The cosmological argument faces the Glendower, Regress, Taxicab, and Gap problems. Cosmological arguments using a sufficiently comprehensive CP or an appropriate PSR are able to overcome the Regress and Taxicab Objections. The Glendower Problem of justifying the explanatory principle is an important one. However, in recent years, a number of arguments for such explanatory principles, as well as weaker versions of these principles still sufficient for the purposes of the cosmological argument, have been produced. There is, of course, still much room for research here: for examining arguments for or against the relevant explanatory principles, and for trying to produce cosmological arguments using yet weaker principles.
What contemporary analytic philosophers have not sufficiently worked on - and what is perhaps the most promising avenue for future research - is the Gap Problem. There are both inductive and deductive approaches here. The deductive ones that are currently known proceed through exciting metaphysical territory of independent interest. The metaphysics of existence/essence composition involved in Aquinas' bridging of the gap is fascinating, and the axiom that the perfections of the effect must be found in the cause is one that needs further exploration, both in connection with the cosmological argument, as well as in connection with emergentist theories of mind.
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