I'm trying to describe the predicate/proposition
n>1
in formal logic notation.
Apparently the answer is
∃y. y=1 $\land$ n>y.
If I try to read this, it reads:
There exists a y. y is 1 AND n is greater than y
I can't quite put my finger on it but there just seems to me something wrong with this solution...
The answer describes y and its property and somehow suddenly the whole thing is also just n>1.
I guess I need some kind of a metaphor to understand this way of converting to formal logic notation smoothly.
I also wonder if others struggle with this too.
Forgot to add this info. To clarify the confusion:
y = 1 has been defined as $\forall$x. x*y = x
but y>1 is not defined.
x>y is defined.
For detailed problem description and solution, click here and look Problem 3 (b)
There is no symbol $1$ in the language. That means we need to come up with a symbol to represent it.
After defining the property "is equal to $1$" by the formula with one free variable $(\forall x)(x \times \cdot =x)$, we can only ever refer to $1$ by this formula. So we make a shorthand for it: the literal string "$z\ \underline{=1}$" is defined to be shorthand for $(\forall x)(x \times z = x)$, whatever $z$ is. I've used an underline to emphasise that this is not an expression in the original language, but a shorthand to be expanded.
Finally, to express $n>1$ without using the forbidden symbol $1$, it's enough to define $1$, assert that it exists at all, and then say that $n$ is bigger than it:
$$(\exists y)(y\ \underline{=1} \wedge n>y)$$
See also the solution to (a) as to how you define "greater than".