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

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V. vi. There is a real number that is greater than any real number. There is a real number that is less than any real number. Given any real number, we can find a greater real number. Given any real number, we can find a lesser real number. If x and y are reals, then x is less than y. There are reals x and y such that x is less than y. 5. Formalize: No elbow is an ankle. 6. Assume that the universe is the set of all fish. Using the information below, formalize each of the given statements. • T (x) means “x is a trout” • S(x) means “x is shiny” • J(x) means “x jumps” (a) Formalize: All fish are trout.

Should it be? 28 CHAPTER 1. PROPOSITIONAL CALCULUS Chapter 2 Predicate Calculus Propositional calculus can express only the simplest of statements. Predicate calculus overcomes this difficulty by introducing variables and quantifiers. Variables will be used to represent an arbitrary object in the set of objects being studied, called the universe. Quantifiers will allow us to talk about a property holding for all objects or that there exists an object for which the property holds. The addition of quantified variables makes the language of predicate calculus sufficiently rich to express almost any mathematical notion.

Also, mathematicians follow the format of L25 to prove biconditional statements. 11. ASSESSING PROPOSITIONAL CALCULUS 27 The big disadvantage of propositional calculus is that it glosses over any fine distinctions. It’s just not very expressive. ” If we let P denote n > 0 and Q denote n+1 > 0, then our formalization is P → Q. This certainly shows us that the statement is an implication, but it hides the fact that the hypothesis and the conclusion are both talking about n. In order to overcome this limitation, we need a logical system that includes variables.