# A question about material implication

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.

• "if p then q" also means "p implies q" – Eliran Jun 16 '19 at 23:47
• oh okay, thanks. so how should I read it in this situation mentioned? the sentence in such a fashion that I don't get it in the normal way. – Daruis soli Jun 16 '19 at 23:51
• "If the logical sentence that corresponds to the number x is true, then M." – Eliran Jun 17 '19 at 0:25

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".

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.