149 Understanding First-order Logic (e) (i) As in (m), but replacing No mathematicians are stupid Some mathematicians are bachelors Some bachelors are not stupid (f) No poor people are politicians Some politicians are crooks Some crooks are not poor people (g) “Some mothers hate watching TV” with “All mothers like watching TV.” (j) No successful politicians are poor Someone is poor only if he is a politician Therefore, nobody who is poor is successful (k) No penguins are white Some penguins are fat All penguins are birds Therefore, no fat bird is white Some penguins are white All penguins are birds Therefore, some white bird is a penguin (h) (l) Every mother loves some child Every mother is a woman Some mothers hate watching TV Some woman loves some child Every bachelor loves a woman he knows Every bachelor is a man Eve is a woman whom every man knows Some bachelor loves some woman (m) No food is better than a good steak Stale bread is better than no food Stale bread is better than a good steak (n) As above, but adding a new premise: For any x, y, z , if x is better than y and y is better than z then x is better than z (Watch out for the quantifiers!) Alonzo Church (14.6.1903–11.8.1995) was an American mathematician and one of the leading logicians of the 20th century, who made profound contributions to mathematical logic and the theoretical foundations of computer science Church studied and had his academic career at the Mathematics Department of Princeton University where he worked until 1967 He then formally retired and moved to the University of California, Los Angeles, as Flint Professor of Philosophy and Mathematics for a further 23 years, until his final retirement at age 87 in 1990 In the early 1930s Church created lambda-calculus as a formal system of defining functions that are effectively computable In connection with his two-part work A Set of Postulates for the Foundations of Logic, published in 1932–1933, he intended to develop a set of axioms that “would lead to a system of mathematical logic free of