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Unformatted text preview: Logic Margaret M. Fleck 19 January 2011 These notes cover propositional logic and predicate logic at a basic level. Some deeper issues will be covered later. 1 A bit about style Writing mathematics requires two things. You need to get the logical flow of ideas correct. And you also need to express yourself in standard style, in a way that is easy for humans (not computers) to read. Mathematical style is best taught by example and is similar to what happens in English classes. Mathematical writing uses a combination of equations and also parts that look superficially like English. Mathematical English is almost like normal English, but differs in some crucial ways. You are probably familiar with the fact that physicists use terms like “force” differently from everyone else. Or the fact that people from England think that “paraffin” is a liquid whereas that word refers to a solid substance in the US. We will try to highlight the places where mathematical English isn’t like normal English. You will also learn how to make the right choice between an equation and an equivalent piece of mathematical English. For example, ∧ is a shorthand symbol for “and.” The shorthand equations are used when we want to look at a complex structure all at once, e.g. discuss the logical structure of a proof. When writing the proof itself, it’s usually better to use the longer English equivalents, because the result is easier to read. There is no hard-and-fast line here, but we’ll help make sure you don’t go too far in either direction. 1 2 Propositions Two systems of logic are commonly used in mathematics: propositional logic and predicate logic. We’ll start by covering propositional logic. A proposition is a statement which is true or false (but never both!). For example, “Urbana is in Illinois” or 2 ≤ 15. It can’t be a question. It also can’t contain variables, e.g. x ≤ 9 isn’t a proposition. Sentence fragments without verbs (e.g. “bright blue flowers”) or arithmetic expressions (e.g. 5 + 17), aren’t propositions because they don’t state a claim. The lack of variables prevents propositional logic from being useful for very much, though it has some applications in circuit analysis, databases, and artificial intelligence. Predicate logic is an upgrade that adds variables. We will mostly be using predicate logic in this course. We just use propositional logic to get started. 3 Complex propositions Statements can be joined together to make more complex statements. For example, “Urbana is in Illinois and Margaret was born in Wisconsin.” To talk about complex sequences of statements without making everything too long, we represent each simple statement by a variable. E.g. if p is “Urbana is in Illinois” and q is “Margaret was born in Wisconsin”, then the whole long statement would be “ p and q ”. Or, using shorthand notation p ∧ q ....
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- Spring '08