basic formalization using given languages

Question

i need to formalize 3 sentences using giving languages, and i want to check if i did them correctly.

1)there's an example that every stupid logician uses.

L(x) - x is a stupid logician

S(x) - x is an example

M(x,y) - x uses the example

2)there are doctors among the the listeneres.

D(x) - x is a doctor

H(x) - x is in the listeners.

3)(consists of three different phrases, variations of the same sentence)

• it is possible to cheat some of the people all of the time
• it is possible to cheat all of the people some of the time
• it is not possible to cheat all of the people all of the time

P(x) - x is a person B(x) - x is a time G(x,y,z) - x can cheat y in time z.

what i did:

1)$\exists x(\forall x L(x) \to \exists y M(x,x))$ (since it's the same example so M(x,x) in my opinion)

2)$\exists x(D(x) \land H(x))$

3)

• $\exists x(\exists y \forall z G(x,y,z) \land P(x) \land \forall x B(x))$
• $\exists x( \forall y \exists z G(x,y,z) \land P(x) \land B(x))$
• $\lnot( \forall x \forall y \forall z(G(x,y,z) \land P(x) \land B(x)))$

please correct me if i've done anything wrong. did my best to elaborate and solve it correctly. however, i am quite sure that on 3) i've done several mistakes that i don't know how to fix, so if you catch them, please show me the correct way to write it.

thank you very much for your help!

Answer 1

The general pattern translating a sentence like 'every person does sth' is to $\forall x:\text{Person} (x) \to \text{Does sth}(x) $, and 'there is a person that does sth' to $\exists x:\text{Person} (x) \land\text{Does sth}(x) $.

1. $\exists y: S(y) \land\forall x: L(x) \to M(x, y) $.
   We have to use different variables for examples and persons.
2. is correct (if plural is not taken strictly, but I think it's not meant)
3. $\exists y:P(y) \land \forall z:B(z) \to\exists x: P(x) \land G(x,y, z) $
   Can you retry the other two?