Strong atheism is the belief that no deity actually exists. To support this position, we have to consider several substantively different definitions, or classes of definitions, of "deity".
The first class, deity1, is the class of contradictory or meaningless definitions of "deity". We can safely affirm that no being exists with contradictory or meaningless properties. For example, the omnimax deity is either contradictory or meaningless because of the problem of evil. It is a contradiction that an omnibenevolent, omniscient, omnipotent, and omnipresent deity would permit evil in the world. Alternatively, we don't know what evil is (we are mistaken in some mysterious way) or there is no such conceptual category as "evil"; in this case, "omnibenevolent" is meaningless. The omnimax deity is offered as an example; finding that some particular definition of deity is not in the class of deity1 does not rebut the idea that we can safely deny the existence of any deity1.
The second class, deity2, is the class of undetectable (i.e. "supernatural") deities. Again, we can safely deny the existence of any deity that is, by definition, completely undetectable. To affirm or deny the existence of such a deity is to say exactly the same thing about the world of experience. An undetectable deity entails its own subtle contradiction: it is exactly the same to say, "Deity2 exists," and to say "Deity2 does not exist."
The third class, deity3, is the class of presently undetected deities. These deities are only detectable under some special circumstances that do not (presently) obtain on Earth. These deities are detectable only after death, or are hiding behind the couch, or on Achernar III, or somewhere else presently inaccessible. The problem is that there are an infinite number of definitions in this class; the probability that any one definition is true, especially a definition that names a finite number of deities, is infinitesimal and warrants disbelief until evidence becomes accessible.
The fourth class, deity4, is, by definition, presently detectable, but strongly paranormal (contradicts our ideas about physics). The evidence presently available, by the definition of paranormality, argues against such a deity. Deities which are detectable only privately fit this definition, because private knowledge (about anything but the content of one's own mind) is itself paranormal. (Note too that having an unusual sensory modality is not private knowledge, since someone who has even a unique sensory modality can prove its existence to someone without it, rendering that modality public.) Of course, the evidence might be sufficient for us to revise our concept of normality, but so far all attempts have fallen flat. Given that human beings have been looking for such a deity for many thousands of years, the failure to find one is itself sufficient evidence to warrant belief that no such deity4 exists.
The fifth class, deity5, is, by definition, presently detectable and not strongly paranormal. This definition includes "God is everything that exists", or "God is the [human emotion of] love." In the atheists' view, a deity5 is no deity at all; the speaker is using metaphorical or figurative language, and we are not literary critics.
All classes of definitions have sufficient warrant for either disbelief, disinterest, or exclusion from consideration. We cannot, of course, be certain that none of these deities (except perhaps deity1), but the preponderance of direct and indirect evidence warrants strong atheism.