Strong atheism is the positive belief that no god exists.
Arguments for strong atheism
It is often said that one cannot prove a negative. However, this is not really correct, and understanding this hinges on understanding the difference between two kinds of truth: contingent and logical. A contingent truth is one whose validity is contingent on other facts which may or may not be true or known. Scientific truth falls in this category. A logical truth is one whose validity depends only on logic, on the definitions and properties of concepts we ourselves define. An example is whether 2 + 2 can equal 5. It follows from the definitions of 2, 4, 5, +, and = that 2 + 2 can only equal 4, not 5. It is not necessary for a reasonable, fair, and open-minded person to remain agnostic on that question. There is nothing contingent or unknown about it.
A strong atheist would argue that the idea of a god is logically contradictory and therefore cannot exist as most theists define the word. The Christian god is defined as an omniscient, omnipotent, intelligent, aware being which created and was responsible for the universe. The problem of evil is one example of a logical impossibility that comes from believing that the god is also omnibenevolent, this can be avoided by relaxing the requirements on God's qualities.
Since its inception, science has follows an empirical method, a combination of theory and observation. We construct our understanding by working from the simple to the complex. We understand atoms in terms of elementary particles and their forces, such as quarks, leptons, gluons, and photons. We understand molecules and chemistry in terms of atoms and their interactions. We understand biology in terms of the underlying chemistry and its emergent properties. We understand intelligence in terms of the complex interactions of the underlying neurological or electronic substrate.
Of all the things we know of in the universe, the most complex is intelligence. To posit an intelligence as the creator and driving force of everything else makes no sense, since it would itself require explanation in terms of simpler underlying entities. It makes no more sense than 2 + 2 = 5. It is not incumbent upon a reasonable, fair, open-minded person to remain agnostic on that point.
Arguments against strong atheism
See arguments for weak atheism.