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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.

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There are really so many way to do this and which one you choose largely depends on your personal preference. Here are some:

Predicate

(∃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.

Set

(∃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).


Sets and predicates

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.


Proof

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.

  • 1
    L ≠ ∅ was not used in the undergraduate logic course I took. I'm guessing the logic taught in most undergraduates in philosophy is equally truncated – virmaior Dec 14 '15 at 10:14
  • That's a pity, it is shorter and (in my opinion) more expressive. – Keelan Dec 14 '15 at 10:36
  • I'm familiar with the empty set, so no problem there, and thanks for confirming my symbolization and rejection of x=L. But yes, I can't imagine proving leprechauns exist logically. Exhibiting the existence of a leprechaun by pointing it out is surely empirical. I'm wondering if this is meant to be a poorly phrased Curry Paradox question? – user115411 Dec 14 '15 at 18:25
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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.

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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 ...

  • Why the down-vote? The question specifically says using syllogistic or predicate logic; one usually doesn't go for the more complex option if a simpler one is possible; and that it's not a logical truth is surely obvious to the scientifically and logically literate; as a mere exercise in using the rules of logic formally expressed one might say more - but there isn't sufficient said in the question to go on. – Mozibur Ullah Dec 14 '15 at 14:45

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