Q&A: Formalization
Formalization
Question
Hello and blessings!
This is not exactly a Torah question, but because the Rabbi also deals with other fields, I wanted to see whether you could help me. I started studying logic and formalization, and I really got tangled up trying to formalize the following sentence:
For every cook, there exists a food that he prepares that is tasty:
P(X) – x is tasty F(Y) – y is a food G(X,Y) – x prepares y tastily
The universal quantifier at the beginning of the sentence got left out for me here….
Option A: x ∃y :[(P(x)∧F(y)) ∧ G(x,y)]
The advantage of this formalization is that if x is not a cook it indeed returns the value F, but the problem is that apparently x could also be other things and not necessarily a cook, so it is hard to make the claim for every x.
I thought of a possibility that in my opinion captures the translation of the sentence:
x: P(x)→[∃y:F(y)∧G(x,y)[
The problem with this sentence is that it gets the value T even when x is not a cook, so maybe it is not equivalent to the sentence I wanted to formalize]
Answer
The formulas here got mixed up. I didn’t understand.
Discussion on Answer
I also don’t know how to write formulas here.
Formalize over the set of all people/objects and not over cooks: if X is a cook, then there exists a food Y that is tasty, P(Y), and that X knows how to prepare, G.
(X): {T(X) -> E(Y):P(Y)^G(X,Y)}
You can move the existential quantifier outside, but in my opinion this is preferable.
But then apparently the implication comes out true even when x is not a cook, because false implies everything—isn’t that a problem?
Not at all. It is entirely possible that someone who is not a cook knows how to make something tasty. The claim is that if he is a cook then he can, not that if someone is not a cook then he cannot.
Sorry, I’m not managing to write the formulas here in the normal way. How do you think one should formalize a sentence like this?