Cat looking in mirror sees lion in mirror" - is that in any way correct?
For example, if I were to write "A cat is a cat. A cat is a lion" and then if someone were to write "A lion is a lion. A cat is a lion" would it be correct to assume that "A lion is a lion. A cat is a lion" would be "correct"?
No, it's not correct. You need to define each term in a context. The definitions are:
A cat is a feline mammal. A cat is also a carnivore. A lion is also a
feline, a predator. A lion is also a mammal.
This is from my textbook on Animal behavior. A very simple definition.
So if you write "A lion is a lion. A cat is a lion" it is incorrect because the terms have different meanings. But if you say "A lion is a cat. A lion is a cat" or "A lion is a cat. A lion is a lion", the sentence is correct.
The most basic way to say it is "A lion is a cat. A cat is a lion".
To add to what @D-Shaw sd, the problem is that, in "A lion is a lion. A cat is a lion" the two predicates are different in nature. One is an essential and categorical property, and the other a more flexible and contingent property.
Essential and categorical properties
A cat is a cat, a lion is a lion.
For every cat, there is exactly one lion, for every lion there is exactly one cat. It is possible for two cats to be alike, but it is impossible for two lions to be alike.
To put it in logical terms, you may use
(Cat) a cat is a cat
(Lion) a lion is a lion
to convey the idea that a cat is just a cat and a lion is just a lion. Note, however, that a cat is not, in this view, a lion.
More than one of each
There are several lions, and several cats. Some cats are lions. We can use the word "lion" to denote the property of being a lion.
(A lion is a cat) every lion is a cat. (A cat is a lion) every cat is a lion.
Contingent and flexible
In the second sentence, "cat" and "lion" stand for different property types. The first says "The cat is a cat", and the second says "The lion is a lion". In the second sentence, the predicates are not categorical predicates, and the sentences would not make sense if they were. Instead, they are what are called contingent properties, and they are different from essential properties.
(Lion is a cat) some lions are cats. (Cat is a lion) some cats are lions.
(Cat is not a lion) some cats are not lions. (Lion is not a cat) some lions are not cats.
The second of the two sentences could be sd, if we wish to give the words "cat" and "lion" the definition "contingent property" (or "flexible property"). The first sentence cannot be sd.
If we want to show that there is at least one cat that is not a lion, it is possible, but we need to use a property which is different from lionhood and is not a lion. So, we use a property which is more flexible than lionhood and which we call "cathood". This property is contingent, and it is possible for a cat to not be a cat.
Some cats are cathood. (Cat is not a cat) some cats are not cathood.
Some cats are not cathood. (Cat is a cat) some cats are cathood.
Note that, in the second sentence, the properties cathood and lionhood are the same (because, strictly speaking, there are no other properties). This is because we can only talk about cathood and lionhood in a contingent way, not in a categorical way.
"A lion is a lion. A cat is a lion" says that a lion is a lion, and a cat is a lion. "A lion is a cat. A lion is a cat" says that the predicates are different in nature. "A lion is a cat. A cat is a lion" says that the pred