9.6.2. Second Statement: Large Birds

Now, let us take another example to make the concepts more clear here you are given an English argument a set of English statements and you have to convert everything into predicates and your domain here is a set of birds because I am stating several properties about birds here, so my domain is set of birds.

In logical reasoning, it is essential to first identify what the domain is when dealing with predicates. A 'domain' refers to the set of items that we are making statements about. In this example, the domain is established as ‘birds’ since all statements that follow will pertain to various characteristics or properties of birds.

Think of it like a classroom environment. If the subject being discussed is animals, then the domain might include all sorts of animals. Similarly, if we switch to discussing just birds, then our 'classroom' is now narrowed down to only birds, just as if we were focusing on only students in a particular grade.

The first statement is all hummingbirds are richly coloured. So, let me introduce predicates B(x) and C(x) here. So B(x) will be true if the bird x is a humming bird. Whereas the predicate C(x) will be true if and only if the bird x is richly coloured that is the definition of my predicates B(x) and C(x) and that is the case and this statement will be represented by for all x, P(x) → C(x).

We introduce two predicates: B(x) signifies that x is a hummingbird, and C(x) indicates that x is richly colored. The statement 'all hummingbirds are richly coloured' is represented logically as 'for all x, if x is a hummingbird, then x is richly coloured.' This fits the form of a universal quantification, which states that a property holds for all members of a specific group.

Imagine that every cat you see is tabby-colored; this is akin to saying 'all cats are tabby.' When we state it in logical form, we make it clear that every member of the 'cat' group possesses the property of being tabby-colored.

The second statement is no large birds live on honey. So I have to introduce a predicate L(x); where L(x) will be true if and only if the bird x is a large bird and my predicate H(x) will be true if and only if the bird x lives on honey. Now again, if you closely see here, there is a universal quantification involved.

In this example, the statement 'no large birds live on honey' means that there aren't any birds that are both large and live on honey. We define L(x) for large birds and H(x) for those that live on honey. The logical expression can be depicted as: 'for all birds x, if x is large, then x does not live on honey.' This again showcases universal quantification where the property must hold for every single bird in the domain.

Consider this scenario: if I say, 'no dogs can drive cars,' it implies that across all dogs, you won't find any who can drive. Similarly, we're asserting a general rule about the attributes of large birds concerning honey.

Now, if you closely see if I apply the rules of equivalence for predicates here if I apply the De Morgan’s law for predicates, which I have discussed in the last lecture. Then I can take this negation inside and when I take negation inside they are exists gets converted into “for all” and this negation will also go with L.

Using De Morgan’s laws allows us to change the structure of logical statements while maintaining their truth-values. For instance, negating the statement 'there exists a large bird that lives on honey' converts it to 'for all birds, if a bird is large, then it does not live on honey.' This conversion emphasizes the universality of the claim, thus reinforcing the assertion about the behaviors of large birds.

Think of it like saying 'not everyone likes pizza.' By rephrasing it as 'if you like pizza, you are not everyone,' you're clarifying and emphasizing that the pizza-lovers and non-lovers exist within the larger ‘everyone’ group.

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