I was quite happy to see the question Pointers for CS applications of logic. Scott Aaronson in his blog sometimes uses the scenario of super-intelligent alien beings having a discussion with us to emphasise the universality of certain notions in mathematics or computer science. This post felt like a very scaled-down but realistic form of that thought experiment. A person from a different, but closely related field comes to us and says, "we do logic, you do logic, and we have all these cool results, but what do you have to show for your logic."
It is difficult to tell from the wording how deep the author's knowledge of classical mathematical logic is, but the question does request for answers that appeal to advanced logical concepts. I was absolutely gung-ho about writing an answer because questions that specifically want and advanced-level survey are rare. A couple of hours later, I realised that this was never going to end. I'm glad to see that the question has now been made community wiki. I think my current answer is too long and should be split into different answers. I have three concrete proposals for answers to that question. Please let me know if you agree with these proposals.
Proposal 1. Every answer discusses a specific sub-area of computational logic and consists of three paragraphs. The first paragraph give a high-level overview of the field aimed at researchers in mathematics or TCS. The second paragraph highlights some results in the field and techniques used to prove those results. The third paragraph mentions current directions and open problems.
Proposal 2. We make a "table of contents" answer that lists different sub-areas of computational logic and hyperlinks to the different answers. Anyone can add to this list, and then add a separate answer for the area mentioned.
Proposal 3. If the OP agrees, we modify the original question a bit to make explicit the structure desired for answers. (We could also do this in the ToC answer I suggest in Proposal 2, but people might not read other answers before replying.
I feel such an advanced research summary would be a valuable resource for the future reference, and that obtaining such a survey via our user base would be effective. If it works here, we could also follow a similar protocol/structure for certain other broad reference requests.
What say ye?