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What are some philosophical defenses of the existence of God?
Specifically, I'd be interested in any philosophers who have provided a thorough and rigorous defense of the existence of a Judeo-Christian God or of Christianity (in particular) to an atheist audience. I would prefer sources that are not church doctrine.
One of the most respected modern apologists is C. S. Lewis. Two of his most famous defenses of Christian faith are Mere Christianity and Miracles. A more recent attempt to marshal rational argument in defense of faith is noted scientist and devout Christian Francis Collins' The Language of God.
Classically speaking, the foremost architect of rational explorations of Christian belief is St. Thomas Aquinas, whose Summa Theologiae comprises 3500 pages of carefully constructed arguments.
Not all Christian thinkers agree that rational arguments for God are desirable. Kierkegaard's approach, for example, might be better described as irrationalist.
This question about a proof of Gods existence is very complicated, because we need to know what exactly one mean by proof.
There are two possibilities of arguments for God's existence. One is a deductive type argument, the other is an inductive type or probabilistic one. Most of the modern arguments like William Payley argument are in the second category, are probabilistic arguments in favor of Gods existence. Many classic arguments like Thomas Aquinas five ways are in the deductive side.
If you want some introduction about this i recommend Edward Feser's book "Aquinas: A beginners guide". In this book, professor Feser draws the Fives ways of Thomas Aquinas and gives some arguments against some wrong interpretations made by modern popular philosophers or scientists that think they are philosophers.
If you want some overview about other than Thomistic arguments, professor Feser have another book called "Five proofs of the existence of God". In this book one have arguments that are very old but explained in a more systematic and clear language for a modern reader.
One more point about deductive arguments. When a mathematician prove a theorem, what he is doing in an axiomatic system is to drawn some conclusions based on some rules having some axioms as the principles they admit as true without a proof. What was done is a necessity link between the premises and the conclusion. If the premises are true, than the conclusion follows necessarily. One can discuss the premises (the axioms) but not the conclusion. The complication in Thomas Aquinas five ways, for example, begins because they are deductive arguments. They prove the logical necessity of a God, if one accept the premises, and the premises thyself are based in the Aristotelian metaphysics, some of then are just obvious conclusions about the doctrine of hylemorphism (like motion/change is the actualization of a potency). So, there are people that don't do a rigorous read in Aquinas arguments and just jump to the conclusions or see some premises as fake because they don't see the aristotelian foundation in then. One have to begin by refuting or rejecting the aristotelian methaphysics for rejecting Aquinas arguments.
There is another bifurcation in this theme. Some modern Christians like Alvin Plantinga and William Lane Craig are what we could call personalists. There is a big difference between the view of a personalist God and the classical simple God. The classical view of God see all the atributes in God like one thing, and the personalist tends to see as different things in one God. There is some profound debates in this topic and a vast literature.
One final point is that Aquinas himself reconises that the fives ways just prove the necessity for a God that is commum to Cristians, Judeos and Muslims. His five ways do not prove exaclty the Cristian God. For this one need more arguments. Is a commum ground that the trinity is something that can't be proved. So, we have some limitations here.
