Is abduction a transcendental method? It can be construed as such, and I occassionally refer to it in that way, but that misses the core of abduction.
If "transcendental" means "arguing from a fact to the conditions of the possibility of that fact"--what must be true for this to occur--then abduction can be understood to be "transcendental." However, that is not the core of Peirce's intention or even more so how its used, because that ignores temporality. Abduction proposes a hypothesis given some facts understood to be evidence, and the hypothesis is true if it allows for the prediction and control of later events and facts. Let me phrase abduction in terms of how it is used to combat modern epistemology.
In short and in Humean terminology, pragmatists no longer say, in a strict sense, that the thing causes the idea and the idea is like the thing, because you cannot beat Hume that way. Instead, we say that an idea is "true" if having and acting on the idea allows us to predict and control the flow of further ideas (experiences). This is abduction, the heart of scientific method, only it forms the basis of a theory of meaning and a phenomenology. However, unlike Hume and the whole modern tradition including Kant, we don't drink the Cartesian Kool-Aid and presume that the mind is radically separate from the body or the world. Hence, we shift the idea of knowledge from correspondence and related issues of representation to semiotics and how to represent ideas such that we can predict and control further experiences. However, the ideas have a real relation to the world even if it is neither direct causation nor representation; they cannot fail to be real without invoking solipsism or paradoxes. In this way we respond to Kant, a topic that I have discussed in more detail previously.
Perhaps you see how this relates to temporality. If an idea is true insomuch as an anticipated future occurs, and we say that reflective thought is abduction (literally), then knowledge and temporality are irreducible in any account of truth, knowledge, meaning, inquiry, etc. Transcendental method does not have this relationship to time.
Leon of After Nature asked me this question, and I thought that I would roll in a few other things while responding.