9.6.3. Third Statement: Dull in Colour

Now what about the third statement? So I do not need to introduce new predicates here because I have already introduced the predicate H(x) over to represent that bird x lives on honey and I have already introduced the predicate C(x) to denote that bird x is richly coloured.

In this chunk, we introduce a new statement about birds, specifically about their coloration. The existing predicates H(x) and C(x) are reused here. H(x) indicates whether a bird lives on honey, while C(x) indicates whether a bird is richly colored. The task is to represent the statement about dullness in color using the already defined predicates, showing a connection between color richness and the habitat of birds.

Consider a situation where you are classifying fruits: you might use one label for 'sweet' and another for 'ripe.' If you find out 'ripe' can refer to whether a fruit is ready to eat (like C(x)) and 'sweet' might talk about a fruit's taste (like H(x)), then these labels can be reused for different kinds of fruits. Thus, you can mention that if fruits are not 'ripe', we can expect them to be 'sour.'

So dull in colour will be negation of C(x). Now the question is, is this universal quantified statement or existential quantified statement? It turns out that it is a universally quantified statement because I am making or asserting this property for all birds, I am not saying it just for some specific bird.

Here we derive what 'dull in color' means, which is defined as the negation of being richly colored (negation of C(x)). The core question is whether this statement about dullness applies to all birds or just some. It is established that the statement applies universally—every bird is considered when making this assertion. This step emphasizes the importance of correctly identifying the scope of our statements—something that influences whether the statement is universal or existential.

Think about a rule applied to all students: 'All students must hand in their homework on time.' This is a universal statement because it includes every student without exception. If we say 'Some students might not hand in their homework on time,' that would be an existential statement, targeting just a subset, not the whole.

So you can imagine that another way to re-interpret this statement is I am making the statement that for all birds x, if bird x does not live on honey then it is dull in colour. So there is “if then” involved here and it is a universal quantified statement.

In this chunk, we redefine the statement about dull coloration. By identifying that dullness is contingent on whether a bird lives on honey, the logical structure reveals an 'if-then' relationship. The statement translates to: for any bird, if it does not reside in a honey-rich habitat, it is then classified as dull in color. This reinforces the presence of logical implications in universal quantification.

Imagine a house rule: 'If it is raining, then you must wear a raincoat.' This is a general rule applied to anyone going out during rain. Here, those not going out in the rain don’t need a raincoat. Similarly, birds not living in specific environments take on specific characteristics, creating a clear logical relationship.

And that is why this will be represented by this expression and what is the last statement that hummingbirds are small, again I do not need any new predicate here, hummingbirds is represented by the predicate B(x) and L(x) was used to represent that bird x is large so negation of L(x) will represent that the bird x is small.

The final point covers how to represent the relationship between hummingbirds and their size. Since we already have predicates for size classification, we simply use the negation of the large bird predicate (L(x)) to classify them as small. Each of these logical representations shows methodical thinking in predicate logic, where existing definitions are utilized for new insights or characteristics.

Consider how we describe various vehicle sizes using simple labels: 'motorbike' for small and 'truck' for large. If we know what a truck is (L(x)), we can directly conclude something is small when it’s not classified as a truck. This represents how using existing knowledge can simplify new classifications.

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