Translating to Predicate Logic with Lexicon

How to translate the following statement into predicate logic:

"Although the examiner hopes that all students meet the requirements for grade E or better, someone will get a lower grade and will be disappointed."


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The first step is to define the alphabet. Take the following first-order alphabet with the required interpretation:

Unary predicates:

  • S (x): "x is a student"
  • E (x): "x is the examiner"
  • G (x): "x - class"
  • D (x): "x disappointed"

Binary predicates:

  • R (x, y): "x is a requirement for y"
  • B (x, y): "x is y or better"
  • O (x, y): "x takes y"

Triple predicates:

  • H (x, y, z): "x hopes y satisfies z"

e: Constant ("class E")

x, y, z, w: Variables

Let's break down the original statement in two parts:

S1: "Assessor hopes that all students will meet the requirements for grade E or better"

S2: "Someone gets a lower grade and will be disappointed"

And use a specific alphabet to spell it in first order:

S1: ∃x (E (x) ∧ ∀y (S (y) ⇒ ∃z∃w (R (z, e) ∧ B (w, z) ∧ H (x, y, w))))

S2: ∃x∃y (S (x) ∧ G (y) ∧ O (x, y) ∧ ¬B (y, e) ∧ D (x))

Finally, we'll compute the original statement, that is:

S1 ∧ S2

Keep in mind that this is only one of the interpretations that will lead to the correct (satisfactory) answer.

I hope this helps



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