The remarkable feature of mathematics is that we may suspend belief like this, and yet rely on the results of a lengthy calculation, even when it has been delegated to a computer. The notation of elementary school arithmetic, which nowadays everyone takes for granted, took centuries to develop. There was an intermediate stage called syncopation , using abbreviations for the words for addition, square, root, etc . The logical calculus is easier to execute than any of the techniques of mathematics itself, yet only in 1934 did Gerhard Gentzen set it out in a natural way.
Examiners fully deserve the garbage that they get in return.
In Sections 1.4 and 1.5 natural deduction is introduced in a way which has been taught successfully to first year informatics undergraduates, for whom reasoning about the elementary details of their programs is a more pressing concern. The manipulation of sets and relations more familiar to mathematics students is treated in Section 1.3 and Chapter II . If you are doubtful of the need for formal logic then I suggest reading the later chapters first. Nevertheless, the ideas which they introduce will play a substantial role in the rest of the book. Today it is mathematics and programming, but in the first instance we shall apply logic to formal manipulation itself.