Write as and do exactly as you did above.
I know that for this statement :
"this equation x²=1 has 2 different solutions only"
the translation will be :
ƎxƎy( x²=1 ʌ y=1 ʌ x≠y ʌ ∀z( z²=1 → (z=xVy=y)).
now it might sound a bit dumb to ask, but what if the equation would be x²-1=2x ?
thanks.
p.s - feel free to correct my freely translated terms, ex : I'm not sure if you say translate or formalize... it would help me be clearer for further questions )
Plato is probably right. In addition, if I wanted to make a statement about the number of roots, I would not use the phrase "the equation x²=1 has 2 different solutions only". Sometimes people say, "There is one and only one solution", where by "there is one" they mean "at least one", and this is not sufficiently strong, so they add the "only one" part. I am wondering if by "only one" they mean "at most one" and if this part can be used alone, i.e., if one can say, "There is only one solution" meaning that there is at most one solution.
So, to make the statement unambiguous, I would say, "The equation has exactly two solutions".
Ah, the vagueness of language.
One of my favorite mathematical stories involves Solomon Lefschetz’s and E.H. Moore. Moore was giving a guest lecture at Princeton. He began his talk “Let a be a point and b be a point”. Leftschetz shouted out “But why don’t you just say ‘let a and b be points’?” Moore replyed “Because a could be b”. Leftschetz got up and left the room.
The best book I've found is 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar.
Indeed it has an especially good treatment of the subject of specific finite quantities expressed in first order logic with identity.
Anyway, it's pretty simple. For any natural number n, and any exprssible predicate P, there is an obvious first order formula to express "there are exactly n number of x such that P holds for x." And it's not much harder to expand this for k-place predicates for any k.
The propositions "for all x y z, P(x) /\ P(y) /\ P(z) -> x = y \/ x = z \/ y = z" and "There exist at most two different x's such that P(x)" are logically equivalent. Indeed, in interpretations where P is true on zero, one or two distinct domain elements, both propositions are true, and on other interpretations, both are false.