# Re-write this sentence in logical notation

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• Feb 19th 2009, 09:37 PM
qtpipi
Re-write this sentence in logical notation
"You can fool all of the people some of the time, and you can fool some of the people all of the time, but you can't fool all of the people all of the time" How do I rewrite this in logical notation? The next question is which statement is true: the original one or its negation? I don't know how to start the logical notation here..
• Feb 20th 2009, 04:09 AM
aliceinwonderland
Quote:

Originally Posted by qtpipi
"You can fool all of the people some of the time, and you can fool some of the people all of the time, but you can't fool all of the people all of the time" How do I rewrite this in logical notation? The next question is which statement is true: the original one or its negation? I don't know how to start the logical notation here..

Let P(x): x is a person; T(t): t is a time; F(x,t): You can fool x at t.

1. You can fool all of the people some of the time.
$\forall x (\text{P(x)} \rightarrow \exists t (\text{T(t)} \wedge \text{F(x,t)}))$

2. You can fool some of the people all of the time.
$\exists x (\text{P(x)} \wedge \forall t (\text{T(t)} \rightarrow \text{F(x,t)}))$

3. You can't fool all of the people all of the time.
$\neg \forall x \forall t((\text{P(x)} \wedge \text{T(t)}) \rightarrow \text{F(x,t)})$

Finally, you need to combine 1,2 &3.