FOL Symbolisation - Symbolising sentence in FOL

Question

I have the following question I have been trying to answer

Symbolise the following sentence in FOL. Remember to provide a translation key.

• All respiring cells have mitochondria, but strictly fermenting cells do not.

I have attempted to answer this in the following way but I am unsure as to if it is correct or not

Domain: Cells

P(x): Respiring Cells S(x): Fermenting Cells l: mitocondria

$ ∀x(Px \to l) (Sx \to ¬ l) $

Am I on the right path with this or am I totally going in the wrong direction?

If I am completely wrong, could someone please breakdown how I could symbolise the sentence provided, just so I can understand a bit more, many thanks.

Answer 1

You'll need to make the part about having mitochondria part a predicate. You can either follow LoMaPh's suggestion and use a $Have(x,y)$ ('$x$ has $y$') predicate together with a $M(x)$ ('$x$ is a mitochondria') predicate, or you can simply use a $M(x)$ predicate that stands for '$x$ has mitochondria'.

You'll also need to use an $\land$ somwhere, since you are basically dealing with two sentences with a 'but' in between, and 'but' in logic simply means 'and'.

Answer 2

You are on the right track to use a conditional to symbolize general sentences like All A's are B or A's are B.

I suggest figuring out what the main connective of the whole sentence is; then symbolizing each part it connects separately.

Look at the first comment by LoMaPh for more direction.