# THE LOGIC NOTES

## Contents

1. Introduction
• What it's all about: basic concepts of logic
• Consequence relations, the minimal requirement
• The idea of propositional logic
• Towards a calculus of derivations
2. Propositional natural deduction
• Natural deduction rules ∧I, ∧E
• The rules →I and →E; discharging assumptions
• Theorems, weakening and contraction
• Proofs using conjunction and implication
• Natural deduction rules ¬I and ¬E; using RAA instead
• Natural deduction rules ∨I and ∨E
• Proofs using negation and disjunction
• RAA is equivalent to ¬I and ¬E

• Sample problems with solutions
• Meanings of the connectives; evaluating formulae
• Systematic reasoning about truth and falsehood
• Normal forms and other regularities
• Equivalent formulae can replace each other
4. Expressing generality
• Naming things, describing things and generalising
• Complex sentences using quantifiers; bad news for goats!
• Natural deduction rules ∀I, ∀E, ∃I, ∃E
• Proofs using the rules for quantifiers
• Defining some important classes of binary relations
• Function symbols, terms and their logic
• Proofs involving relations and function symbols
• Other views of relations: relation algebra and graph theory

• Sample problems with solutions
5. More about first order logic
• Formal definition of interpretations for first order logic
• Rules for analysing ∀x A and ∃x A
• Showing that deducibility and validity coincide
• Propositional logic is complete for its semantics
• Another take on deduction and on semantic reasoning
• Cut is admissible in the sequent calculus
6. Identity and existence
• Reasoning with equations
• The logic of the word 'the'
• The rules ∀IR, ∀ER, ∃IR and ∃ER
• Bringing deduction into line with Logic for Fun
• An outline of free logic
• Proofs with restricted quantifiers and in free logic