Functional Dependencies between Variables

Itisa standard practice to de ne the truth conditionsfora rst-orderform ula with a quanti erin term sofassignm entsto individualvariables. For exam ple,letM be a m odeland s a function (assignm ent)which assignsto variableselem entsin thedom ain ofM .Then a formula 9x (x)istruein M undersi thereisan assignm entswhich di ersfrom satm ostin thevalue itassignsto x,such that (x)istrue on M unders. However,in classical predicate logic assignm entsare notessential;the sam e truth de nition can be reformulated without m entioning them atall. Forexam ple,we can be slightly sloppy and justsay:9x (x)istruein M ifthereisan elem enta in the dom ain ofM ,such that (a)istrue in M . Thisisso because in classicalpredicate logic variablesare justanonym ousplaceholders;every value