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 empty domain

 

What if one's domain of quantification is empty?

 

Then anything that begins "$x" must be false, because it says that there is something such that, which must be false if there isn't anything at all. That means that anything that begins "$x" must be true.

 

Equally, anything that begins ""x" must be true. (It is equivalent to something beginning "$x".) And anything which begins ""x" must be false. (It is equivalent to something beginning "$x".)

 

So, notice that it is possible for ""x x is brown" to be true but "$x x is brown" false: when the domain is empty. (Just as it is possible, as we have seen for ""x[x is a cow x is brown]" to be true and "$x[x is a cow x is brown]" false: when there are no cows.)

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