The big idea: Can you prove God exists just by thinking carefully about what 'God' means — without looking at the world at all? That's the boldest argument in the whole topic.
The ontological argument: The ontological argument claims that once you truly understand the word 'God', you'll see that God has to exist. It runs entirely in the mind, from the definition alone.
How it runs
Define
'God' means the greatest possible being.
Compare
Compare a God that exists in reality with a God that exists only as an idea in your head. The one that really exists is clearly greater.
Conclude
So if God is the greatest possible being, God can't be only an idea — a merely-imagined God wouldn't be the greatest. Therefore God must really exist.
Greatest · Real beats imagined · So real
Checkpoint — ontological: In one line: God is the greatest possible being, and a real God is greater than an imagined one — so God must be real. Hold that; the famous reply is the 'perfect island' objection.
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The next argument does the opposite: instead of staying in the mind, it starts from the plainest fact about the world.
The cosmological / Kalam argument: The cosmological argument starts from something obvious: things have causes. You exist because of your parents, they because of theirs, back and back. The Islamic Kalam version sharpens it.
The Kalam version
Premise 1
Whatever begins to exist has a cause.
Premise 2
The universe began to exist.
Conclusion
So the universe has a cause.
Begins · Caused · First cause
The first cause: That chain of causes can't go back forever, the argument says — so there must be a first cause that started everything without itself being caused. That uncaused first cause is what people call God.
Checkpoint — cosmological: In one line: the chain of causes can't stretch back forever, so there must be a first, uncaused cause — God. Hold that; the killer question is 'then what caused God?'
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The third famous argument doesn't just note that the world exists, but how strikingly ordered it is.
The teleological / design argument: The teleological argument points at how finely-tuned things are: eyes that see, planets that hold their orbits, laws of nature just right for life.
The watch on the beach
The watch
When you find something that complex and well-arranged — say, a watch on a beach — you don't think it fell together by accident.
The maker
You assume a maker.
The world
The world, the argument says, looks even more designed than a watch. So it too must have a designer, and that designer is God.
Watch → maker; world → designer
Go further — higher-level insight: Spot the famous rival to the design argument: evolution. Natural selection can build eyes and finely-tuned creatures with no designer at all — just tiny useful changes kept over huge stretches of time. Naming that the design argument now has to answer evolution is a top-band point.
These aren't only Western arguments — a strikingly similar move appears in classical Indian philosophy.
The Nyāya argument from karma: The Hindu Nyāya school gave a careful argument for God (Ishvara). It's like the cosmological move, but for the moral order rather than the physical one.
How the argument runs
Karma
The law of karma means every action earns a fitting result.
The delay
But the results often come much later, or in another life.
The bookkeeping
Something has to keep track and make sure each action gets its right consequence. A blind, mindless universe couldn't run a fair moral bookkeeping like that.
God
So there must be an intelligent overseer — God — who administers karma.
Karma needs a keeper
Checkpoint — the four arguments: (1) Ontological: God exists by definition. (2) Cosmological: a first cause. (3) Teleological: a designer. (4) Nyāya: an overseer of karma. Three from the physical world and an idea, one from the moral order — all reaching the same conclusion.