Presburger arithmetic does NOT prove its own consistency. Its only function symbols are addition and successor, which are not sufficient to represent Godel encodings of propositions.
However, consistent self-verifying axiom systems do exist -- see the work of Dan Willard ("Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles"). The basic idea is to include enough arithmetic to make Godel codings work, but not enough to make the incompleteness theorem go through. In particular, you remove the addition and multiplication function symbols, and replace them with subtraction and division. This is enough to permit representing the theory arithmetically, but the totality of multiplication (which is essential for the proof of the incompleteness theorem) is not provable, which lets you consistently add a reflection principle to the logic.
Raymond Smullyan gave a very general formulation in terms of representation systems. They appear in his "Theory of Formal Systems", and in the first and last chapters of "Godel's Incompleteness Theorems". They generalise first- and higher-order systems of logic, type theories, and Post production systems.
A representation system consists of:
A countably infinite set $E$ of expressions.
A subset $S \subseteq E$, the set of sentences.
A subset $T \subseteq S$, the set of provable sentences.
A subset $R \subseteq S$, the set of refutable sentences.
A subset $P \subseteq E$, the set of (unary) predicates.
A function $\Phi : E \times \mathbb{N} \rightarrow E$ such that, whenever $H$ is a predicate, then $\Phi(H,n)$ is a sentence.
The system is complete iff every sentence is either provable or refutable. It is inconsistent iff some sentence is both provable and refutable.
We say a predicate $H$ represents the set $A \subseteq \mathbb{N}$ iff $A = \{ n : \Phi(H,n) \in T \}$.
Let $g$ be a bijection from $E$ to $\mathbb{N}$. We call $g(X)$ the Godel number of $X$.
We write $E_n$ for the expression with Godel number $n$.
Let $\overline{A} = \mathbb{N} \setminus A$ and $Q^* = \{ n : \Phi(E_n,n) \in Q \}$.
We have:
(Generalised Tarski Theorem) The set $\overline{T^*}$ is not representable.
(Generalised Godel Theorem) If $R^*$ is representable, then the system is either inconsistent or incomplete.
(Generalised Rosser Theorem) If some superset of $R^* $ disjoint from $T^*$ is representable, then the system is incomplete.
In case it's not clear: in a first-order system, we can take $P$ to be the set of formulas whose only free variable is $x_1$, and $\Phi(H,n) = [\overline{n}/x_1]H$.
Best Answer
See refs at: http://en.wikipedia.org/wiki/Self-verifying_theories