Connotation and denotation are closely interrelated. The following points should be noted with regard to their relation.
(1) Connotation of a term determines its denotation, but denotation of a term does not uniquely determine its connotation. One may know the connotation of a term without knowing its denotation. For instance, even though there are no unicorns in the world, people know the meaning of the word "unicorn" in the sense of knowing its connotation.
'Unicorn' means 'a horse like animal with one long horn on its forehead'. So, one who knows this, knows the connotation of the term, although no one has ever come across any unicorn in the world. Further, two terms having different connotations may have the same denotation. For example, consider the two terms 'equilateral triangle' and 'equiangular triangle'.
The term 'equilateral triangle' means a plane figure enclosed by three straight lines of equal length. The term 'equiangular triangle' means a plane figure enclosed by three straight lines that intersect each other to form equal angles. Notice that these two terms have different connotative meaning but both denote exactly the same set of figures.
So, connotation of a term determines its denotation. In other words, the connotation of a term provides us a set of criteria for deciding whether an object falls within the extension of that term. For example, when we come across an animal in a zoo, we decide whether or not it belongs to the class of leopards by seeing whether or not it has the relevant features of a leopard.
(2) Connotation and denotation vary inversely. When connotation increases denotation decreases, and when denotation increases connotation decreases. Some logicians call it the law of inverse variation. Consider the following sequence of terms:
2. ' aquatic animal', and
3. ' aquatic animal with fins'.
These terms have been arranged in the order of increasing connotation. The connotation aquatic animal' is greater than that of 'animal'. Aquatic animals have all the qualities common and essential for something to be an animal and in addition to those properties they have the property of living in water.
Similarly, the connotation of 'aquatic animal with fins' is greater than that of "aquatic animal', since aquatic animals with fins have all the properties of aquatic animals plus the property of having fins. So you can see that the three terms are in the order of increasing connotation. You will also notice that these terms are in the order of decreasing denotation.
The number of aquatic animals is less than that of animals, and the total number of aquatic animals with fins is less than that of aquatic animals. Thus when we arranged the terms in an order of increasing connotation, the terms also automatically got arranged in an order of decreasing denotation.
Similar relation of inverse variation can also be noticed if we arrange terms in an order of decreasing connotation.
1. 'aquatic animal with fins'
2. 'aquatic animal'
In the above sequence of terms connotation progressively decreases while there is a progressive increase in the denotation of the terms. Thus we have noticed that if we increase the connotation in a series of general terms by including more features, the denotation of the corresponding terms in the series tends to decrease, and if we decrease the connotation in any series of terms, the denotation of the corresponding terms tends to increase.
There are, however, some exceptions to this relation of inverse variation. In some cases an increase in connotation does not result in the decrease in denotation. We observed earlier that the two terms having different connotation might have the same denotation. We cited the case of the two terms 'equilateral triangle' and 'equiangular triangle' to illustrate that point. The same example can be cited to illustrate an exception to the' law of inverse variation'.
Although the term 'equilateral and equiangular triangle' has greater connotation than that of the term 'equilateral triangle', there is no variation in the denotation of these terms. This illustrates the point increase in connotation is not always accompanied by decrease in denotation.