I know that a fallibilist believes that our current beliefs may not be true but i'm trying to argue that one can be a fallibilist and also believe that science progresses towards the truth. Does it make sense to say that because when newer scientific theories disprove our current view, that it is a step towards the truth? like we're getting to the "less wrong" realm right? which implies that we're getting closer to the truth?
There's no contradiction in this view, but you need an account of "approximate truth" for it to work, because you want to say that our current theories might be false, but are still truer than past theories. So theoretical truth cannot be a yes-or-no concept.
As other commentators noted, this hinges on what conception of truth you adopt, and incidentally what conception of meaning you adopt (i.e. how you interpret the content of theories). What is almost certain is that current theories are more empirically successful than past ones, so if you think truth is only a matter of empirical success (following a pragmatic conception of truth, and an empirically based interpretation of theories) then you will have progress. But many philosophers are dissatisfied by such account of truth: they think truth transcends our possibilities of empirical verification, that it is correspondence to the world.
There could be ways to accommodate a notion of approximate truth within a correspondence theory of truth, though. Two aspects of theories must be taken into account: the structural aspects (equations, models) and the referential aspects (the linguistic meaning of physical properties, the fact that they correspond to natural kinds). It's probably more easy to have approximate truth for structural aspect: there could be mathematical resemblance relations. The structure of theories would more and more resemble the structure of the world. The case of reference is more problematic. Is it "approximately true" that there are gravitational forces in the universe, now that we talk about curvature of space time instead? Or is it plain false? Saying that talk of newtonian forces is really disguised talk of space time curvatures seems a bit far fetched and one could be tempted to say that there really are no Newtonian forces in the world...
Now perhaps our theories are more and more true because more and more terms actually refer to natural kinds and properties. Or perhaps theoretical properties can be more or less good carvings of the structure of the world, and current properties are better than previous ones in this respect.
Another solution is to adopt structural realism: there's convergence toward truth, but it only concerns the structure of theories, not the fact that the types of objects and properties that posit really exist independently in the world. The problem of this solution is that it's not obvious that it doesn't boil down to empirical success only. A structure must be structure of something, and if its not structure of natural kinds, it must be structure of observational regularities, but then, saying that the structure of theories reassemble the structure of the world is just saying that our theories are empirically successful.
Popper would say that is exactly what happens. And it is surely what happens short-term in sciences. Folks compare results based upon known theories and the most logical fit wins. Over the longer term, it seems to be overly simple, given certain historical examples.
For example, in the case of Copernicus, people chose to abandon a more effective theory in favor of a simpler one, because the more effective one became so complex that it felt ad-hoc. They kept the known results of the old theory around for daily use, and it still affects our vocabulary and conventions. But they stopped developing it, and study went primarily to the new one, which eventually outstripped it.
To cover such cases, Kuhn proposes that all sciences occasionally decide they are wrong and rewrite their basic principles, he calls these paradigm shifts or 'revolutions'. But he also still believes that science gets more and more powerful in its predictive efficiency and therefore closer in some sense to the truth, even if it does not 'converge' in the sense that yesterday's truth is related to tomorrow's.
Even though contending paradigms cannot necessarily be compared to each other directly, each time we choose a new paradigm, we include more preconditions for taking it seriously. So we can expect all of the next round of candidates to be stronger than most of the rejected options. We would never simply move backward in time and return to a failed paradigm, as the dissatisfaction that caused it to fail is documented. We might choose a paradigm that was in the running last time around and lost out. But overall, there is still a process of elimination taking place.
It is possible for forward progress to be forestalled by the apparent strength of the current paradigm, even though it fails consistently at some things. In particular, if we look at fields like Alchemy or Chinese Medicine, which unified fields too broadly across natural boundaries and lost traction, we can see centuries or millennia of stagnation or asymptotic circling. But in the end, something has always confronted them from outside and set the process of increasing power over time back on track. So whether you can expect new explanations to be better than old ones depends upon what is going on in the 'paradigmatic tectonics' of the sciences involved. But on a grand scale, you can expect progress.