what is Descartes argument for why animals are incapable of reasoning and have no soul? is this argument agreeable?
As a man of science Descartes wanted to give every phenomenon a naturalistic explanation, and hence to explain animals and humans behavior using a reduction to the way machines work. Yet, as a philosopher and a religious man - and not like Hobbes, for example - he couldn't accept a reduction between man and machine. Animals, for Descartes, indeed are complex machines: their organs move and fit themselves to a specific environment and specific actions. In that way he could follow his principle of the universality of the naturalistic explanation. But he couldn't do the same for man himself. Man, for Descartes is different than animals because he holds a much more complex ability: the ability to have a language. As he writes in Discourse on the Method:
[H]ow many different automata or moving machines can be made by the industry of man [...] For we can easily understand a machine's being constituted so that it can utter words, and even emit some responses to action on it of a corporeal kind, which brings about a change in its organs; for instance, if touched in a particular part it may ask what we wish to say to it; if in another part it may exclaim that it is being hurt, and so on. But it never happens that it arranges its speech in various ways, in order to reply appropriately to everything that may be said in its presence, as even the lowest type of man can do
The ability to speak, for Descartes, is unlimited. Every input out of the endless input possibilities, will be answered differently, and hence no machine can get even closer to possess language skills. Man, therefore, is not explainable in a naturalistic fashion.
So his belief that animals have no soul is based on the assumption that everything should be explained naturalistically. Only then he excludes man and only man. Therefore, animals are not more than machines.
This argument is not valid in nowadays Philosophy for several reasons:
Descartes knew only primitive machines like watches. Machines of this kind can actually do one thing only. The invention of the turing machine, though, that is called computer today, changed the perception of a mission-oriented machine. The computer, not like the watch, has a hardware and a software. unlimited types of softwares can be programmed to run on a one hardware, and therefore today's machines can do endless amount of missions. According to this - Descartes argument for the impossibility to implement the unlimitedness of the language is not valid anymore. Other claims for the impossibility to implement the ability to speak are great controversy today. Some might be found in Robert Brandom's How Analytic Philosophy Has Failed Cognitive Science.
While the first argument above claims for the possibility to implement man, and hence to claim that also man has no soul, this coming argument claims that some things are not explainable in a scientific way, no matter what - for animals and for men. This things are called Qualia and refer to the very subjective feeling of, for example, color or pain. These feelings are feelings that cannot be explained to others, and therefore, are not explainable in an objective way. Thomas Nagel argues in What Is It Like To Be a Bat that there is no possibility to explain to a human being the experience of being a bat. Bats use sonar (sound) in order to see, while we use color. Therefore, we are not and will never be able to explain the feeling of "seeing" the world using only sounds. Not being able to explain something, excludes this something from the list of things that can be explained scientifically. If Nagel is correct then we can ascribe to men and to animals abilities that we can include under the definition of "soul", that cannot be imitated by any machine.