Tutorial Contents Tutorial Four: Propositional Calculus: Language - Quantifiers- The Scope of Quantifiers -
The domain of Quantification -
The relation between "x and $x - The Empty Domain -
Predicate Formulae - Scope
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The relation between "x and $x

 

There is a very simple relation between ""xj" and "$xj" (where "j" is some predicate containing "x").

 

 

""xj" is equivalent to "$xj".

"$xj" is equivalent to ""xj".

 

(Think of it like this: when you push a "" through a quantifier in either direction, it flips into a quantifier of the other sort.)

 

So one can translate "Nothing is brown" either as "$x x is brown" or as ""x x is brown".

 

Equally one can translate "Everything is brown" as either ""x x is brown" or as "$x x is brown".

 

We can see also that the two suggested translations of "No cows are brown" are indeed equivalent. "$x[x is a cow x is brown]" is equivalent to ""x[x is a cow x is brown]". But "[jy]" is equivalent to "[jy]". (Check it by a truth-table.) So, ""x[x is a cow x is brown]" is equivalent to ""x[x is a cow x is brown]".

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