Behaviourism closely followed psychology in the early 20th century, and wanted to concern itself only with publicly observable events and processes in the mind. Along with psychological behaviourists, philosophical behaviourists wanted to resolve the metaphysical nature of the mind and show that terms gained from introspection like 'feeling', 'lived experience' and 'will' are either meaningless or can be boiled down to statements about publicly observable, physical events and processes.

They wanted to construct a semantic theory of the mind that would explain the meaning of mental terms, which could only be achieved by being able to verify if a particular term or statement was accurate or false. Since immaterial notions aren't publicly observable and verifiable, behaviourists concluded that psychology and the philosophy of mind should only concern itself with the material... i.e it is a physicalist theory.

Logical behaviourism therefore holds that any mental term can be understood in terms of observable physical processes or events. For example, if I say I have a toothache, a scientist should be able to point to a problem with my gums, the transmission of information regarding this problem through my central nervous system to my mind, and the characteristic changes in the chemical makeup of my brain. Conversely, if those particular physical events and processes were observed, a scientist would be able to say "he has toothache".

In other words, all meaningful psychological statements are translatable into statements which refer only to physical concepts, without any loss of content; all 'conscious experiences' can be reduced to mere behaviour. Psychological concepts, according to Hempel, serve merely to abbreviate "the description of certain modes of physical response characteristic of the bodies of men or animals" (Hempel, 1980, p.19)

Objection: One can feign a mental state (e.g. actors)
But according to Hempel (1980) a full examination of the central nervous system and other physiological conditions will uncover a feign. But then are there not mental terms, all of whose verifiable criteria can be feigned? In this case, Hempel says, the event or process must be genuine, i.e. if Paul shows all the signs of having the flu, then Paul has the flu.

Objection: How can physical tests capture the qualitative nature of an experience, i.e. what is is to be in pain? Surely such experiences are neither behavioural nor dispositional (e.g. being disposed to be angry), and so can have no physical basis? Even if they do have some physical basis, how can we be sure that the mental states exactly correspond with physical states?

This problem is further complicated by the presence of other people. Let's say I experience the sensation of the colour red, and I find that correlates with a brain state X. Another person then says they are also experiencing the sensation of the colour red, and again we find the same correlation in their brain state. One might conclude from this that the two correlations point to an obvious connection. But how can I know that my mental state is the same as that other person's? Our descriptions might coincide whilst our actual mental states are qualitatively very different, suggesting no connection between mental and brain states, or at least no simple correlative connection.

Behaviourists have no satisfactory response to this objection, since they are interpolating the nature of the mental and brain states from the relationship between inputs and outputs, and so can say nothing of the nature of those states themselves if they are qualitatively different but produce the same measurable behaviour. Wittgenstein discussed this problem by way of the 'beetle in the box' analogy:

(paraphrased)Suppose everyone owns a box, and in each box they keep a 'beetle'. Nobody can see inside anyone else's box, nor know anything of its contents, except that its owner says it contains a 'beetle'. How am I to know my 'beetle' is the same as your 'beetle'? If people then used the term 'beetle' in their language, it couldn't refer to any physical object, and so would, gramatically speaking, drop out of consideration. (Wittgenstein)

In other words, not only can we not know anything about these qualitative experiences, they must in fact be considered non-entities, and so any attempt to explain mental states and processes in physical terms merely through demonstrating correlations in input-output scenarios will be completely flawed.

Though logical behaviourism must be wrong about a lot of things, it does correctly identify the non-contingent, conceptual connection between mental and behavioural descriptions. We learn mental concepts in their application to behaviour, and mental phenomena can often be individuated by their behaviour rather than by their subjective internal features. So there does appear to be a strong connection between behaviour and some mental states and processes, but suggesting that behaviour is all that they are seems far fetched.

Indeed, we can know about our own mental states without needing to observer our behaviour, and often without even observing our own mental states; I know I dislike Big Brother without need for any kind of observation. So logical behaviourism is either asymmetrical in terms of how it suggests we understand mental states and processes, or simply unnatural. This criticism was further developed by Malcolm, who pointed out that if one really were a logical behavourist, one would see people's emotions and mental states merely as physical alterations in the three dimensions, and so one wouldn't see someone as being angry, but as having a particular face. This is enough to know that we aren't logical behaviourists.

Finally, one can undermine the behaviourist thesis by arguing from regress. Suppose a logical behaviourist wanted to explain my belief that it is about to rain. Perhaps I bring the clothes in off the washing line, or put on my coat. To explain that behaviour, the logical behaviourist must make further reference to my belief that I will get wet if I don't put my coat on, and my desire not to get wet, and all in terms of behaviour. Clearly to explain the beliefs and desires, the logical behaviourist will have to go in circles, or run out an extremely long string of behavioural explanations without ever leaving mentions of psychological terminology in the final explanation. It would seem that the ambitions of the reductionist, trying to reduce all psychological terminology into verifiable behavioural terms, are simply too ambitious.

References:

C.G.Hempel, 'The Logical Analysis of Psychology' [originally published in 1935], as reprinted in N.J.Block (ed.), Readings in Philosophy of Psychology, Volume I, (London: Methuen, 1980), p.16