4 Expressing Generality
We now turn from the logic of sentential connectives to a more comprehensive system allowing us to analyse logical inferences which turn on the internal structure of atoms and on the behaviour of locutions like 'all' and 'some' which express generality. Consider the following for example:
Example 14 
All logicians are rational Some philosophers are not rational Therefore not all philosophers are logicians. 
Plainly this is valid in virtue of its form, but to demonstrate and explain its validity we need a richer language and more formal machinery than are available in the logic we have studied so far
 First order language Naming things, describing things and generalising
 Formalisation Complex sentences using quantifiers; bad news for goats!
 Quantifiers in Proofs Natural deduction rules ∀I, ∀E, ∃I, ∃E
 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