Reasoning and Logic

It is often useful or even essential to know if a certain statement is true, e.g. Pythagoras' famous a^2 + b^2 = c^2 theorem about right-angled triangles. Knowledge gathered from these statements or theorems can be broadly used to solve more complicated problems. This way of working, i.e., deriving more complex theorems from simpler ones, is useful in many fields, in particular also in computer science.

An argument is a set of premises or assumptions, followed by a conclusion. To be certain of the truth of an argument, the conclusion has to be a logical consequence of the assumptions. To prove this, the conclusion is derived from the premisses. The derivation shows us that once all premises are true, the conclusion is true as well.

The course Reasoning and Logic is about proving the logical validity of arguments. What is a valid argument? When is an argument logically valid and when is it not? How can we determine whether an argument is logically valid? How can we derive a logically valid conclusion from the premises? Or how can we prove that a conclusion is not a logical consequence of the premises?

In this course we will first explain a number of basic proof techniques, such as proof by contradiction, proof by mathematical induction, proof by division into cases, and the use of invariants. The application of these techniques will be practiced by proving and rejecting simple mathematical theorems.

These proof techniques can only lead to a valid argument when the formulation of the premises and the conclusion is sufficiently precise. To express statements precisely, multiple artificial languages exist, of which we will learn two: propositional calculus and predicate logic. For both languages we look at the syntax and semantics and study how to translate expressions from a natural language to the more exact languages. Furthermore we will look at how to establish the logical validity of an argument in both languages.

Moreover, to be able to assign truth values to formulas in predicate logic and because of the importance of this subject in every exact science, in this course we will also pay attention to elementary set theory.

Together, the course provides tools that have important applications across mathematics, computer science, philosophy, and beyond.