The modern interpretation of what we mean by a statement like “all unicorns are one-horned animals” is that we aren’t making the assertion that any unicorns exist. If any did happen to exist, sure, they’d be one-horned animals, if our proposition is true, but we’re reserving judgement about whether they do exist. If we don’t like the way the natural-language interpretation of the proposition leads us, we might be satisfied by saying it’s equivalent to saying, “there are no non-one-horned animals which are unicorns”, and that doesn’t feel quite like it claims unicorns exist. You might not even come away feeling there ought to be non-one-horned animals from that sentence alone.
My understanding is that mathematicians and philosophers have adopted the same view on this issue, that an assertion of “all (something) have (some property)” isn’t also making the implicit assumption that (something) exists. I can venture why mathematical logic would find it convenient to talk about the properties of something independent of whether that thing exists. For example, in many problems, we make the supposition that there is a solution and then try to deduce the properties of that solution. If we’re lucky, this gets us to either the correct solution, or to a range of possible solutions, or to a set of contradictory properties that let us know there is no solution. It’d be tiresome to write out — it’d be tiresome to think — of the whole argument with “any solution to this problem, if it exists” tripping over every line, so, just “any solution” will do if we keep the footnote that we’ll also need some reason to suppose a solution exists.
So we can run with the idea that “all unicorns are one-horned animals”, and since that’s equivalent to “there are no non-one-horned animals which are unicorns” feel confident it’s true since we aren’t going to turn up any unicorns that are non-one-horned animals. Which is all neat enough, until we ask about whether it’s true that “there are no one-horned animals which are unicorns”? Since there aren’t any, the proposition is true. But doing that reversal again gives us “all unicorns are not one-horned animals”, and we claim that’s true again. At this point we start suspecting logicians are just being difficult and wonder why we’re talking to them. Many Intro to Logic students stop listening entirely and never recover.
But it’s where we are. In contemporary logic, we don’t suppose that the proposition “all (something) have (some property)” implies the thing exists, unless that property is “exists”. But if the (something) doesn’t exist, then there can’t be anything which lacks (some property) and yet is also that (something), so, the proposition is true. And it’s true whatever the (some property) is: “all unicorns are one-horned animals”, and “all unicorns are dressed in plaid”, and “all unicorns are never dressed in plaid”, and “all unicorns are not one-horned animals” are all, equally, uniformly true.
There are at least two other convenient ways of rephrasing this, that I think are worth looking at.
(I also understand there is a deep, rich field of philosophy that studies problems about things in fictional worlds, eg, what does it mean to say it’s true that Captain Kirk’s middle name is “Terry”? However, what I know of this I know second-hand, and I know I’m not competent to talk about that past saying that there are people who do have good ideas about it.)
nebusresearch 
Yeah, When people are doing this formally they usually define a ‘universe’ in which the statements, or parts of statements, are made. I guess an simple example would be if I said “it’s red”, you, the listener, would have to be working in a world where ‘it’ referred to the same object. Basically, every statement has underlying assumptions that affect how you interpret it formally. I recommend this book, it was in my bathroom for a while and explains things rather well.
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