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Inference is a sequence of propositions where one of the propositions is identified as the conclusion and the rest of the propositions are called premises such that premises justify or support the truth of the conclusion.

In other words, an inference is a relation between premises and the conclusion. The aim of logic is to characterise the nature of this relation. At the outset, on the basis of this relationship we classify all inferences into two types, namely, inductive and deductive. An inductive inference is a kind of inference where the conclusion is not conclusively justified by the premises.

Moreover, its aim is to make a generalization on the basis of un-contradicted experiences of observation of finite (or limited) number of cases. The set of premises of any inductive inference must be finite and are based on the un-contradicted experiences. The conclusion is just the generalization of such un-contradicted experiences and thus is always probably true.” The form or structure of an inductive inference is:

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Each of a,, a,…….. A has been observed to be S and P.

Nothing has been observed to be S without P.

Therefore, probably all S are P.

The detail discussion on inductive inference, its classification, nature etc. are beyond the scope of this book. In the present work our aim is to study only deductive inferences.

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Deductive inference is a kind of inference where the truth of the premises conclusively support or justify the truth of the conclusion. In a valid deductive argument it is impossible for the premises to be true and the conclusion to be false.

Deductive arguments are primarily of two types depending upon the number of premises the argument contains. These arguments are either immediate or mediate. An immediate inference is a type of deductive inference in which the inclusion is derived from exactly one premise. This type of inference is called immediate because mere is no other premise existing between the initial premise and the conclusion. Or it can be inscribed as an inference where the conclusion is obtained without the use of a middle term.

For example,

All men are mortal. (Premise)

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Therefore, some mortals are men. (Conclusion)

is an instance of immediate inference.

Immediate inferences are mainly of three types namely (i) Conversion, (ii) Obversion, and (iii) Contraposition. Each of these will be studied in detail in the present chapter. On the :her hand, a deductive inference is called mediate if the conclusion is obtained or derived by use of more than one premises. It is of two types (i) Syllogistic and (ii) Non-Syllogistic.

An inference is called syllogistic if and only if the conclusion is derived from two premises taken jointly. In this case, the numbers of premise are exactly two and it is essential to use both the premises to obtain the conclusion. For example,

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All men are mortal. Premise

All kings are men. Premise

Therefore, all kings are mortal. Conclusion

Here, the conclusion cannot be derived if we do not jointly consider these two premises. A mediate deductive inference is called non-syllogistic if the conclusion is derived from more than two premises. In this case we do not have the restriction to use all the premises jointly to derive the conclusion, unlike syllogistic inference.

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Further, syllogistic inferences are again of two types, namely pure and mixed. If all the propositions, including the premises and the conclusion are of one type it is called pure and if it contains different kinds of propositions it is called mixed syllogism.

Pure syllogisms are again of two types, namely (i) Categorical and (ii) Hypothetical. If all the propositions in a syllogism are categorical, it is called categorical syllogism. On the other hand, if all the propositions in a syllogism are hypothetical propositions, it is called pure-hypothetical or hypothetical syllogism. Mixed syllogisms are of three types namely, (i) Hypothetical- categorical, (ii) Disjunctive-categorical and (iii) Dilemma. Hypothetical-categorical syllogism is a kind of syllogism where the major premise is hypothetical, the minor premise is categorical and the conclusion is categorical.

A disjunctive-categorical syllogism is a mixed syllogism where the major premise is disjunctive and minor premise is categorical and the conclusion is also categorical. Similarly, dilemma is a kind of mixed syllogism consisting of a compound hypothetical proposition as its major premise and its minor premise is a disjunctive proposition leading to a categorical or disjunctive proposition as its conclusion. For convenience we give below the various kinds of inferences in a tabular form.

**The Principle of Distribution:**

The principle of distribution states that in any deductive inference, if a term is distributed the conclusion, then it should have been distributed in the premise. Alternatively, no term should be distributed in the conclusion unless it is distributed in the premise. This principle is a valid principle of deductive logic. To see its validity, let us argue as follows.

Assume that the principle of distribution is not valid. Then there exists an argument in which a term say ‘t’ is

distributed in the conclusion but not distributed in the premise. Then by the definition of distribution, ‘t’ is taken in its entire denotation in the conclusion, while only the part of the denotation of ‘t’ is being considered in the premise.

Therefore, the conclusion states more than the premise contradicting the notion of deductive inference. Hence, the principle of distribution must be valid. Now, we will discuss various types of immediate inference namely (i) Conversion, (ii) Obversion. (iii) Contraposition.