s2_1 - CHAPTER 2 Logic 1. Logic Denitions 1.1....

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CHAPTER 2 Logic 1. Logic Definitions 1.1. Propositions. Definition 1.1.1 . A proposition is a declarative sentence that is either true (denoted either T or 1) or false (denoted either F or 0). Notation: Variables are used to represent propositions. The most common variables used are p, q, and r . Discussion Logic has been studied since the classical Greek period ( 600-300BC). The Greeks, most notably Thales, were the first to formally analyze the reasoning process. Aristo- tle (384-322BC), the “father of logic”, and many other Greeks searched for universal truths that were irrefutable. A second great period for logic came with the use of sym- bols to simplify complicated logical arguments. Gottfried Leibniz (1646-1716) began this work at age 14, but failed to provide a workable foundation for symbolic logic. George Boole (1815-1864) is considered the “father of symbolic logic”. He developed logic as an abstract mathematical system consisting of defined terms (propositions), operations (conjunction, disjunction, and negation), and rules for using the opera- tions. It is this system that we will study in the first section. Boole’s basic idea was that if simple propositions could be represented by pre- cise symbols, the relation between the propositions could be read as precisely as an algebraic equation. Boole developed an “algebra of logic” in which certain types of reasoning were reduced to manipulations of symbols. 1.2. Examples. Example 1.2.1 . “Drilling for oil caused dinosaurs to become extinct.” is a propo- sition. 21
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1. LOGIC DEFINITIONS 22 Example 1.2.2 . “Look out!” is not a proposition. Example 1.2.3 . “How far is it to the next town?” is not a proposition. Example 1.2.4 . x + 2 = 2 x ” is not a proposition. Example 1.2.5 . x + 2 = 2 x when x = - 2 ” is a proposition. Recall a proposition is a declarative sentence that is either true or false. Here are some further examples of propositions: Example 1.2.6 . All cows are brown. Example 1.2.7 . The Earth is further from the sun than Venus. Example 1.2.8 . There is life on Mars. Example 1.2.9 . 2 × 2 = 5 . Here are some sentences that are not propositions. Example 1.2.10 . “Do you want to go to the movies?” Since a question is not a declarative sentence, it fails to be a proposition. Example 1.2.11 . “Clean up your room.” Likewise, an imperative is not a declar- ative sentence; hence, fails to be a proposition. Example 1.2.12 . 2 x = 2 + x .” This is a declarative sentence, but unless x is assigned a value or is otherwise prescribed, the sentence neither true nor false, hence, not a proposition. Example 1.2.13 . “This sentence is false.” What happens if you assume this state- ment is true? false? This example is called a paradox and is not a proposition, because it is neither true nor false. Each proposition can be assigned one of two truth values . We use T or 1 for true and use F or 0 for false.
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s2_1 - CHAPTER 2 Logic 1. Logic Denitions 1.1....

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