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I have the following sentence, which I need to write in FOL

There exists a student studying all the subjects of the information technology subject

I don't know how $\forall$ can be combined with the $\land$.

It was written by my teacher as follows

$$\exists x, \forall y: \text{student}(x) \land \text{learn}(x, y) \land \text{ITsubject}(y)$$

However, I was thinking this sentence should be written like this:

$$\exists x \text{student}(x) \land \forall y(\text{ITsubject}(y) \rightarrow \text{learn}(x, y))$$

or maybe

$$\exists x \text{student}(x) \land \forall y(\text{learn}(x, y) \rightarrow \text{ITsubject}(y))$$

Is this right?

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1 Answer 1

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Your two trials are obviously wrong since the occurrence of the latter $x$ are both free if you pay attention to its scope and paranthesis, and your last trial cannot be salvaged since such student may possibly learn non-IT subjects too. You can salvage your first trial as $∃x (student(x) \land ∀y(ITsubject(y) \to learn(x, y)))$, which could be further converted to prenex normal form $∃x∀y (student(x) \land (ITsubject(y) \to learn(x, y)))$ or $∃x∀y (student(x) \land (\lnot ITsubject(y) \lor learn(x, y)))$. And from this it shows your teacher's translation is incorrect or there's some typo in between, which is also apparent semantically since your teacher's version means every subject in your universe is ITsubject which is trivial here and most likely false in most 1-sorted context.

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  • $\begingroup$ sorry i forgot the parenthesis, and you say the statement of my teacher is trivial so that its neither false nor true in some case? $\endgroup$
    – chews
    Nov 27, 2022 at 2:39
  • $\begingroup$ It should be $∃x∀y(student(x)∧(¬ITsubject(y)∨learn(x,y)))$ as shown in the end of my above answer which is obviously not logically equivalent to your teacher's version or entail it (unless you can show how to derive from mine to your teacher's version). Also semantically your teacher's version is wrong translation per common sense, and by "trivial" here I mean your teacher's version is false unless your FOL is 2-sorted and in the universe for $y$ there're only ITsubject(s), nothing else, which is trivial per common sense. $\endgroup$
    – cinch
    Nov 27, 2022 at 3:44
  • $\begingroup$ Also u say the last trial not salvaged but like this sentences: There is an agent who sells policies only to people who are not insured.Translation : (∃x )(Agent(x) ∧ ( ∀ y) SellsPolicy(x,y) => ~Insured(y)) why this wouldnt be (∃x )(Agent(x) ∧ ( ∀ y) ~Insured(y) => SellsPolicy(x,y)) why don't they invert the SellsPolicy(x,y)) after the => $\endgroup$
    – chews
    Nov 27, 2022 at 6:41

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