Geography Lecture

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Geography Lecture

  1. 1. PHIL 201: Introduction to Symbolic Logic Spring 2009 Instructor Information Instructor: Alex Morgan Office: Room 011, Davison Hall, Douglass Campus Office Hours: M 6.00-7.30pm, Scott Hall (locn. TBA) Email: amorgan@philosophy.rutgers.edu Phone: (732) 932 9861, ext.172 Internet: http://eden.rutgers.edu/~amorgo/ Textbook Hardegree, G. ‘Symbolic Logic, A First Course’ (2nd Edition) • Available online here: www-unix.oit.umass.edu/~gmhwww/110/text.htm • • • Also available as hardcopy from bookstores like Amazon I will be referring to the online version Known typos are listed on Hardegree’s website
  2. 2. Course Website www.rci.rutgers.edu/~amorgo/teaching/09s_201/ • • Provides downloads, including the syllabus and these course notes • Allows you to ask questions about the homework (see the site for instructions, or contact me) • Regularly updated throughout the semester, so check often! Provides news and information, including information about the homework and exams Assessment Homework (20%) • A total of 10 bi-weekly homework assignments based on the exercises in the textbook, each worth 2%. Collected at the end of the Monday class. The main point of the homework is to demonstrate that you’re actively working through the material. Exams (80%) • Two exams, a mid-term and a final, each worth 40%. They’ll be held around March 4 and May 4, respectively. I’ll provide more information about the exams later. What to Expect • This course is very different from most other courses in philosophy (and the humanities generally) • We’ll be learning how to use an artificial symbolic language, similar to mathematical ‘languages’ like algebra • The emphasis will be on... ‣ skills rather than facts and ideas, ‣ rigor and precision rather than creativity and interpretation (at least in these early stages)
  3. 3. What to Expect • If you enjoy programming, logic puzzles, Sudoku, etc., then you will probably take to this material quickly, and may even find it fun! • • If not, you should be prepared to put in some extra work • However, some students have difficulty with the kind of abstract, rulebased thinking required in this course. If this sounds like you (e.g. if you have difficulty with algebra or computer programming), please come talk to me after class Either way, so long you put in the work, you’re almost guaranteed a good grade What to Expect • Please note that this is not the ‘easy logic course’ that you might’ve heard about! (that’s 730:101) • Here are some grade distributions from previous semesters: 7 5 6 # Students 8 6 # Students 7 4 3 2 5 4 3 2 1 1 0 0 A B+ B C+ C D F A B+ B C+ C D F Grade Grade Advice • The material we’re covering might seem easy to begin with, but it quickly gets much harder. If you get behind it will be very difficult for you to catch up • The course is more about learning skills than learning facts, so it is crucial that you do lots and LOTS of practice using the exercises in the textbook • If you find yourself struggling with the course, please come see me after class or during office hours
  4. 4. Why Learn Logic? • Symbolic logic will help you to be a better reasoner; it will provide you with a set of tools for analyzing arguments and determining whether they’re any good ‣ Note that the emphasis of the course is not on practical reasoning; if that’s your main interest, take 730:101 • Some understanding of logic is presupposed in virtually all areas of contemporary philosophy. Logic is used to analyze complex arguments, and underlies philosophical theories of meaning, truth and thought • • Logic is used in linguistics to understand syntax and semantics Logic provides the conceptual foundations of computer science, and is studied in its own right as a branch of pure math (heard of Goedel’s incompleteness theorems?) What is Logic? • • Logic is the study of the principles of ‘good’ or ‘correct’ reasoning • Some inferences seem good, while others seem not so good Reasoning involves making inferences from one set of information to another set of information ‣ If I see smoke and infer that there is fire, this seems like a good inference ‣ If I see smoke and infer that the moon is made of cheese, this doesn’t seem like a good inference What is Logic? • Systems of logic were studied in Ancient Greece, China and India • In Ancient Greece, Aristotle developed a system of logic that was based on the analysis of certain kinds of inferences called syllogisms (more on these later) • Aristotle's system became the basis of Wester logic for almost 2,000 years
  5. 5. What is Symbolic Logic? • In the late 1800s, logicians broke from the Aristotelian tradition and attempted bring the rigor and precision of mathematics to bear on logic • They attempted to study logical inference using formal, axiomatic languages • This provided a more precise way of analyzing logical inferences by avoiding the ambiguity of natural languages like English • The main figure in the development of symbolic logic was a German logician named Gottlob Frege What is Logic? • Recall that logic in general is the study of good inferences. In formal logic, we focus on a particular kind of inference, called an argument • An argument means many things in ordinary language, but for us it will mean something quite specific: ‣ An argument is a collection of statements, one of which is the conclusion, and the remainder of which are the premises, where the premises are intended to ‘support’ or justify the conclusion What is an Argument?
  6. 6. Statements • • Recall that an argument is a set of statements • Different kinds of sentences: A statement is a declarative sentence, i.e. a sentence that is capable of being true or false We’re interested in these! ‣ Declarative “The window is shut” ‣ Interrogative “Is the window shut?” ‣ Imperative “Shut the window!” Statements • Which of the following are declarative sentences? ‣ Shut the door ‣ It is raining ‣ Are you hungry? ‣ 2+2=4 ‣ I am the King of France Note that whether or not a sentence is declarative doesn’t depend on whether the sentence is in fact true, but whether it expresses something that could be true Statements vs. Propositions • A statement (i.e. a declarative sentence) is said to express a proposition. You can think of a proposition as (roughly) the meaning of a statement • While a statement is something concrete (e.g. a symbol or a soundwave), a proposition is abstract
  7. 7. Statements vs. Propositions • The distinction is similar to the distinction between mathematical expressions and the numbers they stand for: ‣ ‘4’ and ‘2+2’ and are different mathematical expressions for the same number, namely 4 ‣ Similarly, ‘snow is white’ and ‘der Schnee ist weiss’ are different statements that express the same proposition, namely that snow is white • The distinction is important, but won’t have much of an impact on what we do in this course More on Arguments • Examples of arguments: Are these arguments good? Why? (1). If there is smoke, there is fire There is smoke Therefore, there is fire (2). If there is smoke, there is fire There is smoke Therefore, I am the King of France PREMISES CONCLUSION PREMISES CONCLUSION More on Arguments (1). If there is smoke, there is fire There is smoke Therefore, there is fire (2). If there is smoke, there is fire This seems like a good argument because the conclusion in some sense follows from the premises This seems like a bad argument There is smoke because the conclusion has Therefore, I am the King of France nothing to do with the premises!
  8. 8. Validity • • How can we make this notion of ‘following from’ more precise? With the notion of validity: ‣ To say that an argument is valid means that it is impossible for the conclusion of the argument to be false if the premises are true • Validity has to do with the structure, or form, of the argument, and is independent of whether the premises of the argument are in fact true • An argument that is valid and has true premises is called sound Validity • More examples of arguments: Assume that the premises are true; can the conclusion be false? (3). All cats are dogs NO! The argument is valid All dogs are reptiles Therefore, all cats are reptiles (4). All cats are vertebrates YES! The argument is invalid All mammals are vertebrates Therefore, all cats are mammals Validity T F All dogs are reptiles T Therefore, all cats are reptiles T F • If the premises were true, the conclusion would have to be true, so the argument is valid. • However, the premises are in fact false, so the argument is not sound • In terms of its form, the argument is ‘good’, but in terms of its content the argument is not F (3). All cats are dogs cats dogs reptiles
  9. 9. Validity All mammals are vertebrates F Even though the premises are true, the conclusion could still be false, so the argument is not valid • Even though it has all true premises, it is not valid, so it is automatically not sound In terms of its content, the argument is ‘good’, but in terms of its form, the argument is not T T Therefore, all cats are mammals • • T T (4). All cats are vertebrates cats mammals T vertebrates Validity • Comprehension questions: ‣ Can a valid argument have a false conclusion? Yes ‣ Can a valid argument with true premises have a false conclusion? No ‣ Can anyone give an example of a valid argument with true premises? • Example: (5). All cats are mammals All mammals are vertebrates (premise 1) T (premise 2) T Therefore, all cats are vertebrates (conclusion) T Why is this valid? Why sound? Validity and Logical Form • We saw that arguments (3) and (5) are both valid, and that validity has to do with form. In fact, (3) and (5) have the same form: (3). All cats are dogs All dogs are reptiles Therefore, all cats are reptiles All X are Y All Y are Z (5). All cats are mammals All mammals are vertebrates Therefore, all cats are vertebrates Therefore, all X are Z
  10. 10. Validity and Logical Form • On the other hand, (4) has a different form: All X are Y (4). All cats are vertebrates All mammals are vertebrates All Z are Y Therefore, all cats are mammals Therefore, all X are Z • If an argument is valid, then any argument with the same form is also valid • If an argument is invalid, then any argument with the same form is also invalid Validity and Logical Form • On the other hand, (4) has a different form: (4). All cats are vertebrates All X are Y All mammals are vertebrates All Z are Y Therefore, all cats are mammals Therefore, all X are Z Note that in the textbook, statements like these are called concrete sentences... ...and these are called sentence forms. Sentence forms don’t express a particular proposition Deductive vs. Inductive Logic • The kind of logic that we study in this class is concerned with arguments in which the premises are supposed to logically guarantee the conclusion -- if the premises are true, the conclusion has to be true. This is called deductive logic • There is another kind of logic that is concerned with arguments in which the premises are supposed to make the conclusion more likely, but not necessarily certain. This is called inductive logic, and is a much more complicated subject than deductive logic
  11. 11. Deductive vs. Inductive Logic • • Recall argument (1): Now consider argument (7): ‣ If there is smoke, there is fire ‣ There is smoke ‣ There is smoke ‣ Therefore, there is fire ‣ Therefore, there is fire This is a deductive argument because the truth of the premises logically guarantees the truth of the conclusion This is an inductive argument because the truth of the premise makes the conclusion more likely, but doesn’t guarantee it Syllogisms • A syllogism has two premises and a conclusion • The statements that make up a syllogism contain descriptive terms that refer to sets of things (e.g. ‘cat’, ‘dog’) • The statements also contain logical terms like ‘all’, ‘some’, ‘none’, which describe relations between sets of things (7). Some cats are dogs All dogs are reptiles Therefore, all cats are reptiles dogs cats reptiles Syllogisms • • For example, the first premise in (7) says that some cats are dogs - in other words, that some of the things in the ‘cat set’ are in the ‘dog set’ (7). Some cats are dogs All dogs are reptiles Therefore, all cats are reptiles Questions: ‣ Is (7) valid? Sound? ‣ What is the logical form of (7)? dogs cats reptiles
  12. 12. Next Time... • Please finish Ch. 1 and make a start on Ch. 2

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