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I am struggling to undersand Predicate Logic and my teacher gave me an interesting problem that is just a little confusing. I am unsure what I need to exactly describe these situations. I gave it a go but I would appreciate any insights into what I might be doing wrong.

You are writing a program that keeps track of movies and actors. The situation can be described using the following predicates:

  • R(M,Y) is true if movie M was released in year Y.
  • S(A,M) is true if actor A appears in movie M.
  • H(M) is true if the movie M was a hit!

Use these predicates to state the following facts:

  • (a) Actor “A” has appeared in at least one hit.
  • (b) Actor “A” has appeared in at least two hits.
  • (c) Actors “A1” and “A2” have appeared together in the same movie.
  • (d) All the movies that Actor “A” appears in are hits.
  • (e) Actor “A” appears in all the hits released in ”2020”.

My solutions so far:

  • a. (∃x) S(A,M) → H(M)
  • b. (∃x) S(A,M) → H(M) ∧ H(M)
  • c. (∃x) S(A_1,M) ∧ S(A_1,M)
  • d. (∀x) S(A,M) → H(M)
  • e. (∀x) S(A,M) → H(M) in 2020
ryang
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Destiny
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1 Answers1

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  • Please first read Point $0$ in this other Answer regarding quantification scoping, because a similar issue arises throughout your attempts; for example, $$(\exists x) P\to Q$$ actually means $$(\exists x P)\to Q$$ instead of the intended $$\exists x (P\to Q).$$

  • (a) Actor “A” has appeared in at least one hit.

    a. $(∃x) S(A,M) → H(M)$

    $∃a∃m \big(S(a,m) \land H(m)\big)$

    (b) Actor “A” has appeared in at least two hits.

    b. $(∃x) S(A,M) → H(M) \land H(M)$

    $∃a∃m_1∃m_2 \big(S(a,m_1) \land S(a,m_2) \land H(m_1) \land H(m_2) \land m_1\neq m_2\big)$

    (c) Actors “A1” and “A2” have appeared together in the same movie.

    c. $(∃x) S(A_1,M) \land S(A_1,M)$

    $∃a_1∃a_2∃m \big(S(a_1,m) \land S(a_1,m)\big)$

    (d) All the movies that Actor “A” appears in are hits.

    d. $(∀x) S(A,M) → H(M)$

    $∃a∀m \big(S(a,m) → H(m)\big)$

    (e) Actor “A” appears in all the hits released in ”2020”.

    e. $(∀x) S(A,M) → H(M) in 2020$

    $∃a∀m \big(\big(R(m,2020) \land H(m)\big) → S(a,m)\big)$

ryang
  • 44,428