The strong challenge, to be explained below, is what originally made behaviorism interesting as an approach, but the field of psychology has almost completely avoided it for the past 100 years. Further, the strong challenge points directly to the long-missing piece of the puzzle in creating a naturalistic, non-dualistic, scientific, psychology. It is for lack of dealing with the strong challenge of behaviorism that cognitive neuroscience is moving (albeit slowly) towards a state of crisis. Here is the gist:
The Weak Challenges
There are two "versions" of behaviorism with which most professional psychologists are familiar, each based around a particular challenge to traditional dualism. The first is methodological behaviorism. The second is Watson-style eliminative behaviorism.
Methodological behaviorists claim that we need to study behavior because, well heck, that is all we can study. While this is sometimes played off as an agnostic position with respect to the existence of mind, it is not. When you look at the logic, these researchers are dualists, who believe in a separate world of the mind, and agree with the traditional, Cartesian argument that one person cannot study another person's mind. That is, the methodological behaviorist declares psychology a 'science of behavior' only in the process of surrendering to the cruelty of a world in which it is impossible to truly know another person's mind, and resigns himself to making what little progress he can on other fronts. That this dramatically underestimates the problems that dualism creates is a matter for future discussion. Also, it is unclear why a methodological behaviorist would not be fine with people studying their own minds, working towards a sophisticated introspectionist tradition (which I hear rumor Josh Clegg is trying to reinvigorate). While I know this sounds very negative, it is not all bad. Methodological behaviorism is a perfectly fine position for someone to take, if they are interested being a contributing member to the field of psychological research, and are more interested in doing studies that advancing theory. It is only shoddy when it is viewed as a version of behaviorism. This was a position that helped psychologists get through the day between the advent of the behaviorist challenge and the advent of modern neuroimaging. As a philosophical point, it presents a pretty weak argument.
Eliminative behaviorists represent the position most often presented, in the modern literature, during discussions of behaviorism. The position is most strongly associated with John B. Watson, the first public-face of behaviorism. Though this position is often inappropriately referred to as "radical behaviorism", and Skinner is often lumped into this category, those associations are inaccurate. The eliminative behaviorist denies the existence of the mind, of mental stuff or mental processes, completely. If not for the obvious contradiction, someone supporting this position might be tempted to say that all so-called "mental things" are just figments of your imagination. Watson argued that psychology was 'the science of behavior' as a trumpet call with which to lead the troops into a political battle; the declaration was intended to point out the value of the research he and his friends performed, while simultaneously dismissing the research done by others. For Watson, psychology was about the movement of muscles, because there was nothing else that existed. This is a pretty weak argument though, at least by modern standards. There had been materialist philosophers before, much more sophisticated than Watson; and anyone at the start of the 1900's should have know that you can't get very far after denying the existence of mental stuff.
Neither of these positions pose much of a challenge to the traditional conception of psychology. Methodological behaviorism is a good way of letting people get along as usual, so long as at some point in any longer report they admit that they are only studying behavior and, from that, making educated guesses about the mind. Eliminative behaviorism does not present a strong challenge to traditional psychology for three reasons: First, trite as it sounds, people find it hard to believe that they don't believe. Second, people recognize that psychology cannot really be a science of muscle movements, because that is just not what the term psychology means. It would be like saying that you were going to create a new science of chemistry, but only after denying the existence of chemicals. Third, when you boil things down, the eliminative behaviorists argue that a Science of Psychology can not exist, but even the limited successes of the field in the early 1900's demonstrated that a science was possible. Thus, both of these positions offer little challenge to traditional psychology.
The Strong Challenge
The strong challenge of behaviorism is not a claim about what psychology should be, but about what it always has been. The strong challenge comes in the assertion that: The questions we ask about psychological processes are all, when we examine them closely and phrase them with care, questions about behavior. This is not methodological, nor eliminative, it is the claim that mental things exists within patterns of behavior.
In its modern form this position grows out of pragmatism. It was hinted at in the early work of William James, started to blossom in James's later work,and was picked up by E. B. Holt and his students E. C. Tolman and James J. Gibson. Skinner advocated this position much of the time, though not exclusively. Other advocates (I am lead to believe) include Gilbert Ryle, Wittgenstein, and the Australian Realists, Anderson, O'Neil, and Maze. The connection with William James is perhaps most important, as virtually every American psychologist at the turn of the 20th century had been weaned on James's textbooks. Having those works as foundation readied the field for behaviorism.
When the radical behaviorists claim that psychology is the 'science of behavior', they are making a claim about the original, historic domain of the field. Psychology arose as the experimental study of epistemology and phenomenology, the study of knowledge and experience. While knowledge and experience in the abstract, hypothetical sense can't be studied with the methods of science, individual, concrete, acts of knowing or experiencing can be. How does a person know the capital of North Dakota? When does a person experience lights of different intensities as different? How does a person remember nonsense syllables that were on a list he read earlier? These are all tractable problems, and they are all, when you take it down to a concrete case, questions about behavior. To rephrase: Why does a person correctly answer questions about state capitals? Under what circumstances does a person indicate that one light is brighter than another? How does a person correctly identify which nonsense syllables were on a list he read earlier?
This strong challenge is not limited to psychological science, it works equally for applied cases. Pick any single, concrete example of a person with a psychological problem, watch them and wonder about their problem. Ultimately, isn't your question about why they are doing "those things", whatever those things may be? Note, when we talk about what a person is doing, we are not asking questions about low-level muscle contractions. What you see the person doing might be several types of behavior in several different settings, none of which seem very odd on their own. This might include perfectly reasonable seeming behavior that has no connection to events that occurred in the past. For example, a person who reports others starring at her and following her around, which is only odd because you have observed long enough to know that is not happening. In this example, it is a mismatch in the large-scale behavioral pattern, spread over time and space, that causes concern.
Continuing that example... You might object, "No, no, no, my problem is that she is paranoid. I am interested in why she thinks she is being followed. She doesn't just say she is being followed, she believes it!" My reply is only to point out that we are still dealing with a question about behavior. That she believes she is being followed is that an even wider pattern of her behavior is directed at her not-visible followers. She regularly checks behind her as she walks, she approaches strangers and accuses them of spying on her, she checks her room for surveillance devices each night, etc. Why, you ask, is that larger pattern of behavior happening?
So, that's it. That is the strong challenge that behaviorism offers to mainstream psychology. Though it is 100 years old, it is a challenge that has not yet been fully realized, nevertheless answered. There are many virtues that radical behaviorism can offer, when conceived in this way, and I will have to write about them another time. I will also need to experiment with other methods of presentation. For now, though, hopefully this gives a flavor for where all this is going.