I am reviewing a study guide for an introductory logic course (basic predicate, syllogistic etc.). The problem asks me to symbolize that "leprechauns exist" and prove that it is a logical truth and then critique your proof.
I decided the best symbolization was:
(∃x)(Lx)
where Lx = x is a leprechaun
I am unsure if this is the proper symbolization (would (∃x)(x=L) be better?) and unsure how one can even begin to prove this is a logical truth.
There are really so many way to do this and which one you choose largely depends on your personal preference. Here are some:
(∃x)(Lx), where Lx stands for 'x is a leprechaun' (this is your suggestion).
¬(∀x)(¬Lx), less expressive than the first, just to show that there are so many ways to express this.
(∃x)(x ∈ L), where L is the set of leprechauns
This is equivalent and very similar to saying L ≠ ∅; the set of leprechauns isn't empty.
Personally, I often prefer to use this last way, L ≠ ∅, I find it the most expressive. It isn't very common in logic circles though, I have the impression.
I wouldn't use x=L, because different leprechauns are different. L can be the set of leprechauns, or the predicate of being a leprechaun, but not simply 'a leprechaun'. Because then we could say that x ≠ L, however, x = L2 (another leprechaun).
They're the same, really. Defining a predicate Lx is implicitly defining a set L = {x | Lx} (all x for which Lx is true). And defining a set L is implicitly defining a membership predicate Lx = x ∈ L.
I am [...] unsure how one can even begin to prove this is a logical truth.
The typical way to prove that there exists (at least) one leprechaun is by pointing at it. In the case of leprechauns that may be a little difficult. You could make a claim that they exist in your head, and therefore really exist.
I agree with your first symbolization :
(∃x)(Lx),
because it seems more "natural" to assume that "leprechaun" denotes a species and not an individual; in fact, you are saying : "leprechauns exist".
But in this way we cannot prove anything; the standard semantics for first-order logic assumes that any interpretation has a not empty domain M.
This means that the formula (∃x)(x=x) is valid.
The semantics requires also that to each n-place predicate symbol P an n-ary relation P* on the domain, i.e. a set of n-tuples of members of the domain; for n=1, this must be a subset of the domain.
But nothing prevents that the said subset is the empty set; thus we cannot prove that "leprechauns exist".
Things are different if we use an individual constant l; in this case we can prove, starting from the equality axiom : x=x, the formula :
(∃x)(x=l);
in this case, we are consistent with the semantical specifications, because for each constant symbol c, the interpretation specifies a member c* of the domain M.
This means that, having and individual constant in our language, amounts to assuming that this symbol is a name denotong an object of the domain, and thus assuming the existence of the said object.
The problem asks me to symbolise that 'Leprachauns exist'
If this is all that is asked for; the simplest is:
L: Leprachauns exist
We use the letter L to remind ourselves that Leprachauns are involved in this sentence - it's a mnemonic device.
I'm not sure even how to prove this is a logical truth
Which is already quite close to a good critique; I mean if someone came to you and said 'I saw some leprechauns in the garden' - you'd think perhaps they were starting on a shaggy dog story.
You already know from experience there aren't any. I mean, it's not a logical truth - so your intuition and judgement was already on the right track ...
