Irreducible complexity, as defined by Michael Behe in Darwin's Black Box is a property of a system such that if any part is removed, the system ceases to function. Irreducible complexity is often used as an argument for Intelligent design.
"IF it could be demonstrated that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down. But I can find out no such case."
- "For example, consider the Darwinian assertion that birds evolved gradually from reptiles over long periods of time. This would necessitate a transition from scales to feathers. How could a creature survive that no longer has scales but does not quite have feathers? Feathers are irreducibly complex."
The argument runs: 
- Each stage in evolution must be beneficial or neutral for that feature to propagate to later generations.
- Some organs cannot exist in a simpler form because the change or removal of any part would make them useless at their function.
- Therefore evolution cannot explain the origin of these structures.
- Therefore these structures were designed.
- Therefore a designer exists.
The classic illustration of an irreducibly complex system is a mousetrap: it consists of a base, hammer, spring, catch (or trigger), and fasteners to hold the pieces together. If any of those parts is removed, the mousetrap no longer works. In biology, the eye is often cites as an irreducibly complexed organ.
Construction by Addition, Modification and Removal
The argument falsely assumes that evolution always proceeds by adding parts. Natural selection can modify or remove parts, as well as add them. For instance, whales have no hind legs, but retain vestigial pelvises where their ancestors' legs were attached.
Another example of an irreducibly complex system are naturally occurring rock arches formed by the sea or wind: if any section is removed, the arch collapses. They are formed by the gradual removal of material. Similarly, biological mechanisms often do not exist in the surroundings that allowed them to initially evolve. There is therefore no reason to accept the claim that if a system is irreducibly complex that it cannot be built by natural processes.
Systems incomplete for one function can serve a different function
While it is true that an irreducibly complex system with a missing part loses its nominal function, it may still have some other function. For instance, a mousetrap without a catch is no longer a working mousetrap, but can still work as a tie clip, or a paperweight. A mousetrap without a base can be nailed to the floor. Such a mousetrap would not be as useful, but would still function.
For a biological example, consider the bacterial flagellum, a long spinning hair that functions as an "outboard motor" for bacteria. It is often cited as an example of an irreducibly complex system. But if some of its parts are removed, the resulting system bears a striking similarity to the Type Three Secretory System, a "syringe" that allows bacteria to infect other cells.
Argumentum ad Ignorantiam
No potential example of a supposed irreducibly complex system can, even in theory, demonstrate that it did not evolve from less complex components. One can only demonstrate how a system can be reduced, or claim ignorance as to how it can be. Irreducible complexity is therefore an argument from ignorance and, more specifically, a God of the gaps argument.
A claim that a system is irreducibly complex is not a falsifiable claim. Demonstrating how a complex system can be reduced to less complex components only shows the apologist to be wrong on that particular example. Each 'reduced' component is, in turn, another system susceptible to the same claim of being irreducibly complex, ad infinitum. This lack of falsifiability makes such claims unscientific.
Irreducible complexity is problematic because some natural structures exhibit irreducible complexity (such as a sea or wind arch). The earth has a solid core, which if "removed" would break the Earth's magnetic field which has the "function" of shielding us. The moon stabilises the spin of the Earth, and if removed, would break the spin axis of the Earth which has the "function" of regulating the seasons. The solar system contains the sun but if this were removed, it would break the solar system's "function" of giving the Earth a stable environment. Based on these examples, irreducible complexity concept is an invalid indicator of design in living things.
Appeal to probability
Even if we accept that a certain biological strcutre is improbable, that does not mean it is impossible that it has occurred by chance. Similarly, if we accept that a God would have a higher probability that he would create that biological structure (which is almost impossible to demonstrate), we cannot simply accept that as the explanation. We don't know the events' prior probability (also called its base rate). There may be other explanations that are more probable. For this reason, this argument is sometimes referred to as the lottery fallacy or the base rate fallacy.
- "If we were to flip a fair coin, the odds of it landing heads side up is 50%. What if I were to say that 'I want to place his coin down on the ground heads side up.' What are the odds that it's going to be heads side up based on that? Nearly 100%. I could screw up a little bit, I'm not perfect. But it's way better than 50% as to whether I can set a coin on the ground heads side up. So if the argument is 'this is more likely then you should believe it', then every time you find a coin anywhere in the world that is heads side up, you should believe that it was placed there by a thinking person who intentionally placed it heads side up, [and] that is the most plausible explanation. And I think that we all realize that is actually not the case."