P(x)=x is richly colored.
Q(x)=x is large.
R(x)=x lives on honey.
T(x)=x is a hummingbird.
What are we given?
In English- All hummingbirds are richly colored and No large birds live on honey and Birds that do not live on honey are dull in color.
First, write given three propositions-
"All hummingbirds are richly colored."
∀x(T(x)->P(x))
"No large birds live on honey."
This can be simplified as- All large birds do not live on honey.
∀x(Q(x)->~R(x)) or ∀x(R(x)->~Q(x)) [p->q =~q->~p]
"Birds that do not live on honey are dull in color"
∀x(~R(x)->~P(x)) or ∀x(P(x)->R(x)) [p->q =~q->~p]
Now prove Conclusion-
In English, it should be All humming birds are small.
∀x(T(x)->P(x)) AND ∀x(R(x)->~Q(x)) AND ∀x(P(x)->R(x))
∀x(T(x)->P(x) AND P(x)->R(x) AND R(x)->~Q(x))
∀x(T(x)->P(x) AND P(x)->R(x) AND R(x)->~Q(x)) -> ∀x(T(x)->~Q(x)) [Why? because of Law of syllogism]
So, with this we come to conclusion - ∀x(T(x)->~Q(x)) which means If x is a humming bird then it is not large or All Humming birds are small.
