Logic applications and security protocols references request
I really like the phrase, as well as the book, that logic is calculus of computation. It is kind of hard to find an area in computer science where logic is not used. I will just mention few areas where knowing logic comes in handy. 


Verification. Techniques of logic are heavily used in verification area. Specification of hardware and software systems is usually given as logical formulas. Depending on what properties are being checked, such systems are also modeled by appropriate logic (propositional, FOL, modal logics, linear logic, etc.). Methods like satisfiability and model checking are usually used to check whether a system meets its specification. Moreover, there is an area of decision procedures that tries to come up with efficient procedures for checking satisfiability of formulas in various logics (by most first order theories). In general, you might want to take a look at automated theorem proving.
Programming languages. As pointed earlier, there is a deep connection between programming languages and logic. By Curry-Howard isomorphism, every functional program is isomorphic to some proof in constructive logic. Logic is also used for defining axiomatic semantics of programming languages; see Hoare logic. Prolog is a programming language based on a paradigm where computation is specified by logical formulas. Lambda calculus was first intended as a formal logical system aimed to laying foundations of mathematics, but turn out to be more useful as a programming language. It is now at the core of every functional programming language.
Misc. You might also be interested in the way logic and computability interplay. What might be specially interesting to you is descriptive complexity. Logic is also getting more attention in computational biology. It is also used in AI for knowledge representation.


Hope this gives you some kind of a starting point.