A pseudoscience is a false science, a discipline masquerading as science without its discipline. From the Greek *ψευδής (pseudes), meaning false. Scientists generally use the prefixes para-, quasi-, and semi- to mean the same thing as pseudo-, but the first three prefixes have a long history and are used, especially by non-scientists, with different meanings.
TL;DR:
- para-, aside from science, even above and beyond. The study of things not addressable by science.
- quasi-, an admixture, partly science, possibly extendable to be fully scientific
- semi-, a joining of discernible scientific parts and non-scientific parts
Parascience has an old and distinguished pedigree. The Ngram viewer find its use in the journal Nature from 1869 by Sir Norman Lockyer, Fellow of the Royal Society, astronomer and co-discoverer of helium:
Parascience has all the qualities of a magical system while wearing the mantle of science. Until any significant discoveries are
made, science can justifiably ignore it, but it is important to say
why: parascience is a pseudo-scientific system....
Sir Norman clearly means to denigrate the parascientific, equating para- with pseduo-, but its later users take the prefix to mean alongside or even beyond. Thus from the journal Systematics (Volume 9, 1971)
It is the Change which changes and transforms all things including
birth and death. The Change according to the I Ching, the standard
work of parascience, is the ultimate, which transcends the
category of our description.
Parascience has subcategories -- parapschology, the paranormal, etc. -- the proponents of which all claim some reliable basis for their field.
Semiscience dates from the same era, and we find in the journal The Sanitarian from 1882:
I have no patience with the cold cruelty of semiscience which calmly
contemplates the preventable sickness and death, and hails these as
weeding and pruning processes by which to make the race better and
stronger.
The article discusses the efficacy of "sanitary science," which includes fighting disease with vaccination, antisepsis, etc. The prefix semi- comes from the Latin for half, but its meaning encompasses "partly" or "almost." Here the reference is to the eugenic view of social darwinism that joins one part science (i.e., Darwin's natural selection) to one part social philosophy of non-interventionism.
We can get a more modern flavor of the semiscience from Cosmology and Controversy: The Historical Development of Two Theories of the Universe (1999) by Helge Kragh, which reflects on the discussion of whether the study of the universe as a whole can ever be truly science. This argument goes back at least to 1953, when scientists G. Whitrow and H. Bondi considered in British Journal for the Philosophy of Science whether science was up to the task of explaining a unique object (i.e., the universe). Kragh writes:
This uniqueness had made cosmology "a borderline subject between the
special sciences and philosophy," and Whitrow believed that it would
remain such a borderline semiscience.*
Quasiscience is even older, finding a home in a book review in Economist: Weekly Commercial Times, Banker's Gazette and Railway Monitor (Volume 4, 1847), in which the reviewer regards the book as
... well worthy of being studied, though we should have liked it much
better had it been founded wholly on the ideas and knowledge which the
peculiar circumstance and condition of the Unites Staes call into
being, instead of being founded on a quasi-science of Europe.
The topic is natural rights and property law, and the book apparently bases its conclusion on phrenology, the above-mentioned "quasi-science." A more modern usage occurs in Nonsense on Stilts: How to Tell Science from Bunk (2010) by Massimo Pigliucci:
Thus the demarcation problem remains, and throughout this book we
examined many of it facets, looking for insights from the history of
science, discussions about the alleged difference between soft and
hard science, and critical analyses of quasi-science and downright
pseudoscience.
Quasi- as a qualifier can mean "only apparently" or "partly." In the first cite, phrenology was probably considered partly science, a field which might become fully scientific. It's likely Mr. Pigliucci leans toward the "only apparently" view.
* Forgive me for not following the trail to the BJPS, which is behind a paywall.