# 5   More about first order logic

As in the case of propositional logic, we need to explore the system of first order (quantified) logic semantically as well as syntactically. In this chapter we first define the concept of an interpretation of the first order language, and then consider how to use semantic tableaux to reason about formulae and sequents in terms of their satisfying interpretations. We also look at the sequent calculus, which is an approach intended to unify the deduction-theoretic and the semantic accounts of logic. Finally, we tidy up some loose ends by extending the calculus a little to accommodate restricted (binary) quantifiers along with the unrestricted ones used so far.

• Formal definition of interpretations for first order logic
• Rules for analysing ALLx A and SOMEx A
• Another take on deduction and on semantic reasoning
• The rules ALLIR, ALLER, SOMEIR and SOMEER
• Proofs with restricted quantifiers

• Sample problems with solutions
• Sample natural deduction problems with solutions