The existence of different logics is interesting, but not particularly relevant, because of Gödel/Turing universality. If you have a statement about possibilities, a statement in modal logic, you can encode it in a semantics about possible worlds, in a first order logical form about an expanded universe of discourse (see Kripke semantics). Further, if you have a "fuzzy logic", you can speak about it in an appropriate axiomatic mathematical system which includes real numbers, and makes a map between propositions and fuzzy values.
There are also Bayesian probability calculi which can be thought of as a different logic, some people regard quantum amplitudes as a logic, while other schemes regard permission as in the logic. What you put in the logic and what you put in the axioms is largely up to you.
But the main point is that the ordinary first order logic is complete, it will produce any logical consequence of any axioms, and this was proved by Gödel. This means that if you have a mathematically precise description of some other logic, you can always talk about this logic in terms of first order logic, and consider the other logic as axioms on top of first order logic. This is not a natural point of view, but it is a possible point of view.
The universality of first order logic is the logical analog of Turing universality--- that a finite complexity computer with unbounded memory can simulate any other computer with suitable programming. The formalism of logic is like the instruction set, the axioms are like the instructions of the program, and the deductions are the running of the program. A Turing machine can do Gödel deduction (you can program a computer to deduce in first order logic) and Gödel deduction can describe a computer, so the two results are essentially equivalent, and anything that can be stated in any coherent logical system is something that is meaningful for a computer to analyze and interpret.
So there really is only one type of logic, and Wittgenstein is mostly right. Although it is a mistake to attribute this to Wittgenstein, who was not as mathematically or logically precise at the mathematicians and logicians of the early 20th century in whose footsteps he followed, not as precise, nor as iconoclastic, as Russell, and did not contribute commensurately to the formal developments as Russell did. Further, one may say that Wittgenstein's ideas have a sonority and a lack of mathematical precision that leads those in technical fields to perhaps use the phrase "running one's mouth". The proper attribution of first order logic, the recognition of its importance in philosophy, and the associated logical positivism, better belongs to Hilbert, Frege, Boole, Quine, Gödel, Turing, Russel, Whitehead and others who come before. It continued with those who built the Vienna school, including Carnap, which made logical positivism the ascendant philosophy until the 1970s.
The use of predicate language to remove ambiguity, and the thesis that all statements should be formulated in some predicate language about precise observable criteria, is the central tenet of logical positivism, which flourished in the mid-20th century, but took a beating in the 1970-90s.