# Philosophy 330                                                                                                           Spring 2017

Advanced Symbolic Logic                                                                                   H. Hamner Hill

# Course Description

"Advanced Logic, the final frontier!"  Well, maybe we won't be going where no one has gone before, but we certainly will be going beyond, way beyond, where we went in Logic I.  This course is designed to sharpen students' basic skills in symbolic logic (e.g., translation and proof construction), to deepen students' understanding of basic concepts in logic, to develop advanced analytic and symbolic skills, and to introduce some issues in meta-logic (e.g., completeness, consistency).  Some of the philosophical issues raised by developments in symbolic logic are also introduced (e.g., the problem of counter-factual conditionals).

# Objectives

Students should be able to:

1) Define basic logical terms and concepts.

2) symbolize English sentences and arguments using proper symbolic notation.

3) Transcribe symbolized sentences or arguments into different notation systems.

4) Construct formal demonstrations at both the propositional and predicate levels.

5) Employ several distinct proof construction methods and strategies.

6) Discuss coherently basic issues in meta-logic.

Student Learning Objectives

1)    Students will determine whether arguments are valid using multiple methods (truth  value analysis,   conversion to alternational normal form, Boolean Existential conditionals.

2)    Students  will evaluate the relative  strengths of different analytical methods (e.g. decision procedures vs. proof procedures)

3)    Students will construct falsifying interpretations for relational predicate formulas and arguments.

# Requirements

There will be three in-class examinations, routine (though voluminous) homework assignments, a cumulative final examination.  The homework assignments will count for 25% of the final grade, the three exams, 50%, and the final exam the remaining 25%.  However, completion of each of the assignments is a necessary condition for receiving a passing grade.  Late homework assignments will not be accepted.

Academic integrity is one of the core values of a University.  Integrity involves strict compliance with a set of values, and the values most essential to an academic community are honesty, trust, and respect. Academic integrity is expected not only in formal coursework situations, but in all University relationships and interactions connected to the educational process, including the use of University resources. Instances of academic dishonesty will not be tolerated and they will be punished severely.  The range of punishments the University may impose go from redoing an assignment with a penalty through failure for the class, up to expulsion from the university.   It is my policy any intentional of academic dishonesty results in a zero for the assignment and the assignment will not be treated as having been completed. There will be no opportunity to redo work which is the product of intentional academic dishonesty.  Unintentional and negligent acts of academic dishonesty will be punished according to the severity of the offense and usually there will be an option to redo the assignment with some grade reduction.

Plagiarism is an offense against academic integrity. It is a combination of theft and fraud. The course web page has a detailed definition of what plagiarism is. Plagiarism will not be tolerated, it is a very serious academic offense.  If you are not sure about what plagiarism is, find out. And DON’T DO IT!

Other offenses against academic integrity (cheating, collaboration on individual assignments, use of crib notes, etc.) will be punished similarly.
See http://www6.semo.edu/judaffairs/ for details.

### Attendance

Regular attendance and class participation are expected.  Be prepared to be called upon in class. Do your reading assignments before class so that we can discuss them (instead of me lecturing about them.  I do not think I need to say all this to a group of advanced logic students, but we must keep the higher powers happy.   NOTE  Exam dates are listed on the syllabus.

### Make-Up Exams

Make-up exams generally will not be scheduled unless, A) you have a very good excuse or B) prior arrangements have been made with me.

### Incompletes

Only in extraordinary circumstances will incompletes be given (e.g. major illness, death in the family, etc.).  It has been my experience that students do not benefit from receiving incompletes.

## Notice to Students with Disabilities

Anyone with a disability that requires special assistance or creates special needs should contact me in order to make appropriate accommodations.  No one is required to disclose a disability, but there is no way that I can make special arrangements or modifications to the course unless I know about those disabilities.  I can also assist those of you who may need help in securing assistance and resources from the university.

Complaints

Questions, Comments or requests regarding this course or program should be taken to your instructor.  Unanswered questions or unresolved issues involving this class may be taken to Dean Frank Barrios, College of Liberal Arts.

Required Text:  Quine, Methods of Logic, 4th ed.   Available in the Bookstore.

Additional materials, relevant to the paper/presentation assignment, will be on reserve.

Office:  Carnahan Hall, 211C                                             Office Hours:  MWF 10-11

Office Phone  651-2816                                                     Home Phone    339-0575         hhill@semo.edu

Course Outline

We shall proceed directly through Quine's text, aiming to complete the first 34 chapters thereof.  The first five Units of the class will review truth functions; develop fluency with Quine's system of notation; improve our understanding of consistency, validity , implication, and equivalence; and introduce alternational and conjunctional normal schemata, simplification, and duality.  The second part of the course will examine the basics of predicate logic, syllogisms and Venn diagrams (with expanded Venn techniques) and then move on to Boolean schemata, Monadic schemata, prenexity and purity, and a discussion of some techniques for establishing validity and consistency.  Finally we will extend these concepts to polyadic predicates and a general theory of quantification and introduce new techniques for proof construction, the method of Pure Existentials and the Main Method.  We will round out the course with a discussion of completeness and decidability (that Gödel result).

Unit 1 (Jan. 17)         Through chapter 2

Unit 2                        Through chapter 5

Unit 3                        Through chapter 8

Unit 4                        Through chapter 11

Unit 5                        Through chapter 13      First exam at end of Unit 5

Unit 6                        Chapters 14-17

Unit 7                        Chapters 18-22

Unit 8                        Chapters 23 & 24

Unit 9                        Chapters 25 & 26        Second exam at end of Unit 9

Unit 10                      Chapters 27 & 28

Unit 11                      Chapters 29 & 30

Unit 12                      Chapters 31 & 32

Unit 13                      Chapters 33 & 34        Third exam at end of Unit 13

Unit 14                      Meta-Logical theory

Unit 15                      Review

NOTE:  In this course timely completion of both reading and homework assignments is a necessary but not sufficient condition for success.  For your benefit and for my peace of mind, COMPLETE ALL READING ASSIGNMENTS ON TIME!

PHILOSOPHY 330     ADVANCED SYMBOLIC LOGIC    HOMEWORK ASSIGNMENTS

All homework assignments are due at the beginning of class on the Thursday of the Unit  corresponding to the assignment.

Due          Assignment  Problem Set

#1                   p. 20, #2

p. 32, #3 (1st part); also for (p v q)( pq (r v p));

#4;

#5 (1st 2 parts);

symbolize as a logical schema in dot notation:

If if if p then q then p then p.

p. 40, #1;

p. 45, #1 (except lower left part);

#3 (optional);

Give a truth-value analysis of:

-(pq):qr.. p v r

#2                    p. 52, #2;

#5;

#7 (schemata are on p. 307).

Use fell swoop wherever you can on these problems.

#3                    p. 59, #1;

#2;

p. 67, #2;

#4 (3rd pair only);

#8 (concisely, but cogently).

#4                    p. 74, #1 (2nd and 3rd schemata only), and apply the same instructions to:

-(pq): q v r.p ;

Develop into developed alternational normal form:

pq v r;

p. 74, #3 (concisely).

Investigate for redundant clauses and literals:

pqr v qr v ps v prs  v rs;

Transform into a conjunctional normal schema ( v. p. 83):

pq v pr v pqr  (dualize, distribute, simplify, dualize)

In Chapter 13's axiom system give a proof of:

pq..qp::p.qp

#5                    P. 113, #1 (except second part),  #3.

p. 141, #2 (for the last 3 parts of ex. #1 in Ch. 17 and for ex. # 4 of Ch. 19, giving schemata in all cases)

p. 141, #3

p. 147, #1 for all 4 schemata (but limit universe to just {a,b,c})

p. 154, #4, #5 (for # 4 only)

p. 154, #2; also put 'Fx .xGx v xHx' into prenex form

#6                    Use an appropriate argument to determine the validity or nonvalidity of

'y(Fy ↔∀xFx)' and 'y(Fy ↔∃xFx)'.

Give schemata for the premises of p. 159 #4 (v.p. 312).

Give a falsifying interpretation for each of the following in as small a non-empty universe as possible:

x(Fx-Gx).x-Fx. xGx;

x(Fx Gx).xFx.x(-Gx.Hx).y(-Hy.-Fy). x(Gx .Fx v -Hx);

-xHx v xHx v x(Fx -Gx);

Give a satisfying interpretation for each of the following in as small a non-empty universe as possible:

xyz(Fxy.Fyz.Fxz).xyFxy.xyFyx.x-Fxx;

xy(Fxy-Fyx).xyFxy.

#7                   pp. 173-74, #3, #5, #7 (last three parts only);

supplement #5 by treating the following schemata as it instructs:

xy(Fxy.-Fyx),                                   xFxx xyFxy

xyFxy,                                             x(Fxx yFxy)

Give quantifiational schemata for the following sentences:

(a)        Every logician quotes a logician.

(b)        Not all logicians quote themselves.

(c)        Some, but not all, dodos are logicians.

(d)        No dodo who quotes every dodo is a logician.

(e)        A dodo is a logician if he or she is quoted by a logician.

(f)         Any dodo who quotes all dodos who quote him (her) quotes himself (herself).

Using the method of pure existentials, show the validity of uvxy(Fxu.Fvy..Fxx  v Fyy).

Also, by the same method, determine whether y(xzFzxy zFzyy).  If non valid, give a falsifying

interpretation.

#8                    Using the Main Method, do problems 5, 6, and 7 from p. 189.  Also, by the same method, show that yx(Fx Gy) and xy(Fx.-Gy) are inconsistent.

By either the method of pure existentials or the main method, show that xFxy→∃xGx and y-Gy together imply z-Fzy.  Also, do problem #4 p. 203 (symbolized on p. 315, but try to nail it down yourself), and show, by the main method, that yzx(Fx.Gxy..Fy  v Gzy) is valid.

#9                    (1)        Give an interpretation that falsifies the schema: xyz(Fyz Fxz. .Fxx Fyx).

(2)        Show that "Everybody loves a lover," on its most likely construal, together with "Bob loves Carol" as a further premise implies that "Ted loves Alice."  You may use either the main method or the method of pure existentials (Boolos).

(3)        Use the main method, plus rules of passage (9) and (10) but no others, to show that xuyv(Gy Fx.Fu Gv) implies yx(Fx Gy).

(4)        Show that "Everybody loves my baby but my baby loves nobody but me" implies "I am my baby."  To simplify things a little, you may symbolize the conclusion as if  it read "My baby is me." (Cartwright)

(5)        Use the main method, plus the axioms of identity, to show that xFx and y-Fy imply xy(x≠y).

(6)        Show that "There is a dodo who admires every dodo," "Dodos all admire whatever admire them," and "A dodo who admires any dodo admires every dodo," imply "Each dodo admires every dodo." (Koopman)

#10                   (1)        Show that "I like anyone who laughs at himself but like no one who laughs at all his friends" implies "Anyone who laughs at himself has a friend other than himself. (Solomon)

(2)        Use the main method, plus the axioms of identity, to show that xy(Dx.Dy. x=y) and xy(Dy y=x) imply each other.

(3)        Determine whether "Any fish can swim faster than any smaller  fish" implies "If there is a largest fish, then there is a fastest fish."

(4)        Express the following schema in the clearest English you can, letting 'Qxy' = 'x bears Q to y' and

'Rxy'='x bears R to y'.

xy(Qxy .z(Rzx Rzy))

(5)        Using the predicates 'Q' and 'R' symbolize the following:

(i)         If there is a thing to which everything bears R and a thing to which everything bears Q then  everything bears R to everything.

(ii)        Nothing is borne R by everything unless something is borne Q by everything.

(iii)               Everything bears Q to  something or other.

(iv)              Anything that is borne Q is borne R.

(v)                Nothing that is not borne R is borne Q by anything other than itself.

(6)               Determine which of (i) –(v) are implied by (a).  No justification is necessary.