In predicate logic, we use predicates and quantifiers to express statements about objects in a domain. Today we'll work with a library domain where we have books and other items. We define three predicates: B of x means x is a book, L of x means x is available for loan, and N of x means x is new.
Let's examine statement i: There exists a book that is new but not available for loan. We express this using the existential quantifier exists x, combined with logical connectors. The formula is: exists x, open parenthesis, B of x and N of x and not L of x, close parenthesis. This means there is at least one item x in our domain that satisfies all three conditions: it is a book, it is new, and it is not available for loan.
Now let's examine statement ii: No book is available for loan. We can express this in two equivalent ways. First, using the universal quantifier: for all x, if x is a book, then x is not available for loan. Second, using negation of existential quantifier: there does not exist an x such that x is a book and x is available for loan. Both formulations capture the same meaning that every book in the library is unavailable for loan.
Let's analyze the logical structure of both statements. Statement i uses an existential quantifier to assert that at least one object satisfies the given conditions. It combines three predicates using conjunction, meaning all conditions must be true simultaneously. Statement ii uses a universal quantifier with implication, stating that for every object, if it's a book, then it's not available for loan. These different logical structures reflect the different meanings we want to express in predicate logic.
In summary, we have successfully translated natural language statements into precise predicate logic formulas. Statement i, expressing the existence of a new book not available for loan, uses an existential quantifier with conjunction. Statement ii, expressing that no book is available for loan, uses a universal quantifier with implication, or equivalently, negation of an existential statement. This demonstrates how predicate logic provides a powerful framework for expressing mathematical and logical relationships with precision and clarity.