Is it possible to define a logical system as a set or system of functions? Is it possible to define a logical system as a set or system of functions? If so, which fields should one learn to do this?
 A: I'm not sure if I understand the question correctly, but possibly this is precisely what the field of abstract algebraic logic is all about. In this theory you befenit from the simple fact that logical connectives are basicaly nothing else than operations on algebras, i.e. functions.
In more details let  $\mathcal{L}$ be a set of function symbols, called language and $\mathbf{A}$ an algebra in this language. Moreover let $F$ be a subset of its universe, called filter (or truth set), which serves to inverpret truth values. Together $\langle\mathbf{A},F\rangle$ is called a logical matrix. Then you can define logic, denote its consequence relation by turnstile $\vdash$, in a following way: for every formulas (terms) $\Gamma\cup\{A\} $
$$\Gamma\vdash_{\langle\mathbf{A},F\rangle} A \text{ if and only if for every evaluation } v: Term \to A:\, v[\Gamma]\subseteq F\text{ implies } v(A)\in F, $$
where an evaluation $v$ is a homomorphism from free term (free formula) algera $Term$  to $\mathbf{A}$. In other words from set of formulas $\Gamma$ you can infer a formula $A$ whenever every evaluation satisfying $\Gamma$ also satisfyies $A$ (truth means being member of $F$). 
For a class of such a matrices $\mathbb{K}$ define:
$$\Gamma\vdash_\mathbb{K}A \text{ if and only if for every } \langle\mathbf{A},F\rangle\in\mathbb{K} \text{ it holds }\Gamma\vdash_{\langle\mathbf{A},F\rangle} A$$
This way for example classical logic can be obtained as $\vdash_{\langle\mathbf{2},\{1\}\rangle}$, where $\mathbf{2}$ is the Boolean algebra of two elements. Or you can get intuitionistic logic as $\vdash_\mathbb{K}$, where $\mathbb{K}$ is the class of all matrices based on Heyting algeras with truth sets $\{1\}$. Another famous examples are logics based on algbras over the real unit interval $[0,1]$ (e.g. fuzzy logics).
Every logic defined this way has the following noteworthy features (which in the context of abstract algebraic logic actually serves as a definition of what a (propositional) logic should be).


*

*if $A\in \Gamma$ then $\Gamma\vdash A$,

*if $\Gamma\vdash A$ and $\Gamma\subseteq\Delta$ then $\Delta\vdash A$,

*if $\Gamma\vdash A$ for every $A\in \Delta$ and $\Delta\vdash B$ then $\Gamma\vdash B$,

*if $\Gamma\vdash\ A$ and $\sigma$ is a substitution then $\sigma[\Gamma]\vdash \sigma(A)$.


All of these conditions are rather obvious, maybe except for the last one which  basically asserts that consequence in propositional logic is about form and not about context. So a consequence of a form $\Gamma\vdash A$ will be true under any 'circumstance' provided by the substitution $\sigma$,. This property is usually called structurality. 
