How can the liar paradox be expressed concisely in symbols? In which formal languages?
The Liar is the statement "this sentence is false." It is expressible in any language able to perform selfreference and having a truth predicate. Thus, $L$ is a statement equivalent to $\neg T(L)$. Goedel proved that the usual formal languages of mathematics, such as the language of arithmetic, are able to perform selfreference in the sense that for any assertion $\varphi(x)$ in the language of arithmetic, there is a sentence $\psi$ such that PA proves $\psi\iff\varphi(\langle\psi\rangle)$, where $\langle\psi\rangle$ denotes the Goedel code of $\psi$. Thus, the sentence $\psi$ asserts that "$\varphi$ holds of me". Tarski observed that it follows from this that truth is not definable in arithmetic. Specifically, he proved that there can be no first order formula $T(x)$ such that $\psi\iff T(\langle\psi\rangle)$ holds for every sentence $\psi$. The reason is that the formula $\neg T(x)$ must have a fixed point, and so there will be a sentence $\psi$ for which PA proves $\psi\iff\neg T(\langle\psi\rangle)$, which would contradict the assumed property of T. The sentence $\psi$ is excactly the Liar. Goedel observed that the concept of "provable", in contrast, is expressible, since a statement is provable (in PA) say, if and only if there is a finite sequence of symbols having the form of a proof. Thus, again by the fixed point lemma, there is a sentence $\psi$ such that PA proves $\psi\iff\neg\text{Prov}(\langle\psi\rangle)$. In other words, $\psi$ asserts "I am not provable". This statement is sufficiently close to the Liar paradox statement that one can fruitfully run the analysis, but instead of a contradiction, what one gets is that $\psi$ is true, but unprovable. This is how Goedel proved the Incompleteness Theorem. 


As Joel David Hamkins said, the standard answer to your question is that formal languages like the firstorder language of arithmetic cannot express the liar paradox because they cannot express the predicate "is true" as applied to all its own sentences. Why not? Well, if it could, then we would get a contradiction, following the standard liarparadoxical reasoning. However, this is not the end of the story. For example, there is an interesting paper by Saul Kripke, Outline of a theory of truth, J. Philosophy 72 (1975), 690716, better known among philosophers than among mathematicians, which explains how to define a truth predicate in such a way that the liar paradox can be expressed. The conclusion is just that the liarparadoxical sentence has an undefined truth value. 


There is an extensive discussion of this issue in Vicious Circles by Jon Barwise and Lawrence S. Moss. 


Aladdin M. Yaqūb (1993) The Liar Speaks the Truth, OUP, formalises a very simple language for naturally expressing the liar paradox, consisting of:
Yaqūb carried this out in the usual singlesorted firstorder logic: I think it is more natural to formulate this in a twosorted logic, but Yaqūb's handling of his system is concise and elegant, and this discpline shows that no kind of secondorderness, not even Henkin semantics, is required to model Tarski's Tschema, but only an expansion of the universe to include names, an additional predicate, and axioms sufficient to model the Tschema. It shows, therefore, that an object language can be be its own metalanguage without leaving the realm of the straightforwardly first order. As a consequence, the liar paradox exists within this logic, but it can be "tamed" with a family of possible tweaks to the Tschema. Yaqūb argues for one such tweaking that result in the liar paradox being not selfreferential, but generating a sequence of formulae each involving one more T predicate. By looking at how the interpretation of these formulae evolve in each model, he classifies each formula of the base language —i.e., the subset of sentences that do not use the T predicate— into one of seven classes, depending on whether the sequence converges on a truth value, or whether they oscillate between values, and if so, in what manner. Paradoxical selfreferential sentences are reseolved in a more pleasing manner than Tarski did, by being able to treat them in a unitary formalism that embeds the tower of formulae that are the progressive unwindings within the usual semantics of firstorder logic, and without forbidding sentences that talk about themselves. I found Yaqūb's monograph to be much more readable (I read it in an evening whilst travelling), and his argument much more elegant and compelling than that of Barwise & Etchemendy, and I highly recommend it to anyone who found B&E worth reading. I would be very interested to read an effort to "intuitionise" Yaqūb's theory, by embedding it in intuitionistic firstorder logic in a similarly elegant manner, and using a constructive model theory. 


The liar paradox could be expressed in Church's original lambda calculus of 1932. Let $F$ be the function $\lambda x. \sim x(x)$ Then $F(P) = \sim P(P)$ for any function $P$. In particular, $F(F) = \sim F(F)$ and so $F(F)$ is a sentence that asserts its own falsehood. 


Short Answer (SA). No Paradox, All Liar. Bit Longer Answer (BLA). You don't need a theory of truth to say that a statement is false. You need only deny the statement. And if you tell me that a statement, say, Statement 1 (S1), is identical or even just logically equivalent to a statement that S1 is false, then you are telling me a falsehood, at least, according to principles that both of us probably took for granted beforehand. 

