Phil. 160 Symbolic Logic II

PHIL. 160
SYMBOLIC LOGIC II

Spring Semester 2008, TTh 9:00-10:15

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 old philosophical problems and how they lead to their own new, interesting philosophical issues.

Logic is the theory of good reasoning, and symbolic logic is logic that uses symbols. Those people who devise symbolic logics are trying achieve a more precise and rigorous theory 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 extent to which human knowledge can 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 (that is, propositional logic) and predicate 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. 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 such as an Edge Magazine interview with Rebecca Goldstein where she discusses the philosophical significance of Gödel’s Theorems.

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. 14, 2008

Homework 2: Feb. 28

Test 1: Mar. 13

Homework 3: Apr. 8

Test 2: Apr. 17

Homework 4: May 8

Final exam: May 20, Tues., 8:00a.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 (often 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. When there is room, students on the waiting list will be added in this order: seniors graduating this semester, then all others by random selection. 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 and cell phones:
No photographing, recording or text messaging during class is allowed without permission of the instructor.

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.

Plagiarism: A student tutorial on how not to plagiarize is available online from our library at http://library.csus.edu/content2.asp?pageID=357

Food: Please don't eat and drink during class. You're 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: Liar paradox, Gödel's Incompleteness Theorem, Hilbert's Program, Review of the Syntax of Statement Logic.

Reading: Pages 3-4 of "What is Logic?" by John Nolt, "What is inconsistency?", and chapters 1 and 2 in Bessie [except for page 41, lines 17-26].

topics: Logical vs. factual inconsistency, Barber paradox, Four solutions to the liar paradox (from Russell, Tarski-Quine, Kripke and Priest), informal proof of Gödel's Theorem, classical vs. non-classical logic, statements as sentences with a fixed content, the dispute over whether statements are timeless, deduction vs. induction, truth function, symbolizing arguments, scope of a connective.

Week 2: Semantics of Statement Logic.

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

topics: truth table, interpretation of an uninterpreted language, necessary truth vs. tautology, contradiction, truth tree, truth under an interpretation.

Week 3: Digital Logic and Computer Design

Reading: handouts.

topics: 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.

Weeks 5 and 6: Introduction to Predicate Logic.

Reading: chapter 5.

topics: extension of a language, logical symbol vs. nonlogical symbol, metalanguage, domain, existential quantifier, bound vs. free occurrence of a variable, vacuous quantification, extension vs. intension of a polyadic predicate, interpretation of the language of predicate logic, categorical forms, nested quantifiers, logical validity, logical truth, logical contingency, logical equivalence, logical contradiction, logical consistency, reflexive relation, transitive relation, symmetric relation, equivalence relation.

Week 7: Meta-Logic of Predicate Logic.

Reading: chapter 6, but don't learn the tree method; instead, concentrate on the ideas below.

topics: model of a set, counterexample, monadic predicate logic, Euler diagrams for categorical forms, infinite model, sound and complete deduction system, effective decision procedure, decidable set, decidable language, Gödel’s Completeness Theorem for predicate logic, Church's Thesis, Church-Turing Undecidability Theorem.

Weeks 8-9: Natural Deduction in Predicate Logic with Identity and Functions. More on Meta-Logic.

Reading: chapter 7. Chapter 8 (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.

Week 10-11: Axioms for Arithmetic, and Gödel'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 the Rebecca Goldstein interview "Gödel and the Nature of Mathematical Truth,"

topics: Principia Mathematica, non-logical axioms, Peano arithmetic, Robinson arithmetic, non-standard model, isomorphic model, categorical theory, kinds of completeness, Gödel number, undecidability, logicism, Hilbert's Program. Löwenheim-Skolem Theorem, global truth predicate, liar's paradox, Tarski's Undefinability Theorem, infinitesimal and hyperreal numbers.

Week 12: Axiom Systems for the Philosophy of 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 13 & 14: Intensional Logic and Many-Valued Logic.

Reading: 9.7, 9.9, 9.10, 9.12.

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

Week 15: Review for the final exam.

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.

First reading assignment: By the beginning of the second week of classes, you should have briefly read the first three chapters of the book so that you are familiar with these concepts: connective, truth-function, translation in statement logic, truth table, tautology, consistent set of statements, valid argument, and truth under an interpretation (i.e., truth in a possible world). 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.

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. Be able to translate a symbolic argument into English and vice versa. Be able to determine if a symbolic sentence is logically true, to determine if a set of symbolic sentences is inconsistent, to assess the validity or invalidity of arguments using the techniques of symbolic logic, and to create proofs in both predicate logic and statement 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.

Be familiar with the most important meta-theoretic results of Gödel's Theorems, the Church-Turing Undecidability Theorem, Tarski's Undefinability Theorem, and the Löwenheim-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.

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

PHILOSOPHY DEPARTMENT / PROF. DOWDEN / CSUS
Updated: Apr. 9, 2008


PHIL. 160 SYMBOLIC LOGIC II Spring Semester 2008, TTh 9:00-10:15 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 old philosophical problems and how they lead to their own new, interesting philosophical issues. Logic is the theory of good reasoning, and symbolic logic is logic that uses symbols. Those people who devise symbolic logics are trying achieve a more precise and rigorous theory 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 extent to which human knowledge can 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 (that is, propositional logic) and predicate 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. 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 such as an Edge Magazine interview with Rebecca Goldstein where she discusses the philosophical significance of Gödel’s Theorems. 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. 14, 2008 Homework 2: Feb. 28 Test 1: Mar. 13 Homework 3: Apr. 8 Test 2: Apr. 17 Homework 4: May 8 Final exam: May 20, Tues., 8:00a.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 (often 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. When there is room, students on the waiting list will be added in this order: seniors graduating this semester, then all others by random selection. 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 and cell phones: No photographing, recording or text messaging during class is allowed without permission of the instructor. 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. Plagiarism: A student tutorial on how not to plagiarize is available online from our library at http://library.csus.edu/content2.asp?pageID=357 Food: Please don't eat and drink during class. You're 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: Liar paradox, Gödel's Incompleteness Theorem, Hilbert's Program, Review of the Syntax of Statement Logic. Reading: Pages 3-4 of "What is Logic?" by John Nolt, "What is inconsistency?", and chapters 1 and 2 in Bessie [except for page 41, lines 17-26]. topics: Logical vs. factual inconsistency, Barber paradox, Four solutions to the liar paradox (from Russell, Tarski-Quine, Kripke and Priest), informal proof of Gödel's Theorem, classical vs. non-classical logic, statements as sentences with a fixed content, the dispute over whether statements are timeless, deduction vs. induction, truth function, symbolizing arguments, scope of a connective. Week 2: Semantics of Statement Logic. Reading: chapter 3, sections 1-6, 8. Browse sections 7, 9-11. topics: truth table, interpretation of an uninterpreted language, necessary truth vs. tautology, contradiction, truth tree, truth under an interpretation. Week 3: Digital Logic and Computer Design Reading: handouts. topics: 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. Weeks 5 and 6: Introduction to Predicate Logic. Reading: chapter 5. topics: extension of a language, logical symbol vs. nonlogical symbol, metalanguage, domain, existential quantifier, bound vs. free occurrence of a variable, vacuous quantification, extension vs. intension of a polyadic predicate, interpretation of the language of predicate logic, categorical forms, nested quantifiers, logical validity, logical truth, logical contingency, logical equivalence, logical contradiction, logical consistency, reflexive relation, transitive relation, symmetric relation, equivalence relation. Week 7: Meta-Logic of Predicate Logic. Reading: chapter 6, but don't learn the tree method; instead, concentrate on the ideas below. topics: model of a set, counterexample, monadic predicate logic, Euler diagrams for categorical forms, infinite model, sound and complete deduction system, effective decision procedure, decidable set, decidable language, Gödel’s Completeness Theorem for predicate logic, Church's Thesis, Church-Turing Undecidability Theorem. Weeks 8-9: Natural Deduction in Predicate Logic with Identity and Functions. More on Meta-Logic. Reading: chapter 7. Chapter 8 (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. Week 10-11: Axioms for Arithmetic, and Gödel'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 the Rebecca Goldstein interview "Gödel and the Nature of Mathematical Truth," topics: Principia Mathematica, non-logical axioms, Peano arithmetic, Robinson arithmetic, non-standard model, isomorphic model, categorical theory, kinds of completeness, Gödel number, undecidability, logicism, Hilbert's Program. Löwenheim-Skolem Theorem, global truth predicate, liar's paradox, Tarski's Undefinability Theorem, infinitesimal and hyperreal numbers. Week 12: Axiom Systems for the Philosophy of 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 13 & 14: Intensional Logic and Many-Valued Logic. Reading: 9.7, 9.9, 9.10, 9.12. topics: modal logic, epistemic logic, deontic logic, possible world semantics, tense logic, designated value, three-valued logic, fuzzy logic, paraconsistent logic, truth gaps, sorites paradox. Week 15: Review for the final exam. 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. First reading assignment: By the beginning of the second week of classes, you should have briefly read the first three chapters of the book so that you are familiar with these concepts: connective, truth-function, translation in statement logic, truth table, tautology, consistent set of statements, valid argument, and truth under an interpretation (i.e., truth in a possible world). 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. 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. Be able to translate a symbolic argument into English and vice versa. Be able to determine if a symbolic sentence is logically true, to determine if a set of symbolic sentences is inconsistent, to assess the validity or invalidity of arguments using the techniques of symbolic logic, and to create proofs in both predicate logic and statement 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. Be familiar with the most important meta-theoretic results of Gödel's Theorems, the Church-Turing Undecidability Theorem, Tarski's Undefinability Theorem, and the Löwenheim-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. Understand Hilbert's program and the process of formally axiomatizing a theory. PHILOSOPHY DEPARTMENT / PROF. DOWDEN / CSUS Updated: Apr. 9, 2008