130 Logic as a Tool 3.3.6 A note on renamings and substitutions in a formula Note that renaming and substitution are very different operations; renaming always acts on bound variables, while substitution always acts on free variables Also, as we will see in Theorem 119, renamings preserve the formula up to logical equivalence while substitutions not On the other hand, a suitable renaming of a formula can prepare it for a substitution by rendering the term to be substituted free for such substitution in the renamed formula For instance, the term f (x, y ) is not free for substitution for y in A = ∀x(P (x, y ) ∧ ∃yQ(y )), but it becomes free for such a substitution after renaming A, for example to: A = ∀x (P (x , y ) ∧ ∃yQ(y )) References for further reading For more examples and further discussion on grammar and use of first-order languages and translations to and from natural language, see Tarski (1965), Kalish and Montague (1980), Barwise and Echemendy (1999), Hodges (2001), Smith (2003), Nederpelt and Kamareddine (2004), Bornat (2005), Chiswell and Hodges (2007), and van Benthem et al et al (2014) Exercises 3.3.1 Using the additional predicate I(x) for “x is an integer”, formalize the following sentences in the first-order language for the structure of real numbers R (a) Every square of an integer is greater than (b) Every square of a real number which is not an integer is greater than (c) Some real numbers are not integers (d) Every integer is even or odd (e) No integer is both even and odd (f) For every integer there is a greater integer (g) Every positive integer is a square of some negative real number (h) Not every real number is greater than an integer (i) There is an integer such that not every real number is greater than it (j) No real number is greater than every integer (k) Every real number which is not zero has a reciprocal (l) There is a real number which, when multiplied by any real number, produces that number (m) Between every two different real numbers there is an integer 3.3.2 Using unary predicates T(x) for “x talks” and L(x) for “x listens”, formalize the following sentences in a first-order language for the domain of all humans (a) Everybody talks or everybody listens (b) Everybody talks or listens (c) If John talks everybody listens (d) If somebody talks everybody listens