I am constantly reminded what a challenge the basic building blocks of English are for students who study it. After all, nothing is more basic and few things are more complex than the verb ‘to be’. What sentence can you make up from members of the set A:
A = {lion, white, is}
Trivially:
(1) A lion is white.
(2) There is a white lion.
(3) The lion is white.
Sentence (2) we are told implies existence. I agree fully, but such explanation is misleading without qualification. Students and teachers may be led to believe that (1) and (3) do not imply existence, which is false. They do! OK, (1) is not much of a sentence, and (2) is only a slight improvement on (1), but what is a matter of elegance in grammar may sometimes be an irrelevance in logic.
The question is, what does it mean to say that (2) implies existence? And what about (1) and (3)?
One way to tackle the problem is by responding with ‘So what?’ to someone uttering (2) to you, out of the blue. If your response is indeed ‘So what?’, or ‘Why are you telling me this?’, you already know what it means for ‘there is’ to express existence. The point is existence is so basic we don’t need to talk about it. If there is a white lion, let’im be.
Language, however, wrongfoots you. It gives you (2) and (3), never mind (1), with a verb ‘to be’ sitting squarely in the middle. Some philosophers call it referential tautology. The implication of existence is already present in my uttering the word ‘lion’. The verb ‘to be’ or ‘to exist’ adds nothing.
In first-order logic (2), as (1), can be represented as:
(2.1) (∃x)(Lx • Wx)
where Lx – ‘x’ is a lion, and Wx – ‘x’ is white, and where ∃ is an existential quantifier. The sentence thus reads: there exists an ‘x’ such that ‘x’ is a lion and ‘x’ is white. We have in fact used the word ‘is’ in our paraphrase, but nothing in our symbolic translation corresponds to it! To say that ‘Lions exist’, ‘Some lions exist’, ‘There is at least one lion,’ or, if you can bear it, that ‘Lions are,’ or else that ‘There are lions,’ is simply: (∃x)Lx.
Note that the plural in the preceding paragraph is no more than a variant of ‘There is a lion’ or ‘A lion is,’, which is true. It is a kind of minimal or limiting plural. It definitely does not suggest that ‘All lions exist,’ for when put like this the sentence is meaningless in English. The universal sentence (x)Lx, which can be paraphrased: for any ‘x’, ‘x’ is a lion, or everything is a lion, is clearly false. The existential and the universal would have the same truth value if and only if there was only one object in the universe, and that object was not necessarily a lion either.
(3.1) (∃x)[Lx • (y)(Ly ⊃ y = x) • Wx]
Translation (3.1) is just like (2.1) except for the bit in the middle, and is a representation of sentence (3): The lion is white. Just like (2.1) it is says that there exists an ‘x’ such that ‘x’ is a lion and ‘x’ is white, but it also adds (the bit in the middle) that if any ‘y’ is a lion then that ‘y’ is identical to our ‘x’, and this is what makes our lion ‘unique’, ‘the only lion’, or simply ‘the lion’.
Summing up, all three sentences assert existence, it is just that (1) and (2) assert nothing but the existence of white lions. Sentence (3) tells us more: it tells us that there is a unique lion. Logical notation thus makes explicit to a non-English speaker what the English language doesn’t by merely replacing ‘a’ with ‘the’: it adds one extra piece of information – the uniqueness claim.
And if (1) and (2) do not sound interesting, that is precisely because, when taken out of context, they tell you about nothing other than that somewhere a white lion exits – something you didn’t expect to be told without first asking about it. Sentences about existence become interesting only when we have reason to doubt existence: There is life on Mars; There is a God; and so on.