# Key notes on the five important classifications of Propositions

Propositions may be classified into various classes according to different principles, such as the principles of composition, relation, quality, quantity, modality, significance etc. But for our purpose it suffices to consider the classification of propositions according to relation, significance, quality, quantity and both quality and quantity.

(a) Classification of propositions according to relation:

According to relation, propositions are of two types namely (i) categorical and (ii) conditional.

This distinction depends on the nature of the relation between subject and predicate of the proposition. A proposition is called categorical if the relation between subject and predicate of the proposition is either affirmed or denied without any condition or restriction.

In other words, a categorical proposition is one in which the predicate in either affirmed or denied of the subject unconditionally. For example, "All men are mortal", "Some men are rich", "Gopal is the author of many books" are propositions in which the predicate is affirmed of the subject without any condition. So, they are all categorical propositions.

Similarly, "No man is perfect", some men are not rich", "Gopal is -not the son of Ram" are propositions in which the predicate is denied of the subject without any condition. Hence, they too are categorical propositions.

On the other hand, in case of conditional propositions, the relation between subject and predicate holds conditionally. It is of three types, namely (i) Hypothetical, (ii) Alternative and (iii) Disjunctive. A hypothetical proposition is a conditional proposition, having the form "if P then Q", where "P" and "Q" stand for any propositions.

In this case, the "if-clause" and "then-clause" of the proposition are respectively called antecedent and consequent. The antecedent (or if clause) always states the condition for asserting the consequent (or the then-clause), For example, I shall go if he comes", here "he comes" is the antecedent and 'I shall go' is the consequent of he above proposition.

Moreover, since "my going" depends on "his coming", "I shall go if he comes" is a hypothetical proposition. The same is the case with the propositions "I shall not go to the college if it is raining". Here "my not going to the college" depends on "raining". So "it is raining" is the antecedent and "I shall not go to the college" is the consequent.

An alternative proposition is a kind of conditional proposition having the form "either P or Q", where P and Q are called alternatives representing propositions. In this case, we cannot assert or deny both the alternatives. The assertion of one of them depends on the denial of the other and vice-versa. For example, "Either Gopal is a student of N. C. College or he is a student of City College", expresses an alternative proposition. Here both the alternative cannot be asserted or denied simultaneously, i.e. given that this proposition in true, if we say that Gopal is both a student of N. C. College as well as City College then we would be making a false statement. The same is the case if we deny both the alternatives. Thus in case of an alternative proposition if we assert one alternative then the other alternative must be denied art vice versa.

A disjunctive proposition is a kind of conditional proposition having the form "either P or Q", where P and Q are called disjuncts, representing propositions. In this case, we cannot deny both the disjuncts, though both of them can be asserted. A disjunctive proposition holds good even if one of its disjuncts is asserted. For example, consider the proposition; either Ashok does yoga or pranayam. The proposition is false if Ashok does not do either of the two. But there is no harm if he does both. In other words, a disjunctive proposition holds good if one of its disjuncts is asserted and false if both are denied.

Though propositions according to relation are divided into categorical and conditional yet the ancient logicians, including Aristotle, were interested only with categorical propositions. This is because, any conditional proposition is built out of categorical propositions by use of propositional operations like "If....... then", either..... Or.....” etc.

Hence, categorical proposition is the most basic type of proposition and we shall now develop a system of deductive logic based on the notion of categorical proposition.

(b) Classification of Propositions according to significance:

According to significance, a proposition is either analytic (or verbal) or synthetic (or real). We know that a proposition (or more precisely a categorical proposition) expresses a relation between subject and predicate terms.

A proposition is called analytic (or verbal) if and only if either the subject or predicate terms are equivalent in meaning or the predicate states part of the meaning (or connotation) of the subject.

In other words, an analytic proposition is one in which the predicate either partially or wholly contained in the subject. The truth or falsity of such propositions depends entirely on the meaning of the subject as well as the predicate term occurring in the proposition. For example, "Triangle is a plane figure bounded by three straight lines," "All bachelors are unmarried", "All circular figures are figures", "All red things are colored", Black cats are black" are examples of analytic proposition. The truth of such a proposition depends on the meaning of the subject, predicate and copula present in the proposition.

Thus, to accept or reject such a proposition we don't have to see how the things really are on the world but only to know the meaning of the words and terms present in it. Thus to know the truth of "All bachelors are unmarried", we don't have to look at the world to see whether bachelor is unmarried or not. If we know the meaning of terms ('all', 'bachelor', and unmarried") and the copula (is) then we would be in a position to ascertain its truth.

On the contrary, a proposition is called synthetic (or real) if and only if its truth or falsehood depends on facts. In case of synthetic proposition the subject and predicate terms are non-equivalent in meaning. In such a case, the predicate states something new about the subject in the sense that the predicate is not contained either implicitly or explicitly in the subject.

To know the truth of synthetic proposition say "It is raining", "The grass is green", All men are mortal", we have to look at the world to see how the things really are. Without coking at the world, we cannot either reject or accept the truth of a synthetic proposition.

(c) Classification of propositions according to quality:

According to quality, a proposition is either affirmative or negative. A proposition is called affirmative if and only if the predicate is affirmed of the subject. For example, in the proposition "Ram is a boy", the predicate 'a boy' is affirmed of the subject "Ram".

A proposition is called negative if and only if the predicate is denied of the subject. For example, in the proposition 'Ram is not rich” the predicate ‘rich’ are denied of the subject. In case of affirmative proposition, the copula is affirmative whereas in negative propositions the copula carries the sign of negation. But in case of universal negative propositions, the sign of negation is no: attached to the copula but to the subject. For example in "No man is perfect' or 'No S is P' the sign of negation is attached to the subject of the proposition.

However, this is mere a convention of English language. Note that the quality of the hypothetical proposition depends on the quality of the consequent. For example, the negation of "If he comes, I shall go' would be 'If he comes, I shall not go'. Similarly the negation of 'if A is B then C is D' would be 'if A is B then C is not D. Briefly, we may say that if the consequent of a hypothetical proposition is negative (or affirmative), the hypothetical proposition in question is negative (or affirmative). To repeat, the quality of a hypothetical proposition depends on the quality of the consequent.

(d) Classification of proposition according to quantity:

According to quantity a proposition is either universal or particular. A proposition is called universal if the predicate of the proposition is either affirmed or denied of the entire denotation of the subject. On the other hand, a proposition is called particular if the predicate of the proposition is affirmed or denied of a part of the denotation of the subject. For example, in the proposition "All men are mortal", the predicate 'mortal' is affirmed of the entire denotation of the subject. Here the word 'all' refers to the entire denotation of the subject term 'man'.

Similarly, in the proposition "Some men are rich", the predicate 'rich' is affirmed of a part of the denotation of the subject term 'men'. Further, in the proposition "Some men are not rich" the predicate term 'rich' is denied of a part of the denotation of the subject term. In "No men is perfect" the predicate term is denied of the entire denotation of the subject.

Thus the words like 'All', 'No', 'Some' etc are quantity-words specifying the quantity of the term to which it is attached. 'All' and 'No' are marks of universality and the word 'some' indicates particularity. In a logical proposition the quantity - words (or the quantifiers) are attached only to the subject term. According, "All men are mortal", "No men is perfect" are the samples of universal proposition where as "Some men are rich", "Some men are not rich" are examples of particular propositions.

Note that the classification of propositions that we have introduced so far is mutually exclusive and taken it is jointly are exhaustive. For example, the division of proposition according to quality is either affirmative or negative. This division is mutually exclusive because it is impossible of find a proposition, which is both affirmative as well as negative.

In other words, if a proposition is affirmative then it cannot be negative and vice versa. Similarly, this division is also exhaustive, which means that there is no proposition which is neither affirmative nor negative. These two conditions jointly say that any proposition must have exactly one quality i.e. either it is affirmative or negative. The same is the case with other divisions of propositions like division according to relation and division according to quantity.

(e) Classification of propositions according to both quality and quantity:

We have already discussed the classifications of propositions according to quality as well as quantity that may be repeated for convenience. According to quality, the propositions are of two types i.e. affirmative and negative and according to quantity, propositions are universal and particular. From these two classifications, we obtain a third type of classification of propositions according to the mixed principle of quality and quantity.

Popularly these four types of propositions are respectively known as A, E, I and O propositions. For example, "All men are mortal" is an A-proposition or universal affirmative proposition. It is a universal proposition because the predicate term 'mortal' is affirmed of the entire denotation of the subject term. It is also affirmative as the predicate is affirmed of the subject.

Thus, for an A-proposition we have to verify two conditions i.e. firstly the predicate must be affirmed of the subject and secondly, it must be affirmed of the entire denotation of the subject. The structure of A-proposition is 'All S is P' where 'S' and 'P' stand for subject term and predicate term respectively.

Universal negative propositions are called E-propositions. 'No man is perfect' is an example of an E-proposition. In this case the predicate term 'perfect' is denied of the entire denotation of the subject term 'man'. Since it is a case of denial of the predicate, it is negative and since the denial is about the entire denotation of the subject, it is universal.

Hence, the proposition "No man is perfect" is an example of universal negative proposition or E-proposition. It has the form "No S is P", where 'S' and 'P' stand for the subject and predicate terms respectively.

"Some men are rich" and "Some men are not rich" are examples of I and O-propositions respectively. In the case of former the predicate term 'rich' is affirmed of a part of the denotation of the subject, where as in case of the predicate term is denied of the part of the denotation of subject.

Hence, the former is particular affirmative and the latter is particular negative. The structures of I and O-propositions respectively are "Some S is P" and 'Some S is not P', where S' and 'P' stand for subject and predicate terms. For our convenience we may summaries above discussion in the following table.

 Types of Proposition logical name Structure Concrete example. Universal Affirmative A-Proposition A11S is P All men are mortal. Universal Negative E-proposition No S is P No man is perfect. Particular Negative O-proposition Some S is not P Some men are not rich. Particular Affirmative I-proposition Some S is P Some men are rich.

Note that for Aristotle these four types of categorical propositions (such as A,E,I and O propositions) are called logical propositions because they exhibit two essential or unavoidable features namely quality and quantity.

Further, a categorical proposition in which logicians are interested exhibits its logical constituents explicitly. It states its quantity, quality, subject and predicate in an explicit manner. These four constituents in a categorical proposition should be written in the following order, quantifier, subject term, copula (exhibiting the quality of the proposition) and predicate term.

The categorical proposition where all these constituents are specified is called a logical proposition. The general schema or the skeleton of a logical proposition is as follows:

Quantifier   Subject term    Copula Predicate term

Words such as "all" "none" and 'some' are quantifiers which express universality and particularity. Here we may note that quantifiers are attached only to the subject of a categorical proposition. Moreover, A, E, I and O-propositions are recognized as the most basic for Aristotle because for him any proposition in our ordinary language may be reduced to either of these above four forms without any change of meaning.