## Logic

### Full course description

Given a list of sentences like

- 1 "The next line is false",
- 2 "The next line is true" ,
- 3 "The first line is true"

Then you can use logic or logical thinking to determine which of the previous lines are true or not. As it turns out, these three sentences contradict each other, and you cannot even answer that question. Even while this is just a puzzle, similar thought processes shook the foundation of mathematics and logic in the early 19th century, through things like Russel’s paradox and Godel’s theorem. Next to that, the development of computers and computer science opened up even more the need for a logical framework. Its on these two main topics that this course touches: the development of the foundations of mathematics and the resulting issues, as well as logical techniques of proving. It is to be expected that a student taking this course comes out with a better ability to reason logically.

A brief list of some of the topics covered:

- Syllogistic reasoning, mainly used for linking language to logic
- Axioms, which are the ``things’’ you need to assume to be true without proof
- Proofs, which derive truths from axioms by logical steps.
- Propositional logic: the language of logic which uses formulas like (pàq)àr
- Truth tables, in order to determine which formulas are when true, and which formulas are always true or always false
- Logical deduction: giving proofs using very clear, basic logical steps, leaving no room for discussion or error
- Paradoxes like Russel’s paradox and the Grelling-Nelson paradox.
- Optional topics are, among others, "tableaus", "multivalued logic", "The axiom of choice", and "Gödel’s theorem"

The student will learn how to think rigidly and how to formulate this rigid thinking in a sound and structured way. This will prepare a student for exact thinking used in sciences as well as philosophy. The basics of how to conduct mathematical (and logical) proofs is explained. The students are exposed to the concept of ``axioms’’, and how to use them in order to derive results from them in a logical way. The course exposes students to basic mathematical proofs, and lets them get a first taste of giving proofs themselves. This is elaborated further by working with the more rigorous logical proof systems mentioned above. The course is rounded off by covering Russel’s paradox, how this historically shook the foundations of science, and how it still is something that has to be taken into consideration.

### Course objectives

- To provide the students with a toolbox of logical thinking and reasoning, enabling them to be more sound and rigorous in their argumentations in their respective specialties
- To train the students in certain logical systems of reasoning
- To expose the students to concepts like proofs, axioms, and how to work with them
- To expose the students to science-transcending concepts as axioms and Russel’s paradox.
- To a minor degree embed this in a historical framework

### Prerequisites

SCI1010 Basic Mathematical Tools or substantial high school experience in Mathematics (For an indication of the relevant topics, see SCI-M, p. vi-viii). Students who are unsure if this course is suitable for them can contact the coordinator to discuss their situation.

### Recommended reading

E-reader and possibly additional handouts

Optional material from the Logic in Action website