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Logic statements

  1. Nov 5, 2011 #1
    Hello!

    1) [tex]\forall x (F(x) \rightarrow \forall y (F(y) \rightarrow y=x))[/tex]
    2) [tex]\exists x (F(x) \rightarrow \forall y (F(y) \rightarrow y=x))[/tex]
    3) [tex]\forall x (F(x) \land \forall y (F(y) \rightarrow y=x))[/tex]
    4) [tex]\exists x (F(x) \land \forall y (F(y) \rightarrow y=x))[/tex]

    If 1) is true then 2) is true; if 1) is false then 2) may or my not be true
    If 2) is true then 1) may or may not be true; if 2) is false then 1) is false

    If [tex]\forall y (F(y) \rightarrow y=x))[/tex] is true then:
    If 1) is true then 2) is true; if 1) is false then 2) may or my not be true
    If 2) is true then 1) may or may not be true; if 2) is false then 1) is false

    If [tex]\forall y (F(y) \rightarrow y=x))[/tex] is false then:
    3) and 4) are always false.

    I understand [tex](F(x) \rightarrow \forall y (F(y) \rightarrow y=x))[/tex] to mean that if F(x) is true then [tex]\forall y (F(y) \rightarrow y=x))[/tex] is true. So the first two are determined by whether or not all x or there is some x that make F(x) true. I understand [tex](F(x) \land \forall y (F(y) \rightarrow y=x))[/tex] to mean that they are independent, and F(x) and [tex]\forall y (F(y) \rightarrow y=x))[/tex] can be true or false separately.
    Is this correct?

    Thanks in advance.
     
  2. jcsd
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