A question about material implication

Question

I have a question about reading material implications. I know this read as "if p then q".

like: if the moon is made of green cheese then the earth is round. but I can't read it in this sentence:

F(x) ≡ |x| → M. that is: F(x) is true if and only if the logical sentence that corresponds to the number x implies M.

or when:

the sentence L says: this logical sentence implies M.

This is about Löb’s Paradox and I can't read it as "if p then q", especially the second part of the first sentence, that is: “F(x) is true if and only if the logical sentence that corresponds to the number x implies M.” and the whole part of the second sentence. how should I read these sentences when there is the word implies? I hope I explained it clearly.

Answer 1

In general,

"the logical sentence that corresponds to the number x"

is wrong : "the number x" is not a sentence.

x is a number and also |x| is. Thus, from a formal point of view, |x| is a term (i.e. a name).

The sentence is F(x) = |x|, and thus, F is not a predicate but a function symbol : this means that F(x) is also a term.

In conclusion, we have a first sentence of form a=b and a second sentence M and the expression :

F(x) = |x| → M

is of form : "if p then q".

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If instead the context is that of Löb's theorem, we have to be careful about the syntax of the formula : F(x) can be a predicate (or formula with a free variable) and thus we can correctly use F(x) ≡ A to write a sentence, but if x is a variable, then |x| is not a sentence.