Contents

Introduction
 Inference, truth and validity What it's all about: basic concepts of logic
 Inference in the abstract Consequence relations, the minimal requirement
 Connectives The idea of propositional logic
 Deduction Towards a calculus of derivations

Propositional natural deduction
 Conjunction Natural deduction rules ANDI, ANDE
 Implication The rules IMPI and IMPE; discharging assumptions
 Counting assumptions Theorems, weakening and contraction
 Examples Proofs using conjunction and implication
 Negation Natural deduction rules NOTI and NOTE; using RAA instead
 Disjunction Natural deduction rules ORI and ORE
 Examples Proofs using negation and disjunction
 Extra (math) RAA is equivalent to NOTI and NOTE

Propositional proof exercises Sample problems with solutions

More about propositional logic
 Truth tables Meanings of the connectives; evaluating formulae
 Semantic tableaux Systematic reasoning about truth and falsehood
 Transformations Normal forms and other regularities
 Extra (math) Equivalent formulae can replace each other

Expressing generality
 First order language Naming things, describing things and generalising
 Formalisation Complex sentences using quantifiers; bad news for goats!
 Quantifiers in Proofs Natural deduction rules ALLI, ALLE, SOMEI, SOMEE
 Examples Proofs using the rules for quantifiers
 Properties of relations Defining some important classes of binary relations
 Functions Function symbols, terms and their logic
 Examples Proofs involving relations and function symbols
 Extra (math) Other views of relations: relation algebra and graph theory

Quantifier proof exercises Sample problems with solutions

More about first order logic
 Modelling quantifiers Formal definition of interpretations for first order logic
 Quantifiers in semantic tableaux Rules for analysing ALLx A and SOMEx A
 Soundness and Completeness Showing that deducibility and validity coincide
 Extra (math) Propositional logic is complete for its semantics
 Sequent calculus Another take on deduction and on semantic reasoning
 Extra (math) Cut is admissible in the sequent calculus

Identity and existence
 Identity Reasoning with equations
 Definite Descriptions The logic of the word 'the'
 Restricted quantifiers revisited The rules ALLI_{R}, ALLE_{R}, SOMEI_{R} and SOMEE_{R}
 Manysorted logic Bringing deduction into line with Logic for Fun
 Nonexistence An outline of free logic
 Examples Proofs with restricted quantifiers and in free logic

Challenging the Paradigm
 Paradoxes of Implication Relating logic and language: the case of relevant logic
 Vagueness Thoughts on the sorites paradox and fuzzy logic
 A constructive view of truth A quick look at intuitionist logic
 Modelling nonclassical logic Using 3valued semantics to disprove sequents

Reference: all the rules
 Standard calculus Classical natural deduction and variants
 Sequent calculus Left and right rules for general sequents
 Substructural logics Rules of the nonclassical deductive systems
 Bibliography