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Although fundamental, the logic of truth-functional connectives has very limited expressive power. In this chapter we go further, explaining the basic ideas of quantificational (alias first-order or again predicate) logic, which is sufficiently expressive to cover most of the deductive reasoning that is carried out in standard mathematics and computer science.
We begin by presenting its language, built around the universal and existential quantifiers, and the ways they can be used to express complex relationships. With no more than an intuitive understanding of the quantifiers, some of the basic logical equivalences involving them can already be appreciated. For a deeper understanding, we then present the semantics of the language, which is still bivalent but goes beyond truth tables. This semantics may be given in two versions, substitutional and x-variant, which, however, are equivalent, under a suitable condition, for the basic logical relations. After explaining the distinction between free and bound occurrences of variables, and the notion of a clean substitution, the chapter ends with a review of some of the most important logical implications with quantifiers and with the identity relation.
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The same books as for propositional logic, with different chapters. In detail:
Hein J (2002) Discrete structures, logic and computability, 2nd edn. Jones and Bartlett, Boston, chapter 7
Huth M, Ryan M (2000) Logic in computer science. Cambridge University Press, Cambridge
Howson C (1997) Logic with trees. Routledge, New York, chapters 5–11
Hodges W (1977) Logic. Penguin, Harmondsworth, sections 34–41
A more advanced text, despite its title, is:
Hedman S (2004) A first course in logic. Oxford University Press, Oxford
- Something About Everything: Quantificational Logic
- Springer London
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