Logic

Why Do We Need Proofs?

One aspect of mathematics that students often struggle with, particularly in geometry (which traditionally has been where proof is introduced), is writing proofs. Why do we need to learn about proofs? Why are proofs needed in the first place? Here are a few answers we’ve given to these questions. Why does math need proofs? First, …

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Patterns of Logical Argument

We’ve looked at various aspects of turning English sentences into logical statements, and modifying them by negation, converse, and so on. Let’s finish by looking at some questions about standard rules of inference, such as Modus Ponens and the Law of Syllogism. Four ways to argue We can start with a question about the basics, …

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Complicating the Converse

(An archive question of the week) Usually when we discuss converses (and inverses and contrapositives) we use clear, idealized examples. But statements in real life — even in real math — are not quite so straightforward. The difficulty is not merely in the language, but in the complexity of our statements. A question in the …

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Negating Logic Statements: How to Say “Not”

Last time, I started a series exploring aspects of the translation of English statements to or from formal logical terms and symbols, which will lead to discussions of converse and contrapositive, and eventually of logical arguments. We’ve looked at how to translate concepts of “or” (disjunction) and “if” (conditional); but our goals will also require …

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Translating Logic Statements

The next few posts will examine aspects of logic, both symbolic logic, and how we talk about theorems in general. We’ll start here with issues in interpreting the wording of logic, and some of the semantic difficulties we face. English isn’t logical. (Well, I suppose humans in general aren’t logical.) Which kind of OR? We’ll …

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Counting Kings

(A new problem of the week) Having discussed counting earlier this week, let’s take a look at a different kind of counting. The subject of combinatorics (the study of counting) arises in many guises: probability, sets, geometry. Here, we look at a relatively basic type of problem that involves the same sort of organized thinking …

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Venn Diagrams: Language Issues

(A new question of the week) I mentioned that we have had a number of questions related to Venn diagrams recently. Here I would like to show a couple of these, from a Philippine student. Even fluent English speakers can get confused in these problems; observing how a student new to the language misinterprets details …

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Venn Diagrams: Over the Top

(An archive question of the week) Last time we looked at various 2- and 3-set Venn diagram problems (and alternative methods). One discussion I found while collecting them deserved to be set aside for special examination, if only because it would scare the beginner. A mixture of tools will make the work easier. Here it …

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