The Traditional Square of Opposition: Contradictories, Contraries, Sub-contraries, and Sub-alternation The Square of OppositionThe Traditional Square of Opposition: Standard-form categorical propositions having the same subject terms and the same predicate terms may differ from each other in quality, or in quantity, or in both. Any such kind of differing has been traditionally called OPPOSITION. The various kinds of OPPOSITION are correlated with some very important truth relations, as follows:1) Contradictories2) Contraries3) Sub-contraries4) Sub-alternation1) CONTRADICTORIES(A:O and E:I)2) Two categorical propositions having the same subject and predicate terms but differing in both quantity and quality are contradictories. E.g.a. All judges are lawyers. b. Some judges are not lawyers. 3) Here, exactly one is true and exactly one is false. 4) They cannot both be true; they cannot both be false.5) Similarly, a. No politicians are idealists. b. Some politicians are idealists.6) Hence, A : O and E : I are contradictories.2) CONTRARIES (A : E) The categorical propositions (A : E) having the same subject and predicate terms but differing in quality are contraries. E.g. All poets are dreamers. No poets are dreamers. They cannot both be true, but can both be false. E.g. Pakistan will win the coming game with India. India will win the coming game with Pakistan.3) SUB-CONTRARIES (I : O) Two propositions (I : O) having the same subject and predicate terms but differing in quality are sub-contraries. E.g. Some diamonds are precious stones. Some diamonds are not precious stones. They cannot both be false, although they may both be true.4) Sub-alternation Two propositions have the same subject and predicate terms agreeing in quality but differing in quantity, are called sub-alternation. E.g. (for A : I) All spiders are eight-legged animals. Some spiders are eight-legged animals. Likewise, (for E : O) No whales are fishes, Some whales are not fishes. Here, the A and E (universal) proposition are called the SUPER-altern, and I and O (particular) called the SUB-altern. In sub-alternation, the super-altern implies the truth of the sub-altern. E.g. from the following A proposition: All birds have feathers. We can hold to follow the corresponding I proposition: Some birds have feathers. Similarly, From E proposition: No whales are fishes. We can hold to follow the corresponding O proposition: Some whales are not fishes. However, the implication does not hold from the particular to the universal. E.g. From I proposition: Some animals are cats. We cannot infer A proposition: All animals are cats.The Square Of Opposition: So, we have 4 ways in which propositions may be opposed, i.e. contradictories, contraries, sub-contraries, and as sub-and super-alterns. These are represented by a diagram called:

The Square of Opposition

The Use of Square of Opposition Provides the logical basis for validating certain elementary forms of argument (immediate inferences). What is meant by immediate and mediate inferences?Mediate vs. Immediate Inference Inference: drawing a conclusion from one or more premises. Mediate Inference: when more than one premise is relied on. The conclusion is drawn from the first premise through the mediation of the second. Immediate Inference: However, when a conclusion is drawn from only one premise there is no such mediation, and the inference is said to be immediate.Immediate Inference through Traditional Square of Opposition: If an A proposition is the premise, the one can validly infer that the corresponding O proposition is false. If an A proposition is the premise, the corresponding I proposition is true. If an I proposition is the premise, its corresponding E proposition, which contradicts it, must be false.ExercisesA. If we assume that the first proposition in each of the following sets is true, what can we affirm about the truth or falsehood of the remaining propositions in each set?B. If we assume that the first proposition in each set is false, what can we affirm?1. a. All successful executives are intelligent people. b. No successful executives are intelligent people. c. Some successful executives are intelligent people. d. Some successful executives are not intelligent people. 2. a. Some college professors are not entertaining lecturers. b. All college professors are entertaining lecturers. c. No college professors are entertaining lecturers. d. Some college professors are entertaining lecturers.