Phil. 160 Symbolic Logic II

PHIL. 160
Symbolic Logic II

Spring Semester 2009

Prof. Dowden

Catalog description: Further study of deductive logic. Topics include: principles of inference for quantified predicate logic; connectives; quantifiers; relations; sets; modality; properties of formal logical systems, e.g. consistency and completeness; and interpretations of deductive systems in mathematics, science, and ordinary language.

Prerequisite: MATH 031, or PHIL 060, or instructor permission. 3 units.
Further course description: Our course will go beyond Phil. 60 to explore advances in symbolic deductive logic, showing how these advances deepen our understanding of the nature of good reasoning, including its scope and its limits.

Logic is the theory of good reasoning, and symbolic logic is logic that uses symbols. Those people who devise symbolic logics are trying achieve more precise and rigorous theories than can be achieved when they limit themselves to ordinary words.

Our course will survey the deep results yielded by this study of symbolic logic. These results concern the surpising extent to which human knowledge can not be freed of contradictions, whether our knowledge can be expressed without loss of content inside of a formal language, and what our civilization has learned from the field of symbolic logic about the limits to what people can know.

We will begin our course with a review of Phil. 60 and the rigorous development of both statement logic (also called propositional logic) and predicate logic (also called quantificational logic). Then we will learn about their applications, extensions and meta-theory.

Textbook: Elements of Deductive Inference: An Introduction to Symbolic Logic by Joseph Bessie and Stuart Glennan. Here's a picture of the cover:

We won't be using the CD (compact disk) that accompanies the textbook, so you can confidently buy a used copy of the book that doesn't contain the CD. There will also be some required reading from leaflets handed out in class, and from web pages.

Grades: Your grade will be determined by four homework assignments (each 12%), two tests (each 13%), and a comprehensive final examination (26%). Tests are open book. Homework questions will be handed out a week in advance of the due date. Class attendance is optional, but you are responsible for material covered in class that isn't in the book.

Due dates:

Homework 1: Feb. 12, 2009

Homework 2: Feb. 26

Test 1: Mar. 12

Homework 3: Apr. 7

Test 2: Apr. 21

Homework 4: May 7

Final exam: May 21, Thursday, 10:15 a.m.

Late assignments, and make-up assignments: I realize that during your college career you occasionally may be unable to complete an assignment on time. If this happens in our course, contact me as soon as you are able. If you provide me with a good reason for missing a test or homework assignment (illness, accident, etc.), then I'll use your grade on the final exam as your missing grade. There will be no make-up tests nor make-up homework. I do accept late homework with a grade penalty of one-third of a letter grade per 24-hour period beginning at the class time the assignment is due. Examples. If you turn in the assignment a few hours after it is due, then your A becomes an A-. Instead, if you turn in the same assignment 30 hours late, then your A becomes a B+. Weekends count. No late work will be accepted after the answer sheet has been handed out (normally this will be at the next class meeting) nor after the answers are discussed in class, even if you weren't in class that day.

Add-Drop: To add the course, try to do so by using the CMS system. If the course is full, then see me about signing up on the waiting list. To drop the course during the first two weeks, use the CMS system. No paperwork is required. After the first two weeks, it is harder to drop, and a departmental form is required, the "Petition to Add/Drop After Deadline." As with any university course, make sure you are dropped officially (by CMS or by the instructor or department secretary); don't simply walk away into the ozone or else you will get a "U" grade for the course, which is counted as an "F" in computing your GPA (grade point average).

Laptops, cell phones: No photographing or recording during class is allowed without permission of the instructor. During class, turn off your phone. Your laptops may be used only for note taking, and not for browsing the web, reading emails, or other activities unrelated to the class.

Testing protocol: For in-class tests, you may use your books and notes but not your computer or phone.

Disabilities: If you have a documented disability and require accommodation or assistance with assignments, tests, attendance, note taking, etc., please see me early in the semester so that appropriate arrangements can be made to ensure your full participation in class. Also, you are encouraged to contact the Services for Students with Disabilities (Lassen Hall) for additional information regarding services that might be available to you.

Honesty: See the University's policy on honesty and cheating. A student tutorial on how not to plagiarize is available online from our library.

Food: Except for water, please do not eat or drink during class. You are welcome to leave class anytime if the need arises.

Professor: My office is in Mendocino Hall, room 3022, and my weekly office hours will be announced in class on the first day. Feel free to stop by at any of those times, or to call. If those hours are inconvenient for you, then I can arrange an appointment for an alternative time. You may send me e-mail at dowden@csus.edu or call my office at 278-7384 or the Philosophy Department Office at 278-6424. The fastest way to contact me is by email. My personal web page is at http://www.csus.edu/indiv/d/dowdenb/index.htm

Prof. Dowden

Schedule of topics and readings:

Week 1: Survey of What is Ahead, and a Review of the Syntax of Statement Logic.

Reading: Pages 3-4 of the handout "What is Logic?" by John Nolt, and the webpage "What is inconsistency?", and browse chapters 1 and 2 in your Bessie textbook. Note the assumption about fixed context that your authors make in the middle paragraph on p. 22, and note the difference between material implication and strict implication on pages 49 and 50. If you don't get all these pages read by the first week, don't worry; read them in the weeks ahead. Besides, you probably already know everything in these chapters.

topics: Review of deduction vs. induction, how to symbolize an argument, and the scope of a connective. Survey of these new topics: non-classical logic, solutions to the liar paradox (from Russell, Tarski, Quine, Kripke, and Priest), and Gödel's Incompleteness Theorems.

Week 2: Semantics of Statement Logic.

Reading: chapter 3, sections 1-6, 8. Browse sections 7, 9-11.

topics: truth tables and connectives, interpretation of an uninterpreted formal language, tautology, contradiction, truth tree, truth under an interpretation.

Week 3: Digital Logic and Computer Design

Reading: handouts.

topics: applications of Statement Logic to switching algebra, series-parallel electrical switching circuits, logic gates, resistor-transistor logic for inverters, NAND-gates, combinational circuits, disjunctive normal form and sum of products, half-adders, sequential circuits, feedback, pattern recognition.

Week 4: Natural Deduction in Statement Logic.

Reading: chapter 4.1-4.8.

topics: conditional proof, nested subproof, reductio ad absurdum, derived rule, truth-functional validity, truth-functional truth, truth-functional contingency, truth-functional equivalence, truth-functional contradiction, truth-functional consistency, sound vs. unsound inference rules, weak vs. strong completeness.

Week 5: Introduction to Predicate Logic.

Reading: Chapter 5 in this order: 5.1, 5.2, 5.3, 5.6, 5.7, 5.4, 5.5, 5.8.

topics: wff, existential quantifier, bound vs. free occurrence of a variable, vacuous quantification, nested quantifier, intension vs. extension, domain, symbol key vs. interpretation, truth under an interpretation, expansions of quantifiers, counterexample to a statement, counterexample to an argument, categorical forms, Euler diagrams, valid argument, logical falsehood, equivalent pair, contradictory pair, consistent set, logical symbol vs. nonlogical symbol, metalanguage, polyadic predicate, reflexive relation, transitive relation, symmetric relation, equivalence relation.

Week 6: Meta-Logic of Predicate Logic.

Reading: chapter 6, but don't learn the tree method; instead, concentrate on the ideas below from pages 265, 278-81, 284-5, 287, 290-1, question 3 on 297, and 298-303.

topics: model of a set, restriction to monadic predicate logic, infinite vs. finite model, sound and complete deduction system, effective decision procedure, decidable set, decidable language, G�del�s Completeness Theorem for predicate logic, Church's Thesis, Church-Turing Undecidability Theorem.

Weeks 7-8: Natural Deduction in Predicate Logic with Identity and Functions. More on Meta-Logic.

Reading: chapter 7 and sections 8.1-8.4, 8.6-8.9, 8.11.

topics: creating proofs, compactness, consequences from an infinite set of premises, forcing an infinite model, weak mathematical induction, compound functional expressions.

Weeks 9-11: Definite Descriptions, Axioms for Arithmetic, and G�del's Incompleteness Theorems.

Reading: 9.1-9.3 [skip bottom half of p. 394 and top two lines of p. 395], 9.4, p. 354 Part V, and "G�del and the Nature of Mathematical Truth,"

topics: Meinong and Russell's theory of definite descriptions, Principia Mathematica, non-logical axioms, Peano arithmetic, Robinson arithmetic, non-standard model, isomorphic model, categorical theory, kinds of completeness, G�del number, undecidability, logicism, Hilbert's Program. L�wenheim-Skolem Theorem, global truth predicate, liar paradox, Tarski's Undefinability Theorem.

Week 12: Compactness.

Reading: handouts.

topics: George Berkeley's criticism of calculus. Robinson's use of the Compactness Theorem to justify Leibniz's Infinitesimals. Hyperreal numbers.

Week 13: Axiom Systems for Theories in Physical Science, Higher-Order Logic.

Reading: 9.5, 9.6.

topics: analysis of a law of nature, Hempel's Raven Paradox, theory reduction, bridge law, identity of indiscernibles, indiscernibility of identicals, many-sorted first-order language, second-order logic.

Weeks 14-15: Intensional Logic and Many-Valued Logic.

Reading: 9.7, 9.9, 9.10, 9.12.

topics: Brief survey of modal logic, epistemic logic, deontic logic, possible world semantics, tense logic, designated value, three-valued logic, fuzzy logic, paraconsistent logic, truth gaps, sorites paradox.

The above schedule of course topics may be changed somewhat as we progress through the semester, but these changes, if any, are not expected to affect the schedule of the homeworks and tests.

Relevance of logic to other subjects: Click on the ticket:

Student outcome goals: The hope is that by the end of the semester you will have achieved the following goals:

Be able to reason more effectively.

Be able to describe the scope of logic, that is, what symbolic logic can be used to do; and be able to describe the limits of logic, that is, what symbolic logic cannot be used to do.

Build on the abilities you learned in Phil. 60 to recognize when an English argument is capable of being analyzed with symbolic techniques, to translate a symbolic argument into English and vice versa, to determine if a symbolic sentence is logically true, to determine if a set of symbolic sentences is consistent, to assess the validity or invalidity of arguments using the techniques of symbolic logic, to create proofs in both predicate logic and statement logic, and be capable of creating and analyzing rigorous proofs using the methods of classical symbolic logic.

Know the important extensions of these logics to non-standard logics such as deontic logic, free logic, intensional logic, modal logic, many-valued logic, second-order logic, many-sorted logic, paraconsistent logic, and fuzzy logic.

Understand Hilbert's program and the process of formally axiomatizing a theory.

Be familiar with the most important meta-theoretic results such as G�del's Theorems, Church-Turing Undecidability Theorem, Tarski's Undefinability Theorem, and the L�wenheim-Skolem Theorem.

Be able to say how symbolic logic has deepened our knowledge of some important philosophical issues, and how it has led to new issues of its own.

Know the extent to which human knowledge can be freed of contradictions.

Be able to say what our civilization has learned from the field of symbolic logic about the limits to what people can know.

PHILOSOPHY DEPARTMENT / PROF. DOWDEN / CSUS
Updated: May 10, 2009


PHIL. 160 Symbolic Logic II Spring Semester 2009 Prof. Dowden


	Catalog description: Further study of deductive logic. Topics include: principles of inference for quantified predicate logic; connectives; quantifiers; relations; sets; modality; properties of formal logical systems, e.g. consistency and completeness; and interpretations of deductive systems in mathematics, science, and ordinary language. Prerequisite: MATH 031, or PHIL 060, or instructor permission. 3 units. Further course description: Our course will go beyond Phil. 60 to explore advances in symbolic deductive logic, showing how these advances deepen our understanding of the nature of good reasoning, including its scope and its limits. Logic is the theory of good reasoning, and symbolic logic is logic that uses symbols. Those people who devise symbolic logics are trying achieve more precise and rigorous theories than can be achieved when they limit themselves to ordinary words. Our course will survey the deep results yielded by this study of symbolic logic. These results concern the surpising extent to which human knowledge can not be freed of contradictions, whether our knowledge can be expressed without loss of content inside of a formal language, and what our civilization has learned from the field of symbolic logic about the limits to what people can know. We will begin our course with a review of Phil. 60 and the rigorous development of both statement logic (also called propositional logic) and predicate logic (also called quantificational logic). Then we will learn about their applications, extensions and meta-theory. *Textbook: Elements of Deductive Inference: An Introduction to Symbolic Logic* by Joseph Bessie and Stuart Glennan. Here's a picture of the cover: We won't be using the CD (compact disk) that accompanies the textbook, so you can confidently buy a used copy of the book that doesn't contain the CD. There will also be some required reading from leaflets handed out in class, and from web pages. Grades: Your grade will be determined by four homework assignments (each 12%), two tests (each 13%), and a comprehensive final examination (26%). Tests are open book. Homework questions will be handed out a week in advance of the due date. Class attendance is optional, but you are responsible for material covered in class that isn't in the book. Due dates: Homework 1: Feb. 12, 2009 Homework 2: Feb. 26 Test 1: Mar. 12 Homework 3: Apr. 7 Test 2: Apr. 21 Homework 4: May 7 Final exam: May 21, Thursday, 10:15 a.m.** Late assignments, and make-up assignments: I realize that during your college career you occasionally may be unable to complete an assignment on time. If this happens in our course, contact me as soon as you are able. If you provide me with a good reason for missing a test or homework assignment (illness, accident, etc.), then I'll use your grade on the final exam as your missing grade. There will be no make-up tests nor make-up homework. I do accept late homework with a grade penalty of one-third of a letter grade per 24-hour period beginning at the class time the assignment is due. Examples. If you turn in the assignment a few hours after it is due, then your A becomes an A-. Instead, if you turn in the same assignment 30 hours late, then your A becomes a B+. Weekends count. No late work will be accepted after the answer sheet has been handed out (normally this will be at the next class meeting) nor after the answers are discussed in class, even if you weren't in class that day. Add-Drop: To add the course, try to do so by using the CMS system. If the course is full, then see me about signing up on the waiting list. To drop the course during the first two weeks, use the CMS system. No paperwork is required. After the first two weeks, it is harder to drop, and a departmental form is required, the "Petition to Add/Drop After Deadline." As with any university course, make sure you are dropped officially (by CMS or by the instructor or department secretary); don't simply walk away into the ozone or else you will get a "U" grade for the course, which is counted as an "F" in computing your GPA (grade point average). Laptops, cell phones: No photographing or recording during class is allowed without permission of the instructor. During class, turn off your phone. Your laptops may be used only for note taking, and not for browsing the web, reading emails, or other activities unrelated to the class. Testing protocol: For in-class tests, you may use your books and notes but not your computer or phone. Disabilities: If you have a documented disability and require accommodation or assistance with assignments, tests, attendance, note taking, etc., please see me early in the semester so that appropriate arrangements can be made to ensure your full participation in class. Also, you are encouraged to contact the Services for Students with Disabilities (Lassen Hall) for additional information regarding services that might be available to you. Honesty: See the University's policy on honesty and cheating. A student tutorial on how not to plagiarize is available online from our library. Food: Except for water, please do not eat or drink during class. You are welcome to leave class anytime if the need arises. Professor: My office is in Mendocino Hall, room 3022, and my weekly office hours will be announced in class on the first day. Feel free to stop by at any of those times, or to call. If those hours are inconvenient for you, then I can arrange an appointment for an alternative time. You may send me e-mail at dowden@csus.edu or call my office at 278-7384 or the Philosophy Department Office at 278-6424. The fastest way to contact me is by email. My personal web page is at http://www.csus.edu/indiv/d/dowdenb/index.htm Prof. Dowden Schedule of topics and readings: Week 1: Survey of What is Ahead, and a Review of the Syntax of Statement Logic. Reading: Pages 3-4 of the handout "What is Logic?" by John Nolt, and the webpage "What is inconsistency?", and browse chapters 1 and 2 in your Bessie textbook. Note the assumption about fixed context that your authors make in the middle paragraph on p. 22, and note the difference between material implication and strict implication on pages 49 and 50. If you don't get all these pages read by the first week, don't worry; read them in the weeks ahead. Besides, you probably already know everything in these chapters. topics: Review of deduction vs. induction, how to symbolize an argument, and the scope of a connective. Survey of these new topics: non-classical logic, solutions to the liar paradox (from Russell, Tarski, Quine, Kripke, and Priest), and Gödel's Incompleteness Theorems. Week 2: Semantics of Statement Logic. Reading: chapter 3, sections 1-6, 8. Browse sections 7, 9-11. topics: truth tables and connectives, interpretation of an uninterpreted formal language, tautology, contradiction, truth tree, truth under an interpretation. Week 3: Digital Logic and Computer Design Reading: handouts. topics: applications of Statement Logic to switching algebra, series-parallel electrical switching circuits, logic gates, resistor-transistor logic for inverters, NAND-gates, combinational circuits, disjunctive normal form and sum of products, half-adders, sequential circuits, feedback, pattern recognition. Week 4: Natural Deduction in Statement Logic. Reading: chapter 4.1-4.8. topics: conditional proof, nested subproof, reductio ad absurdum, derived rule, truth-functional validity, truth-functional truth, truth-functional contingency, truth-functional equivalence, truth-functional contradiction, truth-functional consistency, sound vs. unsound inference rules, weak vs. strong completeness. Week 5: Introduction to Predicate Logic. Reading: Chapter 5 in this order: 5.1, 5.2, 5.3, 5.6, 5.7, 5.4, 5.5, 5.8. topics: wff, existential quantifier, bound vs. free occurrence of a variable, vacuous quantification, nested quantifier, intension vs. extension, domain, symbol key vs. interpretation, truth under an interpretation, expansions of quantifiers, counterexample to a statement, counterexample to an argument, categorical forms, Euler diagrams, valid argument, logical falsehood, equivalent pair, contradictory pair, consistent set, logical symbol vs. nonlogical symbol, metalanguage, polyadic predicate, reflexive relation, transitive relation, symmetric relation, equivalence relation. Week 6: Meta-Logic of Predicate Logic. Reading: chapter 6, but don't learn the tree method; instead, concentrate on the ideas below from pages 265, 278-81, 284-5, 287, 290-1, question 3 on 297, and 298-303. topics: model of a set, restriction to monadic predicate logic, infinite vs. finite model, sound and complete deduction system, effective decision procedure, decidable set, decidable language, G�del�s Completeness Theorem for predicate logic, Church's Thesis, Church-Turing Undecidability Theorem. Weeks 7-8: Natural Deduction in Predicate Logic with Identity and Functions. More on Meta-Logic. Reading: chapter 7 and sections 8.1-8.4, 8.6-8.9, 8.11. topics: creating proofs, compactness, consequences from an infinite set of premises, forcing an infinite model, weak mathematical induction, compound functional expressions. Weeks 9-11: Definite Descriptions, Axioms for Arithmetic, and G�del's Incompleteness Theorems. Reading: 9.1-9.3 [skip bottom half of p. 394 and top two lines of p. 395], 9.4, p. 354 Part V, and "G�del and the Nature of Mathematical Truth," topics: Meinong and Russell's theory of definite descriptions, Principia Mathematica, non-logical axioms, Peano arithmetic, Robinson arithmetic, non-standard model, isomorphic model, categorical theory, kinds of completeness, G�del number, undecidability, logicism, Hilbert's Program. L�wenheim-Skolem Theorem, global truth predicate, liar paradox, Tarski's Undefinability Theorem. Week 12: Compactness. Reading: handouts. topics: George Berkeley's criticism of calculus. Robinson's use of the Compactness Theorem to justify Leibniz's Infinitesimals. Hyperreal numbers. Week 13: Axiom Systems for Theories in Physical Science, Higher-Order Logic. Reading: 9.5, 9.6. topics: analysis of a law of nature, Hempel's Raven Paradox, theory reduction, bridge law, identity of indiscernibles, indiscernibility of identicals, many-sorted first-order language, second-order logic. Weeks 14-15: Intensional Logic and Many-Valued Logic. Reading: 9.7, 9.9, 9.10, 9.12. topics: Brief survey of modal logic, epistemic logic, deontic logic, possible world semantics, tense logic, designated value, three-valued logic, fuzzy logic, paraconsistent logic, truth gaps, sorites paradox. The above schedule of course topics may be changed somewhat as we progress through the semester, but these changes, if any, are not expected to affect the schedule of the homeworks and tests. Relevance of logic to other subjects: Click on the ticket: Student outcome goals: The hope is that by the end of the semester you will have achieved the following goals: Be able to reason more effectively. Be able to describe the scope of logic, that is, what symbolic logic can be used to do; and be able to describe the limits of logic, that is, what symbolic logic cannot be used to do. Build on the abilities you learned in Phil. 60 to recognize when an English argument is capable of being analyzed with symbolic techniques, to translate a symbolic argument into English and vice versa, to determine if a symbolic sentence is logically true, to determine if a set of symbolic sentences is consistent, to assess the validity or invalidity of arguments using the techniques of symbolic logic, to create proofs in both predicate logic and statement logic, and be capable of creating and analyzing rigorous proofs using the methods of classical symbolic logic. Know the important extensions of these logics to non-standard logics such as deontic logic, free logic, intensional logic, modal logic, many-valued logic, second-order logic, many-sorted logic, paraconsistent logic, and fuzzy logic. Understand Hilbert's program and the process of formally axiomatizing a theory. Be familiar with the most important meta-theoretic results such as G�del's Theorems, Church-Turing Undecidability Theorem, Tarski's Undefinability Theorem, and the L�wenheim-Skolem Theorem. Be able to say how symbolic logic has deepened our knowledge of some important philosophical issues, and how it has led to new issues of its own. Know the extent to which human knowledge can be freed of contradictions. Be able to say what our civilization has learned from the field of symbolic logic about the limits to what people can know. PHILOSOPHY DEPARTMENT / PROF. DOWDEN / CSUS Updated: May 10, 2009