Notes on Reduction of Propositions into Logical Form

There are four standard forms of categorical propositions such as A, E, I and O-propositions having the structure of the form, 'All S is P' 'No S is P’, 'Some S is P' and 'Some S is not P' respectively. Thus, we know that the logical structure of any categorical proposition exhibits the following four items in the order as given below.

Quantifier (Subject term) copula (predicate term)

Here the first item is the 'quantifier' (or more precisely the words expressing the quantity of the proposition). It is attached to the subject term only. The second item in any logical proposition is the subject term. The predicate term, that expresses something about the subject, comes after the copula. The copula is placed in between the subject and predicate term.

Further, the quality of the proposition is expressed in and through the copula. The copula and the predicate term are respectively the third and fourth logical elements of a categorical proposition. Thus, a categorical proposition which is in standard form must exhibit explicitly the subject, the predicate, the copula, its quality and quantity. Let us call a categorical proposition regular if it is in its standard form, otherwise it is called irregular.

In our ordinary language most of the categorical propositions are irregular in nature. Even though there are irregular categorical propositions they can be put in their regular form. It should be noted that while reducing an irregular categorical proposition into its standard form, we should pay enough attention to the meaning of the proposition so that the reduced proposition is equivalent in meaning to its irregular counterpart.

Before describing the method of reduction of irregular propositions into their regular forms, it is profitable to understand the reasons for irregularity of a categorical proposition: The irregularity of any categorical proposition may be due to one or more of these following factors.

(i) The copula is not explicitly stated; rather it is mixed with the main verb which forms the part of the predicate

(ii) Though the logical ingredients of a categorical proposition are present in the sentence yet are not arranged in their proper logical order.

(iii) The quantity of a categorical proposition is not expressed by a proper word like 'all', 'no' (or none), 'some' or it does not contain any word to indicate the quantity of the proposition.

(iv) All exclusive, exceptive and interrogative propositions are clearly irregular.

(v) The quality of the proposition is not specified by attaching the sign of negation to the copula.

Keeping these factors in mind, let us describe systematically the method of reduction of an irregular categorical proposition into its standard form (or into a regular proposition). Below we describe the method of reduction.

I. Reduction of categorical propositions whose copula is not stated explicitly

In our ordinary use of language, very often the copula is not explicitly or separately expressed but is mixed with the main verb. The main verb in such a case forms the part of the predicate. The moment copula is identified; the other items of a logical proposition are brought out in a usual manner. We know that the copula of any logical proposition must be in present tense of the verb "to be" with or without the sign of negation.

Now let us consider an example of an irregular proposition, where the copula is not explicitly stated. "All sincere students deserve success". This is an irregular proposition, as the copula is clearly mixed with the main verb of the proposition. The method of reducing such irregular sentences into regular ones is as follows. The subject and the quantifier of the irregular proposition should remain as they are, while the rest of the proposition may be converted to a class forming property (i.e. term) which would be our logical predicate.

In our above example 'All' is the quantifier attached to the subject 'sincere students'. We should not touch the quantifier nor the subject term of the proposition, they should remain where they are. On the other hand, the rest of the proposition 'deserve success' should be converted into a class forming property 'success deserving'. This should be our logical predicate. Then we link the subject term with the predicate term with a standard copula. Thus,

"All sincere students deserve success." Irregular proposition.

"All sincere students are success deserving." A - Proposition.

"All people seek power." Irregular proposition.

"All people are power seekers." A - Proposition.

"Some people drink Coca Cola." Irregular proposition.

"Some people are Coca Cola drinkers." I - proposition

II. Irregular propositions where the usual logical ingredients are all present but are not arranged in their logical order.

Consider the following examples of irregular propositions. "All is well that ends well" and "Ladies are all affectionate." In these cases, first we have to locate the subject term and then rearrange the words occurring in the proposition to obtain the regular categorical proposition. Such reductions are usually quite straight forward. Thus we reduce the above two examples as given below.

"All is well that ends well." Irregular proposition

"All things that end well are things that are well." A - Proposition

"Ladies are all affectionate." Irregular proposition

"All ladies are affectionate." A - Proposition

III. Statements in which the quantity is not expressed by proper quantity words. Some propositions do not contain word like 'All', 'No', 'some' or contain no words to indicate the quantity. We reduce such a type of irregular proposition into its logical form as explained below.

Here we have to consider two sub-cases : sub-case (i) where there is indication of quantity but no proper quantity words like 'All', 'No', on 'Some' are used and Sub­ case (ii) where the irregular proposition contains no word to indicate its quantity.

Sub-case (i): Affirmative sentences that begin with words like 'every', 'any', 'each' are to be treated as A-propositions, where such words are to be replaced by the word "all" and rest of the proposition remains as it is or may be modified as necessary. The followings are some of the examples of this type.

"Every man is liable to commit mistakes." Irregular proposition.

"All men are persons who liable to commit mistakes." A - Proposition.

"Each student took part in the competition." Irregular proposition.

"All students are persons who took part in the competition." A - Proposition.

"Any one of my students is laborious." Irregular proposition.

"All my students are laborious." A - Proposition.

A negative sentence that begins with a word like 'every', 'any', 'each', or 'all' is to be treated as an O-proposition. Any such proposition may be reduced to its logical form as shown below.

"Every man is not honest". Irregular proposition

"Some men are not honest." O - Proposition

"Any student cannot get first class." Irregular proposition.

"Some students are not persons who can get first class." O - Proposition.

"All is not gold that glitters." Irregular proposition.

"Some things that glitter are not gold." O - Proposition.

Sub- Case (ii):

"Sentences with singular term or definite singular term without the sign of negation are to be treated as A-proposition. For example, "Ram is mortal.", "The oldest university of Orissa is in Bhubaneswar." are A-propositions.

Here the predicate is affirmed of the whole of the subject term. On the other hand, sentences with singular term or definite singular term with the sign of negation are to be treated as E-propositions. For example, "Ram is not a student" and "The tallest student of the class is not a singer" are to be treated as E-propositions. These are cases where the predicate is denied of the whole of the subject term.

IV. “Sentences beginning with the words like 'no', 'never', 'none' are to be treated as E-propositions. The following sentence is an example of this type.

"Never men are perfect." Irregular proposition

"No man is perfect." E - Proposition

V. Affirmative sentences with words, like 'a few', 'certain', 'most', 'many' are to be treated as I-propositions, while negative sentences with these words are to be treated as

O-propositions. Since the word 'few' has a negative sense, an affirmative sentence beginning with the word 'few' is negative in quality. A negative sentence beginning with the word 'few' is affirmative in quality because it involves a double negation that amount to affirmation. The following are examples of above type.

"A few men are present." Irregular proposition.

"Some men are present." I - proposition.

"Certain books are good." Irregular proposition.

"Some books are good." I - proposition.

"Most of the students are laborious." Irregular proposition.

"Some students are laborious." I - proposition.

Here we may note that 'most' means less then 'all' and hence it is equivalent to 'some'.

"Many Indians are religious." Irregular proposition.

"Some Indians are religious." I - proposition.

"Certain books are not readable." Irregular proposition

"Some books are not readable." O - Proposition

"Most of the students are not rich." Irregular proposition.

"Some students are not rich." O - Proposition

"Few men are above temptation." Irregular proposition

"Some men are not above temptation." O - Proposition

"Few men are not selfish." Irregular proposition

"Some men are selfish.' I

VI. Any statement whose subject is qualified with words like 'only', 'alone', 'none but', or 'no one else but' is called an exclusive proposition. This is so called because the term qualified by any such word applies exclusively to the other term. In such cases the quantity of the proposition is not explicitly stated.

The propositions beginning with words like 'only', 'alone', 'none but' etc are to be reduced to their logical form by the following procedure. First interchange the subject and the predicate, and then replace the words like 'only', 'alone' etc with 'all'. For example,

"Only Oriyas are students of this college." Irregular proposition.

"All students of this college are oriyas." A - Proposition.

"The honest alone wins the confidence of people." Irregular Proposition.

"All persons who win the confidence of people are honest." A-proposition.

VII. Propositions in which the predicate is affirmed or denied of the whole subject with some exception is called an exceptive proposition. An exceptive proposition may be definite or indefinite. If the exception is definitely specified as in case of "All metals except mercury are solid" then the proposition is to be treated as universal and if the exception is indefinite, as in case of "All metals except one is solid", the proposition is to be treated as particular.

"All metals except mercury are solid." is a universal proposition which means

"All non-mercury metals are solid."

Now let us consider an example where the exception is indefinite. For example, "All students of my class except a few are well prepared", it is to be reduced to an I-proposition as given below.

"All students of my class except a few are well prepared." Irregular proposition.

"Some students of my class are well prepared." I - proposition.

VIII. There are impersonal propositions where the quantity is not specified. Consider for example, "It is cold", "It is ten O'clock". In such cases propositions in question are to be reduced to A-proposition because the subject in each of these cases is a definite description.

"It is cold". Irregular proposition

"The whether is cold." A - Proposition.

"It is ten O'clock." Irregular proposition.

"The time is ten O'clock." A - Proposition.

There are some propositions where the quantity is not specified. In such cases we have to examine the context of its use to decide the quantity. For example, consider following sentences (1) "Dogs are carnivorous", (2) "Men are mortal", (3) "Students are present." In first two examples, the quantity has to be universal but in the third case, it is particular. Thus, their reductions into logical form are as follows.

"Dogs are carnivorous." Irregular proposition.

"All dogs are carnivorous." A - Proposition.

This is so because we know that "being carnivorous' is true of all dogs.

"Men are mortal." Irregular proposition.

"All men are mortal." A - Proposition

Here 'being mortal' is generally true of men. But in the proposition "Students are present", we mean to assert that some students are present". So the proposition "Men are mortal" is reduced to "All men are mortal" But in the example "Students are present", 'being present' is not generally true of all students.

So the proposition "Students are present" is reduced to "Some dents are present" which is an I-proposition. Thus the context of use of a proposition determines the nature of the proposition.

IX. Problematic propositions are particular in meaning. For example "The poor may be happy" should be treated as a particular proposition, because what such a proposition asserts is that it is sometimes true and sometimes false.

Thus, "The poor may be happy" is reduced to "Some poor people are happy", which is an I-proposition

X. Similarly, there are propositions where the quantity is not specified but their predicates are qualified by the words like 'hardly', 'scarcely', 'seldom'. Such propositions should be treated as particular negative. For example, "Businessmen are seldom honest", is an irregular proposition. It is reduced to "Some businessmen are not honest". If such a proposition contains the sign of negation that these proposition is to be treated as an I-proposition.

For example, "Businessmen are not seldom honest." is to be reduced to "Some businessmen are honest", which is an I - proposition. This is so because it involves a double negation which is equivalent to affirmation.