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# 2: First Order Logic and Automated Reasoning in a Nutshell

Perhaps more foundations in modelling may be useful before delving into how to represent what you want to represent, but at the same time, one also needs to understand the language to model in. In this case, this means obtaining a basic grasp of logic-based ontology languages, which will help understanding the ontologies and ontology engineering better, and how to formalise the things one wants to represent. Therefore, we shall refresh the basics of first order logic in Section 2.1 (comprehensive introductions can be found elsewhere, e.g., [Hed04]), which is followed by a general idea of (automated) reasoning and two examples of tableau reasoning in Section 2.2. If you have had a basic course in logic from a mathematics department, you may wish to just skim over Section 2.1; my experience is that automated reasoning is typically not covered in such a mathematics or standard basic course in logic and you should therefore still engage with Section 2.2.

• 2.1: First Order Logic Syntax and Semantics
To be able to study those aspects of logic, we need a language that is unambiguous; natural language is not. You may have encountered propositional logic already, and first order predicate logic (FOL) is an extension of that, which enables us to represent more knowledge in more detail. Here, I will give only a brief glimpse of it. Eventually, you will need to be able to recognise, understand, and be able to formalise at least a little bit in FOL.
• 2.2: Reasoning
How does one find out whether a formula is valid or not? How do we find out whether our knowledge base is satisfiable? The main proof technique for DL-based ontologies is tableaux, although there are several others. The following subsections first provide a general introduction, the essential ingredients for automated reasoning, and then describes deduction, abduction, and induction.
• 2.E: First Order Logic and Automated Reasoning in a Nutshell (Excercises)
• 2.S: First Order Logic and Automated Reasoning in a Nutshell (Summary)