I'm currently writing a paper on Descartes argument for God based on his third meditation. One premise of the argument that seems fairly important is the claim that "no effect can be greater than its cause". Putting aside how Descartes can know this on his own epistemology, have there been any strictly logical arguments made for this position? Or are the arguments generally made from induction?
I assume that the passage you rephrase as Descartes’ claim “no effect can be greater than its cause” is
Iam vero lumine naturali manifestum est tantundem ad minimum esse debere in causa efficiente et totali, quantum esse in eiusdem causae effectu; nam quaeso, undenam posset assumere realitatem suam effectus, nisi a causa? (Med III. 14)
Now, it is manifest by the natural light that there must at least be as much reality in the efficient and total cause as in its effect; for whence can the effect draw its reality if not from its cause?
The crucial word is the Latin “tantundem (= as much than)” which refers to the term “realitas objectiva (= objective reality)” in the passage just before the quote. Hence your question reads:
How to argue that the cause has at least as much objective reality than the effect?
Descartes does not speak about the relation between cause and effect if both are physical events or material objects. Descartes speaks about the relation between ideas. In the section before he conludes that his idea of God can only origin from an idea which has equal or even more objective reality:
[…] illa [idea], per quam summum aliquem Deum, aeternum, infinitum, omniscium, omnipotentem rerumque omnium, quae preaeter ipsum sunt, creatorem intelligo, plus profecto realitatis objectivae in se habet quam illae, per quas finitae substantiae exhiberunt. (Med III.13)
[…] the idea by which I conceive a God [sovereign], eternal, infinite, [immutable], all-knowing, all-powerful, and the creator of all things that are out of himself, this, I say, has certainly in it more objective reality than those ideas by which finite substances are represented.
Now Descartes has to argue why the existence of the God-idea in his mind, being the effect, implies that the God-idea has a referent as its cause, i.e. that God exists. That’s the content of the rest of the meditation and it seems to me the proper problem.
In my opinion Descartes attempts to derive his solution from the meaning of the technical term realitas objectiva. But that requires to dive deeply into scholastic ontology and the scholastic principles of argumentation. It’s not enough to consider only the new principles introduced by Descartes’ epistemology - it's not an easy task.
I know next to nothing about Descartes, so I will trust Jo Wehler's answer to be the correct one. As an aside, you might want to consider a parallel with the principle of conservation of energy in physics, combined with the fact that no process is 100% efficient. That tells you in a different sense that the magnitude of effects, when quantified in a very narrow way purely in terms of energy, cannot be greater than their causes. Here you have to take a holistic view of what the causes are. For example, if I illuminate the tasteless facade of my billion-dollar mansion, you might say that the effect of countless gazillions of photons bouncing off the building and into space is greater than the cause, if you consider the cause to be my lazy flick of a switch that triggered the thousands of floodlights to blaze. However, if you take a broader view that the cause of the blazing light includes half the output of my local power-station, then you will see that the causes do indeed exceed the effects.