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By Hirst H.P., Hirst J.L.

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In the table below, an OK appears if the term is free for x in the formula. If not, then an X appears. x y f(x,y) 3 ∀y(A(x, y) ∨ B(z)) OK X X OK A(x) ∨ ∀z(C(z, z) ∧ A(z, y)) OK OK OK OK B(y) → ∀y(A(x, z) ∧ ∃xC(x, y)) OK X X OK ∀x∃yD(x, y, z) OK OK OK OK Let’s summarize some shortcuts. We can plug x in for x in any formula, and not worry. We can plug a constant symbol in for x in any formula, and not worry. Note that we only ever plug terms into free occurrences of variables. We never plug terms of any sort into bounded occurrences of variables.

1. ∀xC(x) “All natural numbers are multiples of 10” is false, since 9 is not a multiple of 10. 6. TRUTH AND SENTENCES 41 2. ∀xL(x) “All natural numbers are even” is false, since 3 is not even. 3. ∃x(C(x) ∧ L(x)) “There is a natural number that is both even and a multiple of 10” is true. For example, 20 is such a number. 4. ∃x(L(x) ∧ ¬C(x)) “There is a natural number that is both even and not a multiple of 10” is true. For example, 4 is such a number. 5. ∀x(L(x) → C(x)) “For all natural numbers, being even implies being a multiple of 10” is false, since 4 is even but not a multiple of 10.

XL(x) “All people live on university campuses” is false. 3. ∃x(C(x) ∧ L(x)) “There is someone who is both a university chancellor and lives on a campus” is true. 4. ∃x(L(x) ∧ ¬C(x)) “There is someone who both lives on a campus and is not a chancellor” is true. 5. ∀x(L(x) → C(x)) “Living on a campus implies one is a chancellor” is false. Alternately, “every person who lives on a campus is a chancellor” is false. Example. Let M be the model where the universe is the natural numbers, C(x) means x is a multiple of 10, and L(x) means x is even.

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