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Logic07

# Logic07 - Introduction to Logic Lecture 7 Brian Weatherson...

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Unformatted text preview: Introduction to Logic Lecture 7 Brian Weatherson, Department of Philosophy September 23, 2009 Logic 201 (Section 5) Lecture 7 1 / 65 Sentence Forms Every sentence we’ve seen so far is of one of the following four forms: 1 An atomic sentence; i.e., a predicate followed by a list of names 2 ¬ X 3 X ∧ Y 4 X ∨ Y Since X and Y need not be atomic sentences, sometimes it is hard to tell what the form of a sentence is. Logic 201 (Section 5) Lecture 7 2 / 65 The Main Connective In Chapter 4, they talk a lot about the “main connective” of sentences. If the sentence is atomic, it has no connective, and hence no main connective. If the sentence is of the form ¬ X , then ¬ is the main connective. If the sentence is of the form X ∧ Y , then ∧ is the main connective. If the sentence is of the form X ∨ Y , then ∨ is the main connective. Logic 201 (Section 5) Lecture 7 3 / 65 Main Connective: ¬ A ¬ is the main connective in two cases. 1 The sentence is of the form ¬ A , where A is an atomic sentence. 2 The sentence is of the form ¬( 1 ... ) 1 . In the first case, we say that the scope of the negation is the atomic sentence A . In the second case, we say that the scope of the negation is everything between ( 1 and ) 1 . Logic 201 (Section 5) Lecture 7 4 / 65 Main Connective: ∧ A ∧ is the main connective if two conditions are met. 1 The sentence is of the form X ∧ Y . 2 Both X and Y are complete sentences, with every bracket paired up in the appropriate way. Note that but X and Y may have brackets around their extremes. That’s OK. Both of these two strings are sentences. 1 Cube ( a ) ∧ Large ( a ) 2 ( Cube ( a ) ∧ Large ( a )) Logic 201 (Section 5) Lecture 7 5 / 65 Main Connective: ∨ A ∨ is the main connective if two conditions are met. 1 The sentence is of the form X ∨ Y . 2 Both X and Y are complete sentences, with every bracket paired up in the appropriate way. Note that but X and Y may have brackets around their extremes, like with ∧ . Logic 201 (Section 5) Lecture 7 6 / 65 Truth Tables Logic 201 (Section 5) Lecture 7 7 / 65 Representing Truth Conditions The truth conditions of a complex depend on the parts. Truth tables are used to display the dependencies Logic 201 (Section 5) Lecture 7 8 / 65 Representing Truth Conditions This is a truth table. We will use it to represent how the truth values of large sentences depend on their parts. This picture comes from the program ‘Boole’, which we’ll be using a lot today. Logic 201 (Section 5) Lecture 7 9 / 65 To the left are columns for each atomic sentence that’s a part of the sentence we’re considering. If there are n atomic sentences, then there are 2 n rows. So if there is one atomic sentence, there are two rows; if there are two atomic sentences, there are four rows; if there are three atomic sentences, there are eight rows, and so on....
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Logic07 - Introduction to Logic Lecture 7 Brian Weatherson...

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