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Rules of inference problem

  1. Apr 20, 2014 #1
    1. The problem statement, all variables and given/known data

    Consider the following statements.
    All engineers either like computer or like power tools. All engineers who like computers
    like video games. All engineers who like power tools like camping out. Some engineers
    like literature.
    Based on these given statements, show that you can make the following inferences. Show
    all steps in your work.
    There is at least one engineer who likes video games and literature.

    2. Relevant equations

    3. The attempt at a solution

    Let E(x) be the proposition 'x is an engineer,'
    Let C(x) be the proposition 'x likes computer,'
    Let P(x) be the proposition 'x likes power tool,'
    Let V(x) be the proposition 'x likes video games,'
    Let O(x) be the proposition 'x likes camping out,'
    Let L(x) be the proposition 'x likes literature,'
    where x is the domain of all people.

    Steps and corresponding reasons:
    1. [itex]\forall[/itex]x E(x) → C(x) [itex]\vee[/itex] P(x) premise
    2. E(a) → C(a) [itex]\vee[/itex] P(a) universal generalisation
    3. [itex]\forall[/itex]x (E(x)[itex]\wedge[/itex]C(x)) → V(x) premise
    4. (E(a)[itex]\wedge[/itex]C(a)) → V(a) universal generalisation
    5. [itex]\forall[/itex]x (E(x)[itex]\wedge[/itex]P(x)) → O(x) premise
    6. (E(a)[itex]\wedge[/itex]P(a)) → O(a) universal generalisation
    7. ∃x E(x)[itex]\wedge[/itex]L(x) premise
    8. E(a)[itex]\wedge[/itex]L(a) existential generalisation


    I'm not sure about the statements for universal and existential generalisation, i.e. if they should all refer to the same a.
     
  2. jcsd
  3. Apr 20, 2014 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You can't prove that inference, it isn't true. Suppose for example, our "universe" consists of three engineers, "A", "B", and "C". "A" likes computers and, so, video games. "B" and "C" like power tools and, so, camping out. "C" likes literature.

    That satisfies all the given conditions but does not satisfy "There is at least one engineer who likes video games and literature." A is the only engineer who likes video games and he does nor like literature.

    "I'm not sure about the statements for universal and existential generalisation, i.e. if they should all refer to the same a."

    There is no "same a". "a" is a variable.
     
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