Example:Everybody loves somebody

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When writing down quantified statements, the order of the quantifiers matters. For example, the following are very different:

  • [math]\href{/cs2800/wiki/index.php/%5Cforall}{\forall} x, \href{/cs2800/wiki/index.php/%5Cexists}{\exists} y, x \text{ loves }y [/math] means "everybody loves somebody"
  • [math]\href{/cs2800/wiki/index.php/%5Cexists}{\exists} y, \href{/cs2800/wiki/index.php/%5Cforall}{\forall} x, x \text{ loves }y [/math] means "there is somebody that everyone loves"

A proof of the first fact might proceed by choosing an arbitrary [math]x [/math], and then defining [math]y [/math] as [math]x [/math]'s mother; then reasoning that everybody loves their mother, so [math]x [/math] loves </math>y</math>.

A proof of the first fact would have to start by specifying a value of [math]y [/math]; this can't depend on [math]x [/math] because [math]x [/math] isn't defined yet. So, there would have to be a single person (for example, let [math]y := [/math]Raymond), and argue that since loves raymond, [math]x [/math] must love [math]y [/math].