Formalising the sentence "someone is plotting against me"

Question

Someone is plotting against me.

Can the above sentence be translated into predicate logic without using the existential quantifier? If not, is it because the sentence is self-referential?

Answer 1

@Snared provided the right answer but I thought I'd give a bit of additional information to clear up seeming confusion. I am by no means an expert so others may chime in too.

You asked to formalise without the existential quantifier, but let us start by doing it so that we know what we are not allowing ourselves to do.

When we say "Someone is plotting against me" we probably mean "There exists a person who is plotting against me."

More verbosely, we could say "There exists an x for which x is a person and x is plotting against me."

That begs the question whether "x is a person" is needed? To quantify something you need to have some domain of concepts in mind over which you are quantifying. (Further reading on Domain of Discourse)

So let's assume, for argument's sake, that the domain you are discussing is people. Then you only really need to say "There exists an $x$ for which $x$ is plotting against me." because in the context of this discussion, we assume x is a person.

$$\exists x, plotting(x, me)$$

This statement cannot be self-refferential (i.e. because the things that the statement refers to are people not statements). This is in stark contrast to a statement about statements (e.g. this statement is true).

Now that we have that out of the way, we can rephrase the statement using the universal quantifier: "It's not true that for every x, x is not plotting against me."

$$\lnot\forall x,\lnot plotting(x, me)$$

Other things you may have had in mind

You might be refering to something like the Barber's Paradox. Perhaps you heard someone say something like "There exists a person x for whom the following is true for each person y: if-and-only-if y is not plotting against themself, then x is plotting against y." This is self-refferential in a different way but you would need to clarify what you meant by it.

Perhaps you were confused about the ambiguity of Someone. Is someone generally out to get you or is there a specific person out to get you. This difference is called De re and de dicto