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Formal Logic: Propositional Logic
  时间:2014-01-09 10:08:15  作者:

Course Title: Introduction to Formal Logic / Propositional Logic

 

Course Group applied to: 3. 3 – “Quantitative Reasoning” (See related comment in “Course Goals” below.)

 

Course Code:

 

Credits: 2

 

Hours: 2 per week

 

Student Requirements:  In addition to the usual requirements of attendance and classroom focus, the course requires four exams, equally weighted and scheduled roughly at even intervals throughout the semester.  Note is made also of individual student participation, primarily in the form of presenting solutions to text problems on the whiteboard.

 

Course Goals: Logic, it was demonstrated by recent logicians like Alfred Tarski, Gottlob Frege, and Bertrand Russell, provides the foundation of quantitative reasoning.  In fact, by some accounts, it is historical accident that logic today is most conspicuous within the field of philosophy, and not mathematics.  Its first systematic development is found in the surviving works of Aristotle, who tied it closely to rhetoric and counted it as one of three branches of “theoretical” knowledge along with theology and natural science. 

 

One of the several virtues of the text (as noted below, the main text is volume I of Howard Pospesel’s Introduction to Logic) s that it demonstrates the role of logic not only within these disciplines, but within the enterprise of daily life.  Logic, as the student learns from the wealth of examples provided from this text, and the adjunct “Problems” compilation that accompanies this two-volume set, permeates life.  Examples that I provide below will serve as illustration.  It is essential to virtually any kind of human endeavor, academic or otherwise.  In fact, it is so deeply fundamental to human activity that it may be hard to categorize it in terms of a “course group” – 

 

one of the benefits of this course, I believe, is that it will make the student more mindful of the argumentative structure that figures into a variety of activities, such as the composition of papers, articles, exam responses, and persuasive editorials.  Another benefit, as I see it, is one of overall academic “hygiene” – its exercises are meant to instill you with a sense of rigor, clarity, precision, and logical subtlety that will aid him or her in a range of academic activity.

 

Over the span of the term, the student will demonstrate, by means of classroom exercises and exams, the ability to identify components of deductive logical discourse in natural (English) language; to translate fairly complex passages of natural language into expressions of formal logic; to construct formal proofs using the exercises in the text and in brief photocopy exercise using the logical apparatus provided; to construct tests of validity and invalidity of argument using the truth table method; and to identify and demonstrate the modal status (tautology, contradiction, and contingent formula) of propositions in both natural and formal language.

 

 

 

Pedagogical Methods:  The primary method of this course is informal lecture, with explanation of text passages and demonstration of the inference rules provided therein.  Students are encouraged to present their own solutions of problems on the whiteboard, especially when they diverge (not necessarily for the worse) from those of the instructor. 

 

Evaluation Methods: As noted, four exams are the main part of the grading total.   Students also are encouraged to take a brief hand in the teaching process by presenting solutions of problems on the whiteboard and explaining the rationale that they used.   

 

Textbooks and References: The main text for this course is Howard Pospesel’s Introduction to Logic:  Propositional Logic (Upper Saddle River, New Jersey, 2000).  It is the first of two volumes in the set, the latter of which I have used in a shared independent project with two graduating students this (spring 2013) term, and which I wish to develop soon into a separate course. 

 

Other Resources:  I sometimes use a few of the problems in an adjunct text, also by Pospesel, which provides samples of argument from the traditions of science, literature, and philosophy.  

 

Course Contents and Class Schedule: 

 

1)  Introduction:  Course content, requirements, and policy

 

The principal task of this chapter, which typically occupies the first class session, is to introduce students to the most basic concepts of deductive formal logic.  So, for example, what is an argument?  (Answer:  It is a collection of statements, one of which is claimed to follow from the others.)  The statement inferred is called the conclusion, while the statement(s) supporting it are called the premises.  One point of introduction is the distinction between deductive and inductive reasoning.  Each is counted as logic, but how are they different?  

 

An old “saw” has it that “deductive goes from general to particular”, while “inductive goes from particular to general” - and there is some truth in this sing-song comment, as witness the following examples:  

 

Argument (1)  “Socrates is moral, Callicles is mortal, Critias is mortal, therefore all men are mortal.”  Argument (2)  All men are mortal, Socrates is a man, therefore Socrates is mortal.”

 

The first argument “goes” from the particular instances of man to a conclusion about men as a whole.  The second, by contrast, goes from the whole class of men to a particular case.  Hence the particular / general claim.  Yet recent texts make the point more effectively by saying instead that inductive reasoning is ampliative - meaning that it makes an inference that goes beyond the information that is contained in its premises.  Hence it is strong in degree - that is, it makes its conclusion probable, in some degree or other.  (Examples are provided in class to show how an argument of this kind might gain strength - the addition of further information, an increase in the variety of information, a modification of its conclusion to make it less bold, less sweeping in its scope, etc.) Deductive reasoning, on the other hand, is “cleaner” in its structure - if the premises of a deductive argument are true (and this, of course, can be an issue unto itself), then its conclusion cannot be false.  A deductive argument that is rightly structured, in regard to this point, is said to be valid, and if not, then invalid.  An argument that is valid and has true premises is said to be sound.

 

I will not go further here with the points made in this section of the course, but will assume that its general direction is by now evident.  Instead I will concentrate in the next few pages on concrete examples of the activity that the course contains.  

  

2)  This week’s activity introduces the first of ten basic or “primitive” rules of inference that make up the first eight chapters of the text.  In this text, which is fairly representative of the discipline in recent decades, there are five connectives that logically modify or link together the propositions that make up the units or building blocks, as it were, of the formal language.  

 

A formula is any grammatically coherent expression within the language.  A formula can have any finite length, from a simple component statement to a complex expression that involves many symbols.       

 

These units are represented by block letters of the English alphabet.  The connectives are if ... then, and, not, if and only if, and or.  They are expressed as the arrow, the ampersand, the dash (or negation symbol), the double arrow, and the wedge (like a small-case “v”).  As is the case in mathematics, grouping symbols (initially parentheses, brackets, and braces) serve to provide clarity and definite content when an expression becomes complicated.  

 

Each connective has corresponding to it an “in” and an “out” rule - hence the ten rules total.  The rule introduced in this week’s reading is called “arrow out” - traditionally known as modus ponens (“in the mode of affirming”), it works as follows:

From a “conditional” statement (“If A, then B”) and the further affirmation of its antecedent (the “A” portion), infer the consequent (the “B”).  

(And note that here, as always, the inference rules concern formulas of every complexity, hence the cursive font.)

    

This rule has very simple instances (e. g., “If the lights are on, the shop is open; the lights are on; hence the shop is open), which help the student to see how logic permeates activities of myriad kinds, and help likewise to break down the imagined barrier between philosophy and “real life” - once a student has a grasp of its structure, he or she can arguments more complex.  

 

To cite one example from Pospesel’s Chapter 2 “Exercises” section, extrapolated from Thor Hyerdahl’s Kon-Tiki:  

(Often these exercises contain direct quotation from the original text, followed by Pospesel’s paraphrase, which helps to make explicit it logical content.)  

Peruvians sailed to Polynesia before Columbus.  If so, then they must have traveled to Polynesia in balsa boats if the boats available at that time were all made of balsa.  The boats at that time were all constructed of balsa. Conclusion:  The Peruvians traveled to Polynesia in boats made of balsa.  

 

  (In a method similar to the text author’s, I am marking the key letters that represent the component propositions:  Block letter B will represent the first sentence; block letter T will represent the claim that the Peruvians traveled to Polynesia in balsa boats; block letter A will represent the claim that the boats available at that time were in fact made of balsa.)  

 

Since it would be laborious to multiply these examples too many times, I will limit myself to this example and a couple of more below to show the sort of cognitive “workout” that the course provides.  It is important to note that in an exercise of this kind, students have to read not only the words of these sentences, but the logical content that these words provide.  In one passage, a key idea might be expressed by the word ‘made’, in another, ‘constructed’.  (This example, of course, is elementary, but it serves to make a point that can be illustrated with others more complex.)  The sentences must be captured, wholly and consistently, in the formal language, after which the rules of inference are applied in order to furnish a “proof” that is analogous to those constructed in classical geometry.  (The proof, in this case, runs optimally five lines, but by the end of the term, students are providing solutions to problems fare more involved and running perhaps three to five times this length.)  

     

3)  This chapter introduces the “and” symbol as represented by the ampersand.  Two more rules of inference are given, and the result is a step forward in the complexity and interest level of the arguments treated. 

 

Example:  [According to Archimedes’ principle], a solid body floating in a liquid displaces a volume of liquid that has the same weight as the body itself.  If this is true, the chunk of ice has the same weight as the water displaced by its submerged portion.  The weight of a substance remains constant through melting.  If (i) the ice has the same weight as the water displaced by its submerged portion and (ii) the weight of a substances is unaffected by melting, then the ice cube turns into a mass of water having the same weight as the water initially displaced by its submerged portion.  Provided that the cube turns into a mass of water having the same weight as the water initially displaced by its submerged portion, it turns into a mass of water having the same volume as the water initially displaced.  The level of water in the glass will remain constant if the cube becomes a mass of water having the same volume as the water initially displaced.  Hence, the water level remains constant.  

 

 (In this case, there are six component propositions, and the optimal proof length is several lines more than was the case in the example preceding.)  Here again, the exercise requires a “depth” reading of the English, as well as deductive work to symbolize and make inferences.     

 

4) Chapter 4 takes a significant step forward in this overall development - it introduces the method of what is traditionally called conditional proof -  The method of “provisional assumption” – deriving a conditional formula by means of introducing a new and un-given assumption into the derivation to see what might follow from it.  

 

Complex problems are contained in the exercise section, some of which require not only deductive work, but ingenuity - and this demand for ingenuity is increased by the insistence that all inferences are made within the tight structure of the given rules.  (To cite one example, an exercise in the “Challenge” section asks the student to infer a formula - say, “G” - when the formula itself is the only premise provided.) 

 

Subsequent chapters introduce the remaining connectives, and a lengthy set of “derived” rules, i. e., each one formally derivable from the ten rules initially covered.  The final part of the course is devoted to (i) truth tables and (ii) properties and relationships between the logical structure of various statement types (tautology, contradiction,and continent formula).  

 

Hopefully these preceding pages will speak to university concerns about the content of the course, and its claim to inclusion in the university “core” curriculum.  There is nothing complex about the reading assignments of this course - again, nearly all the reading comes from the “propositional” volume of the Pospesel set.  In order to provide further information, I am appending a schedule of readings from my current logic course, along with the course policy statement and the set of “Q and A” that is designed to speak to student interests and concerns in the opening week of the course.  This material begins on the page below.

 

 

 

 

 

 

 

Formal Logic:  Propositional Logic

Wednesday   10:00 – 12:00 noon   E 201

Shantou University   Fall Semester   2013

Dr. Kelly Nicholson   Center for International Studies

 

 

                                                        Course Readings

 

During 

this class session  –   We will cover  –  

 

Sept 18th introduction and course policy

 

Sept 25th the conditional statement and the “arrow out” rule; the “ampersand” rules (Pospesel, Propositional Logic, chs 2 & 3)

 

Oct 2nd   [Autumn Festival Week]

 

Oct 9th conditional proofs and “provisional” assumptions (ch 4)

 

Oct 16th review and first exam

 

Oct 23rd   conditional proof involving negation (ch 5)

 

Oct 30th   biconditional statements (ch 6)

 

Nov 6th review and second exam

 

Nov 13th disjunctions and completion of the ten “primitive” rules (ch 7)

 

Nov 20th   summation, chs 1 – 8, and the expanded rules (chs 8 & 9)

 

Nov 27th the expanded rules, continued (ch 9)

 

Dec 4th review and third exam

 

Dec 11th truth tables and validity (ch 10)

 

 

 

Dec 18th truth tables, continued (ch 10)

 

Dec 25th the logical status of propositions (ch 12)

 

Jan 1st   extra credit projects and fourth exam

 

 

 

Course Policy and Requirements:  Q & A

 

 

What is this course about?  

 

Logic (Greek logos, meaning truth, reason, or principle) is the branch of philosophy that examines the structure of reasoning.  This course serves as an introduction to contemporary formal logic, and it will acquaint you in some detail with propositional logic, the system that uses propositions (roughly, and for our purposes, complete declarative sentences) as its basic units of expression.  

 

Formal logic examines what is sometimes called the depth structure of deductive reason.  The examples in this text have been drawn from material that ranges from the comic strip and the sports page to treatises of history, science, and philosophy itself.  It will require (1) the construction of formal proof, according to the rules and strategies provided in the text, and (2) the translation of conventional English discourse and argument into the language of symbols.  

 

How will the class be graded?

 

There are several criteria of grading for the course.  First, and primarily, you will be graded on the basis of your performance on the four exams, which will be given at fairly regular intervals throughout the term.  Owing to the nature of the material, the exams will not be re-set to accommodate personal schedules.  If you miss an exam, you will be able to compensate by means of in-class effort, mainly by taking a hand in the solving of text problems on the board for the benefit of the class.  This effort can also be used for extra credit.

 

You should plan, as a general rule, to be present for the entire duration of the class session.  This means not only physical presence, but attentiveness to the task at hand.  Please be punctual.  Also, out of respect for the learning environment, please refrain from wondering in and out of the class while we are in session.  A short break will be allowed at the end of the first hour.  

 

 

This material looks awfully abstract.  Why should I study it?

 

Logic, as you will soon see, is essential to virtually any kind of human endeavor, academic or otherwise.  One of the benefits of this course, I believe, is that it will make you more mindful of the argumentative structure that figures into a variety of activities, such as the composition of papers, articles, exam responses, and persuasive editorials.  Another benefit, as I see it, is one of overall academic “hygiene” – its exercises are meant to instill you with a sense of rigor, clarity, precision, and logical subtlety that will aid you in a range of academic activity.

 

 

Expected Learning Outcomes

 

Over the span of this term, the student will demonstrate, by means of classroom exercises and exams, the ability

· to identify components of deductive logical discourse in natural (English) language;

· to translate fairly complex passages of natural language into expressions of formal logic;

· to construct formal proofs using the exercises in the text and in brief photocopy exercise using the logical apparatus provided;

· to construct tests of validity and invalidity of argument using the truth table method;

· to identify and demonstrate the modal status (tautology, contradiction, and contingent formula) of propositions in both natural and formal language.

 

 

Class sources

 

This course will make use of a variety of sources, including (1) sections of Howard Pospesel’s Introduction to Logic:  Propositional Logic, and (2) brief photocopy materials relevant to our classroom activity.

 

Note:  In keeping with university concerns with self-assessment, the Center for International Studies will engage in periodic exercises in order to gauge the effectiveness of its methods and its overall curriculum.  To this end, students will be asked, on occasion, to provide (apart from the actual course exams) demonstration of their proficiency in reading, writing, and critical thinking over the span of the course term.  These exercises will be noted with respect to the final course grade.

This semester’s assessment theme:  Gather, assess, interpret, and synthesize information and ideas from printed sources, electronic sources, and observation. 

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