Some of our earliest experiences of the conclusive force of an argument come from school mathematics: Faced with a mathematical proof, however we try to twist the matter, there is no possibility of denying the conclusion once the premisses have been accepted.

Behind the examples from mathematics, there is a more general pattern of'demonstrative arguments' that is studied in the science of logic. Logical reasoning is applied at all levels, from everyday life to the most advanced sciences. As an example of the former, assume that under some specific conditions, call them A,something, call it B, necessarily follows. Assume further that the conditions A are fulfilled. To deny B under these circumstances would lead to a contradiction, so that either B has to be accepted or at least one of the assumptions revised – or at least that is what the fittest thinker would do to survive.

A remarkable level of complexity is achieved in everyday logical reasoning, even if the principles behind it remain intuitive. We begin our analysis of logical reasoning by the observation that the forms of such reasoning are connected to the forms of linguistic expression used and that these forms have to be made specific and precise in each situation. When this is done, it turns out that a rather limited set of first principles is sufficient for the representation of any logical argument.