While there are many objections raised against Aquinas's Third Way, one of the most damning identifies two fallacies in his argument: a "modal fallacy" and a "quantifier shift fallacy."
The first part of Aquinas's argument relies heavily on the following set of premises:
1) Contingent beings exist.
2) For every contingent being, there is some time t when that being did not exist. ("For a thing that need not be, once was not.") (Modal Fallacy?)
3) If there are only contingent beings, there was some time t when nothing existed. ("And if everything need not be, once upon a time there was nothing.") (Quantifier Shift Fallacy?)
Regarding premise 2: Does it follow that just because it is possible for something not to exist, that it cannot always exist? Why can't it exist eternally, but not necessarily? Likewise, why does time t need to be in the past? Why couldn't time t be just as easily in the future? The argument does not seem to mandate a past or future time, yet it is rather essential for Aquinas's argument that this time t was in the past.
Regarding premise 3: Does it follow that just because it is possible for everything not to exist, that everything did not exist at the same time? It is frequently noted that this is like saying: "Everyone has a mother, therefore there is one mother of everyone."
The following articles will take you further into these critiques and subsequent attempts to defend and rehabilitate Aquinas's argument.