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logicquantifiersfirst-order-logic

Could anyone give a human language example which could test ∀x.(px=>q) does euqal to ∃x.px=>q


I have problem to understand the equivalence of the following sentence intuitively for long time

∀x.(px=>q)

∃x.px=>q

I understand the truth table below does says they are equivalent:

p(a) p(B) q  p(a)=>q p(b)=>q (p(a)=>q)&(p(b)=>q) p(a)|p(b)    p(a)|p(b) =>q
 T    F   T      T      T            T               T              T
 F    T   T      T      T            T               T              T
 T    T   T      T      T            T               T              T
 F    F   T      T      T            T               F              T
 T    F   F      F      T            F               T              F 
 F    T   F      T      F            F               T              F
 T    T   F      F      F            F               T              F
 F    F   F      T      T            T               F              T

But what i'm looking for is an human language example to verify the validity of the equivalence, so I could understand more intuitively, could anyone give a example?


Solution

  • The second one means "there is a px value which implies a q result".

    The first one means "for each px value, there is a q result"

    This could be the difference between "a driver has a driving licence" and "every drivers must have one".