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Math Help - Logics Help!!

  1. #1
    Junior Member wpcp15's Avatar
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    Exclamation Logics Help!!

    I have a couple a problems I'm having a hard time defining ... Please Help!!

    1. An enemy of Al is a friend of Bill. Therefore, anyone who knows the enemy of Al knows a friend of Bill. (Exy <-> x is an enemy of y; Fxy <-> is a friend of y; Kxy <-> knows y, a= Al, b= Bill)

    2. Ptah is the father of all gods. But nothing is the father of itself. Therefore, Ptah is not a god. ( Fxy <-> x is the father of y; Gx <-> is a god, p= Ptah)

    Please any help at all!!
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  2. #2
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    What do you need to do: just translate those statements into symbols, or do you need to prove the conclusion as well? If so, do you need a formal proof?
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  3. #3
    Junior Member wpcp15's Avatar
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    I have to translate it into symbols and then prove it

    Yes all are proofs have to be formal

    I figured if I could get some help translating it, then I could start trying to prove it
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  4. #4
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    OK, I'll give English equivalents of sentences in (a) that can be converted into symbolic form using search/replace.

    1. An enemy of Al is a friend of Bill.
    For any x, if x is an enemy of Al, then x is a friend of Bill.

    anyone who knows the enemy of Al knows a friend of Bill.
    For any x, if there exists an y such that x knows y and y is an enemy of Al, then there exists a z such that x knows z and z is a friend of Bill. Note: the scope of the first "exists" is until comma.

    Another way to put it:
    For any x and y, if x knows y and y is an enemy of Al, then there exists a z such that x knows z and z is a friend of Bill.

    Note #1. The scope of "for any y" is to the end of the sentence.

    Note #2. I am not sure if "the" in "anyone who knows the enemy of Al" is intentional. The translations above assume "knows some enemy". I don't necessarily see any educational value in stressing "the".

    Note #3. A\land B\to C is equivalent to A\to(B\to C). The latter form is often preferable. For example, to deduce C from A, B, and (A\land B\to C) one has to first form A\land B and then apply MP, whereas in the second form one just uses MP twice.
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  5. #5
    Junior Member wpcp15's Avatar
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    Unhappy

    still confused
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  6. #6
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    As I said, these English sentences can be converted into symbolic form using search/replace operation according to the rules given in the table below. Some symbols and words in the table are set in italic. They are patterns and stand for subexpressions. They have to be in turn converted first and then substituted into the text in the "Replace with" column.

    For example, to convert "For any x, if x is an enemy of Al, then x is a friend of Bill", do the following. This sentence matches the pattern "For any x, y", where x is "x" and y is "if x is an enemy of Al, then x is a friend of Bill". So the expression "if x is an enemy of Al, then x is a friend of Bill", for which y stands, has to be converted first; its translation is ((E(x,a))→(F(x,b))). Then this intermediate result has to be substituted into (∀ x. y), resulting in (∀ x. (((E(x,a))→(F(x,b)))). The final result has superfluous parentheses, but this is OK; it is better than not having enough parentheses. Those that are definitely not needed can be removed. In this example, we get ∀ x. (E(x,a) → F(x,b)).

    Code:
    Search for                        Replace with
    ---------------------------------------------
    For any x, y                      (∀ x. y)
    
    There exists an x such that y     (∃ x. y)
    
    If assumption, then conclusion    ((assumption) → (conclusion))
    
    x and y                           ((x) ∧ (y))
    
    x or y                            ((x) ∨ (y))
    
    x is an enemy of y                E(x,y)
    
    x is a friend of y                F(x,y)
    
    x knows y                         K(x,y)
    
    Al                                a
    
    Bill                              b


    A general remark. The difficulty of a problem can often be assessed by thinking about how hard it would be to write a computer program that solves the problem. Most mathematical problems require creative thinking, which is extremely hard to capture in computer code. The task we are dealing in this post, however, can be done automatically in any more-or-less sophisticated text editor.
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