. A is equal to B, C is equal to B; A is equal to C. A is greater than B, C is equal to D; AC is equal to B + D. Mathematical reasonings are usually regarded as valid, if they conform to the axioms of mathematics. By taking the axioms as major premisses, and the data of the reasonings as minor premisses, they may, however, be reduced to the syllogistic form. Examples 6 and 7 given above may be stated syllogistically as follows: 6. Things which are equal to the same thing are equal to one another; the two things A and C are equal to the same thing (B); therefore the two things A and C are equal to one another. 7. A thing which is greater than a second, which is greater than a third, is greater than the third; the thing A is greater than a second (B), which is greater than a third (C); therefore the thing A is greater than the third (C). Similarly, other mathematical reasonings may be reduced to fully-expressed syllogisms. 3. 5. 6. 7. Water expands by heat, Mercury expands by heat, All material bodies expand by heat. Cocoanut oil is solidified by cold; .. All liquids are solidified by cold. ... ... The friction of the palms of our hands against each other produces heat, The friction of two pieces of wood produces heat, The friction of all material bodies produces heat. Many men whom I knew have died, All the men in the past ages have died; All men will die. The three angles of this triangle are together equal to two right angles; .. The three angles of any triangle are together equal to two right angles. These two straight lines cannot inclose a space, .. No two straight lines can inclose a space. An equilateral triangle can be constructed upon this finite line, An equilateral triangle can be constructed upon any finite line. Inductive reasonings conform to the canons and rules of Induction. By taking the canons and rules as major premisses, and the data of the reasonings as minor premisses, Inductive reasonings, like mathematical, may be reduced to the syllogistic form1. 1 See below, Appendix D. CHAPTER II. OF IMMEDIATE INFERENCES. § 1. IMMEDIATE Inference, as a process of reasoning, is the process of deriving or deducing a proposition from a given proposition or premiss. As an argument or reasoning expressed in language, it consists of the given proposition, and the proposition necessarily following from it. As an inference or conclusion, it is the proposition thus following,-the result of the process. The derivation of a proposition from a term may also be regarded as a kind of Immediate Inference. Every attribute connoted by a term may be affirmed of the term. Thus there are two kinds of Immediate Inference. (1) In the first kind, a proposition is inferred from a term. Take the connotative term 'man,' and let its connotation consist of the two attributes ‘rationality' and ‘animality.' From this term it is evident that we may at once infer the following two propositions: (i) 'Man is rational,' (ii) 'Man is animal.' This kind of immediate inference depends on the axiom that every attribute connoted by a term may be predicated of it. This axiom is the basis of the formation of verbal propositions by the analysis of the connotation of terms. This mode of immediate inference is really equivalent to the affirmation of an attribute of an aggregate of attributes, or of a thing or things, of which the attribute affirmed is known to form a part. |