So I asked this question a month back but I realised something that was stated in the question stem and I am now confused:
Consider the predicate language with predicate symbol $<$, where $x<y$ means that "$x$ is an object different from $y$ that comes before $y$", and $=$ is the usual equality symbol.
Choose the first-order logic formulas that correctly translate the sentence:
"There exists an element that comes after all others".
A. $∃x∀y(y<x)$
B. $∃x∀y(¬(y<x)→(y=x))$
C. $∀x∃y(x<y)$
D. $∃x∀y((y<x)∨(y=x))$
The correct options are B and D
I have two questions:
You see where I am confused is that people said that A and C are incorrect because A and C can mean the same object, however if you read the question stem it says in bold "where $x<y$ means that "$x$ is an object different from $y$ that comes before $y$" so surely the fact that they may point to the same object shouldn't be an issue because it clearly states that $x$ and $y$ are different objects
My second question is that don't you need to find just one element that comes after all others? So for A and C you just need to find a single element that comes after all others because of the $∃x$ and the $∃y$ part, so why is the issue that $x$ and $y$ may be the same object arising if you just need 1 element and there will always be an element greater than all others, say we consider the set of natural numbers for example
I have no idea what you are trying to say or ask in part 2, but for part 1: Yes, the English sentence says that you ste are looking for an element that is greater than all others … but the logical sentences A and C do not specify that $x$ and $y$ are different …. which is exactly why they are wrong!