Modula-3.Language definition by Cardelli L.

By Cardelli L.

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If y is bound in p, then the substitution p[y / x] might include new bound instances of y in place of free instances of x. This may change the meaning of p in a way that is not intended. 16 Let Person denote the set of all people, and let m LooksLike n mean that person m looks like person n. The following predicate is a statement about a person o; it states that there is some person who does not look like o: ∃ p : Person • ¬(p LooksLike o) We may make the same statement about person m by substituting m for o: ∃ p : Person • ¬(p LooksLike m) However, if we substitute p for o, we obtain a different statement entirely: ∃ p : Person • ¬(p LooksLike p) The expression substituted for o contains a free occurrence of p, which is then bound by the quantifier.

23 If n denotes a natural number, then the predicate n≥5 is satisfiable. There are natural numbers greater than or equal to 5. A predicate that is false for all choices is said to be unsatisfiable. Valid, satisfiable, and unsatisfiable predicates are the analogues of tautologies, contingencies, and contradictions in the language of propositions. Chapter 4 Equality and Definite Description In this chapter we extend our language of mathematics by adding a theory of equality between expressions. The language of predicate calculus with equality is strictly more expressive than without, since it allows us to assert the identity of two objects, or to distinguish between them.

This will be made easier later in the book when we have introduced finite sets. Then we shall be able to say, for example, that there are 29 distinct things with property p. 13 The statement ‘there is exactly one book on my desk’ may be formalised as ∃ b : Book • b ∈ Desk ∧ (∀ c : Book | c ∈ Desk • c = b) where ‘Book’ denotes the set of all books, and ‘x ∈ Desk’ means that ‘x is on my desk’. Specifying that there is exactly one object with a given property occurs so often that there is a special notation for it: the unique quantifier.

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