Logic06h

# Logic06h - Introduction to Logic Lecture 6 Brian Weatherson...

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Unformatted text preview: Introduction to Logic Lecture 6 Brian Weatherson, Department of Philosophy September 21, 2009 Logic 201 (Section 5) Lecture 6 1 / 44 Translation Logic 201 (Section 5) Lecture 6 2 / 44 Conditions on Translation We’re going to spend a bit of time looking at translations back and forth between English and FOL. Learning FOL is something like learning a new language. And one way to tell whether you’ve learned a new language is to check whether you can produce good translations between it and a language you’re familiar with, in this case English. But to do this, we need to say a little about what makes a translation good. Logic 201 (Section 5) Lecture 6 3 / 44 Conditions on Translation When we produce translations, we mostly care about truth and falsity. A good translation is true in exactly the same circumstances as the sentence it translates. Logic 201 (Section 5) Lecture 6 4 / 44 Conditions on Translation A good translation is true in exactly the same circumstances as the sentence it translates. That doesn’t just mean we have to preserve truth value in the actual world. Imagine that a really is a cube in the world we’re considering. Now consider the following two sentences, (1) in FOL and (2) in English. 1 Cube ( a ) 2 New York is large. (1) and (2) have the same truth value, i.e. true, but that doesn’t mean (2) is a good translation of (1). Logic 201 (Section 5) Lecture 6 5 / 44 Conditions on Translation A good translation is true in exactly the same circumstances as the sentence it translates. That means if (1) is a good translation of (2), the following conditions must be met. 1 (1) and (2) must have the same truth value; and 2 (1) and (2) must keep having the same truth value even when we change the world around. If we changed a to a tetrahedron, then Cube ( a ) would be false, but “New York is large” would still be true. So they aren’t good translations of one another. Logic 201 (Section 5) Lecture 6 6 / 44 Pronouns FOL doesn’t have any pronouns, as English does, so if you want to translate sentences into FOL from English, you need to replace the pronouns with names. Consider this example. 1 a is a cube and it is large. To translate it into FOL, we need to first replace the pronoun, the bolded it , with the name of the object it refers to. That would give us: a is a cube and a is large. And that is easy to translate as Cube ( a ) ∧ Large ( a ) . Logic 201 (Section 5) Lecture 6 7 / 44 Pronouns This is the first of several translation rules we’ll use. Rule 1: Replace all pronouns Since FOL doesn’t have pronouns, this will make translations clearer. Logic 201 (Section 5) Lecture 6 8 / 44 Condensed Conjunctions There are many ways of shortening a conjunction in English....
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Logic06h - Introduction to Logic Lecture 6 Brian Weatherson...

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