Applications of logic in sciences i understand math as the study and description of the behavior of mathematical structures, and as you know this math structures could include rings, fields, metric spaces, propositions, categories, numbers, sets, operators, differential equations..., a big part of this structures was born under the need of the description of a problem. For example the study and solution of the problem of the Brachistochrone curve gives to us the calculus of variations, or the study of the behavior of the heat and waves was the main column of the development of the Fourier series expansion, and as you know this is useful in physics, electrical, mechanical, and in general engineering. So other structures such as differential equations, tensors, matrices are useful for physics, chemistry, economics, engineering, and even abstract ones such as linear spaces, groups, rings, operators, Banach spaces, Hausdorff spaces are useful in physics. But in general logic, understanding it as the classification of truth parametrized but several specifications using several structures such as languages, binary operators, models, this to proof under what conditions a given expression id true, so it has several applications in number theory, algebra, topology, but this ones are mathematical fields, so i want to know if besides computer science foundations, type theory in CS, programming languages fundamentals, design and analysis of algorithms, digital logic, computer architecture, (that by itself is a huge approach of logic in life), are there any applications of logic in physics, economics, engineering, biology..
 A: Take a look at this book (if you can find it, that is):


*

*The Axiomatic Method in Biology
One of the authors is, surprisingly, Alfred Tarski! Is it a typo? Did Tarski really work on Biology? A passage from the Fefermans' biography of Tarski:

In England he stayed at the home of his biologist friend J. H. Woodger;
  at the 1935 Paris congress, Woodger had given a short exposition of
  his efforts to axiomatize biology in the language of Principia
  Mathematica. Tarski then and there offered to help by reading and
  criticizing his further work on that project. In the following years
  they had considerable correspondence and Woodger visited him in Poland
  three times, with many enjoyable discussions and hiking excursions.

A: Of course, physicists go in for logical inferences, so in a very boring sense, logic applies to physics in the way it applies of any other body of reasoned propositions. But I take it the question was more about the possibilities of what might be called 'carry-over' of modelling techniques.
To explain: Mathematical physics deals with idealised models of various physical structures and processes. Mathematical logic deals with idealised models of structures of inference. (To be sure, advanced mathematical logic courses and texts will often deal with more, e.g. the basics of the theory of computation and the basics of set theory, but that's largely because of a chapter of historical accidents.) 
Now, it can happen that a toolkit of applicable mathematics initially developed for use in one area of enquiry can turn out to be useful elsewhere, so get carried over. Could it turn out that mathematical tools developed by logicians are useful to physicists? Well, in this case, the home territories of the physicists and the logicians are so different, we perhaps shouldn't expect very much carry-over. 
There have been occasional suggestions, e.g. baffled quantum theorists musing whether their foundational problems might be generated by adherence to classical logic, and wondering whether something like a revised logic might help them. But such cases seem to be few and far between. 
