▁and ▁is ▁instead ▁a ▁specific ▁element ▁of ▁the ▁universe ▁of ▁disc ourse , ▁then ▁P ( c ) ▁only ▁implies ▁an ▁exist ential ▁quant ification ▁of ▁the ▁propos itional ▁function . ▁ ▁The ▁empty ▁set ▁▁ ▁By ▁convention , ▁the ▁formula ▁ ▁is ▁always ▁true , ▁regardless ▁of ▁the ▁formula ▁P ( x ); ▁see ▁vac uous ▁truth . ▁ ▁Universal ▁closure ▁▁ ▁The ▁universal ▁closure ▁of ▁a ▁formula ▁ φ ▁is ▁the ▁formula ▁with ▁no ▁free ▁variables ▁obtained ▁by ▁adding ▁a ▁universal ▁quant ifier ▁for ▁every ▁free ▁variable ▁in ▁ φ . ▁For ▁example , ▁the ▁universal ▁closure ▁of ▁ ▁is ▁. ▁ ▁As ▁ad joint ▁ ▁In ▁category ▁theory ▁and ▁the ▁theory ▁of ▁elementary ▁to po i , ▁the ▁universal ▁quant ifier ▁can ▁be ▁understood ▁as ▁the ▁right ▁ad joint ▁of ▁a ▁fun ctor ▁between ▁power ▁sets , ▁the ▁inverse ▁image ▁fun ctor ▁of ▁a ▁function ▁between ▁sets ; ▁like wise , ▁the ▁exist ential ▁quant ifier ▁is ▁the ▁left ▁ad joint . ▁ ▁For ▁a ▁set ▁, ▁let ▁ ▁denote ▁its ▁powers et . ▁ ▁For ▁any ▁function ▁ ▁between ▁sets ▁ ▁and ▁, ▁there ▁is ▁an ▁inverse ▁image ▁fun ctor ▁ ▁between ▁powers ets , ▁that ▁takes ▁subsets ▁of ▁the ▁cod om ain ▁of ▁f ▁back ▁to ▁subsets ▁of ▁its ▁domain . ▁The ▁left ▁ad joint ▁of ▁this ▁fun ctor ▁is ▁the ▁exist ential ▁quant ifier ▁ ▁and ▁the ▁right ▁ad joint ▁is ▁the ▁universal ▁quant ifier ▁. ▁ ▁That ▁is , ▁ ▁is ▁a ▁fun ctor ▁that ,