As we have seen that there are four types of categorical propositions namely A, E, I and 3-propositions having respectively the logical structure of the form “All S is P”, “No S is P”, “Some S is P” and “Some S is not P”.

Since our main goal is to develop a theory of inference involving these propositions, it is necessary to know the relations that hold between these 7 repositions. The square of opposition of propositions exhibits all the possible relations that obtain between A, E, I and O-propositions.

We may define opposition of propositions as a kind of relation obtaining between two propositions having the same subject and same predicate but offering with respect to quality or quantity or both (i.e. both in quality as well as in quantity).

To obtain all types of oppositions, let us draw a square representing all the four types of repositions namely A, E, I and O. This is called the square of opposition of propositions.

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Here, our representation of propositions by use of a square is such that the upper two tips of the square represent the two universal propositions ‘A’ and ‘E’, while two lower tips of the square represent the two particular propositions ‘I’ and ‘0’.

Further, two affirmative propositions would be on one side of the square (in the present case they occupy the left side of the square) and other two negative propositions are on the right side of square. As it is shown by use of the square, there can be four types of oppositions of propositions namely (I) Contradictor (ii) Contrary, (iii) Sub contrary and sub alternation. Let us discuss each of these.

(i) Contradictory Opposition

This kind of opposition obtains between two propositions having the same subject and same predicate but differing both in quality as well as in quantity. Thus, the pair, A and O is contradictorily related. Similarly, E and I are also contradictorily related. Note that A and 0- propositions and so also E and I propositions are different from each other with respect to both quality and quantity.

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A and O-propositions are qualitatively different because A-proposition is affirmative and O-proposition is negative. They are also quantitatively different as A-proposition is universal and O-proposition is particular. Hence, if one is true then the other must be false and also conversely. This means that they cannot be true together nor false together.

For example, “All judges are lawyers” and “Some judges are not lawyers” cannot both be true together and they cannot be false together. If one is true then the other must be false and vice versa. Thus we derive the following logical principle from the denial of an A-proposition. The denial of an A-proposition yields an O-proposition with the same subject and predicate.

For example, “It is false that all S is P” will be equivalent to “Some S is not P”. Similarly, the denial of an O-proposition yields an A-proposition. For example, “If is false that some S is not P” will be equivalent to “All S is P” Similar remarks can also be made wish regard to the relation between E and I propositions having the same subject as predicate which are also contradictorily related.

ii) Contrary Opposition

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Contrary opposition is a kind of opposition that holds between two universal propositions having the same subject and same predicate but differing only in quality. Thus, A and E- propositions are contrarily related. Here, the truth of A-proposition implies the falsity of E- proposition and the truth of an E-proposition implies the falsity of an A-proposition. But from one falsity of either one, nothing about the other can be inferred. The truth of either one implies lie falsity of the other. This means in case of contrary opposition, both cannot be true together but can be false together.

For example, “All poets are emotional” and “No poets are emotional” are contrarily related. Because the truth of “No poets are emotional” implies the falsity of “All poets are emotional” i.e. the truth of either one implies the falsity of the other. But, if one is raise, the truth value of the other remains undecided.

Thus we have the logical principle namely the truth of A-proposition implies the falsity of E-proposition but not vice versa, and similarly me truth of E-proposition implies the falsity of A-proposition but not vice versa, (of course in both the cases the subject and the predicate remains the same).Thus we see that A and E- propositions are contraries that can be false together but they cannot be true together.

Sub-Contrary Opposition

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Sub-contrary opposition is a kind of opposition holding between two particular impositions having the same subject and same predicate but differing only in quality. Thus, I and O-propositions are sub-contrarily related. Here the falsity of one implies the truth of the other. In other words, if an I-proposition is false then the corresponding O-proposition is true. Similarly, from the falsity of an O-proposition, we can infer the truth of the corresponding I- proposition.

On the other hand, from the truth of one nothing can be inferred with regard to the truth or falsity of the other. Thus, I and O-propositions can be true together but they cannot be raise together.

Sub-Alternation Opposition

Sub alternation opposition is a kind of opposition between two propositions having the same subject and same predicate that have the same quality but differ in quantity. This opposition holds between a universal proposition and its corresponding particular proposition. This opposition obtains between A and I-propositions as well as between E and O-propositions.

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In other words, the opposition between a universal proposition and its corresponding particular proposition is called sub-alternation. Technically speaking the universal proposition is called super-altern and the corresponding particular proposition is called subaltern. In this case, we say that (I) from the truth of super-altern, the truth of subaltern follows but not vice versa.

Hence, the logical principle with respect to sub alternation is (i) the truth of A-proposition implies the truth of I-proposition but not vice versa and (ii) Similarly the truth of E-proposition implies the truth of O-proposition but not vice-versa.

Thus, we can notice that the principles so obtained from the square of opposition of propositions provide the basis for validating certain immediate inferences. For example, given an A-proposition to be true, we can deduce that the corresponding O-proposition is false. Again from the truth of A-proposition we can deduce that the corresponding I-proposition is true.

Immediate inferences based on the square of opposition of propositions are given at the end of the section in a tabular form. Given the truth or falsehood of any one of the four types of categorical proposition, the truth or falsehood of some or all of the other can be immediately deduced. For listing of immediate inferences based on the square of opposition of proposition, we may introduce these following notations and conventions.

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i) We may rewrite the standard forms of categorical propositions such as ‘All S is P’, ‘No S is P ‘Some S is P and some S is not P as ‘S a P’, S e P’, ‘S i P’ and ‘S o P respectively.

ii) Let A be any categorical proposition. Then we express ‘A is true’ just by writing ‘A’ and we express ‘B is false’ just by writing ‘~B’. For example if ‘S a P is true’ then we express it just by writing ‘S a P’. Similarly if ‘S a P is false’, then it can be expressed b\ the expression ~(S a P), where ‘-‘ is the symbol of negation.

iii) Let the symbol ‘|—’ stand for the relation of logical consequence or implication. Thus, we may read the expression “S a P |— S i P” as ‘the truth of S i P follows from the truth of S a P |— S a P true, the truth of ‘S i P’ follows. Now we may list the immediate inferences based on the square of opposition of propositions.