Can someone give me a natural deduction proof for this argument?

Question

I am defending the usefulness of modern logic to a class and I came across this modalized version of a Thomistic argument (https://www.bu.edu/wcp/Papers/Reli/ReliMayd.htm). However, although I have studied both predicate and modal logic before, some of this gentleman's notations confuse me immensely. Namely, I have never seen (3[]t) for example, or a modal quantifier within the exact same parentheses as an existential quantifier.

P.S. with respect to natural deduction proofs, I am looking for something like this:

1. [](P -> Q)
2. <>(P)
3. W1: P .... 2, Possible Instantiation.
4. W1: P -> Q ..... 3, Necessary Instantiation
5. W1: Q ..... 3,4 Modus Ponens
6. <>Q ..... 5, Possible Generalization.

Answer 1

Read the footnotes. He points to a paper with a deduction of the third way. It is found here: "The Modal Third Way" It's a 57 step deductive proof, found on page 9-10. Followed by another 17 step proof for something else (I didn't read the paper). I don't think anyone else will want to do this in their free time at this time of the academic year unless they have really unhealthy procrastination habits like myself.

As a bonus, here is a quote from the conclusion of the paper:

I have argued that Aquinas’ original Third Way can be modally transformed into an arguably sound demonstration of the existence of a temporally neces- sary being who is supreme. The theist understandably might take some measure of comfort from this argument by interpreting the temporally neces- sary supreme being as God. One should be cautious, however, about reading too much into the Modal Third Way. The god of this proof is one that need not exist in every possible world. Worse, the theistic victory will be pyrrhic if it turns out that the universe is uncaused and unexplained. The supreme being of the Modal Third Way could also be the god of deism, the god of pantheism, or even a demiurge.