PROPOSITIONAL LOGIC

Introduction

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We are now about to embark on the most math-like portion of the course. We will once again be abstracting the form from the content of statements and then we will be using symbols to express relations between statements and parts of statements. Eventually we will be constructing proofs very similar to geometric proofs.

Some people are taking logic specifically to avoid calculus, so this section might look scary. If it isn't scary to you--good!!--please don't let me talk you into being scared with this introduction! But for some people symbolizing and abstraction feel like foreign procedures that no normal person would do.

If you are one of these people, relax. Is not impossible to do well in these sections. As a matter of fact, it is VERY possible to ace them. Even for "normal" people.

The first piece of advice is to not let yourself be intimidated by the strangeness of what you see when you flip through chapters six and seven of the Hurley text. You will very quickly become familiar with the symbols and the language. (This is never a problem.)

The next bit of advice is to not move on to each new lesson until you are very comfortable and proficient with the preceding one. From Lesson #20 through the end of the course there will be a gradual build-up of skills covered in each section. These skills will grow out of and depend upon each preceding section. (This is where most students have trouble.)

Try to take things step by step. This guide will tell you in simple language how to do each task you need to learn. You can get the why from Hurley. Hopefully you will grasp both the how and the why, but notice this:

Some students get stuck trying to understand why and never learn how to do much of propositional logic. If students can temporarily let go of their need to know why, this will eventually come to them quite naturally from the actual practice of how.

My final suggestion is to practice reading the symbols out loud. Or, at least get used to thinking them correctly. IE: "A É (B · C)" is "if A then both B and C" not "A, um squiggly thing parentheses-doohickie B dot C close parentheses" Much of what you learn from now on is intuitive, and your common sense will help you a lot, if you give it something to work with.

LESSON #20

Symbols and Translation

Reading Assignment: 6.1 (pp. 301-309) and pp.319-322 (Further Comparison With Ordinary Language)

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The Language P

In this lesson, we will learn the language P, the simplest language used in symbolic logic. If you take the Logic 320 course you will learn several modifications to this language, to make it less clumsy. But even though it is very simple it will suit our purposes. What it does is iron out all the fine points of distinction and ambiguity that make English so expressive, but so complicated. The simplicity of P makes it very easy to use to see the logical structure of statements and arguments.

For this lesson we will be abstracting the form from the content of statements. We will again be using capital letters when we abstract, just as you did in the last two chapters, but notice the difference. The basic elements which will be replaced by letters are here whole simple statements, or propositions, instead of terms. Recall that a statement is a sentence that is either true or false. A simple statement is the smallest possible chunk of language which can be called true or false. A complex statement is made up of two or more simple statements. Any letter can substitute for any simple statement, just make sure you don't use the same letter for more than one statement, and you use the same letter for each instance of the same statement.

Simple statements (represented by capital letters) can then be combined using one of five connectives (symbols) ~ · v É  º (see p. 302) to represent complex statements.

Of the five connectives , the tilde ~ (negation) is a little different from the others in that it does not really "connect" anything, but simply negates whatever comes after it.

We will also be using parentheses ( ), (and brackets [ ], and braces { }), to construct well-formed formulas--WFF's (pronounced "woofs". . . and you thought logicians had no sense of humor. . . (!)) These are the punctuation of the language P.  WFFs are the sentences of this language. Whenever you have more than two terms or you wish to negate more than a simple statement you will need to use parentheses to clarify the meaning of your complex statements. Outside of the parentheses you may need to use brackets and occasionally you may even need braces. There should never be more than two simple propositions and one connective other than ~ without a set of parentheses separating them. 

~ A · B  is not the same as  ~(A · B)

(A · B) v C is not the same as A · (B v C)  so the parentheses are necessary to avoid ambiguity.

It is important, even in a correspondence course, to verbalize these translated complex statements:

IMPORTANT: The type of statement is determined by the symbol that is outside any parentheses. For example, a statement is classified as an "and" statement or a "conjunction" if the dot symbol · is not covered by any parenthesis.

Study carefully the blue boxes in the Hurley text as well as all the sample translations. Here are some highlights that I have observed can cause students trouble. But this can't substitute for reading the chapter. This chapter is useful for flipping back to as you try to do the homework. Prof. Hurley does a good job of giving an example of each problem that comes up. 

*Note that what follows the "if" in a conditional sentence will be placed in front of the É symbol, no matter where it occurs in the English statement. Example: A if B = B É A. The phrase "only if" is treated differently. "Only if" always precedes the consequent in English. Example: A only if B = A É B.

**Notice that "neither are" is the same as saying "both are not," which again is the same as "not either are." (Indeed, think about this for a while and you will probably understand why this is so.) These can be translated by the following two equivalent formulas:

~(A v M) and ~A · ~M

On the other hand, the phrases "either one or the other is not" is the same as "it is not the case that they both are" or "not both." Both of these statements can be translated by the following two equivalent formulas:

~A v ~B and ~(A · B)

It may take some clear thinking to understand the difference between "not both" and "both not," but there is a very important difference. Reflect on this for a while.

***Also spend some time thinking about the piece of the reading from section 6.2.***

You must think about these things, making up examples for yourself, and listening to yourself and others speak as you go about your business, until they make sense. If you do this, the following exercises will be a piece of cake.

Logic Coach Assignment: 6.1 I all. Do II too if you think you need more practice.

ASSIGNMENT 1 (Eight points each)

Translate the following statements into symbolic form (the language P) using capital letters to represent affirmative English statements. Be sure to use well-formed formulas. (e.g., see Assignment 2).

1. If Jesse is late Bekki will be mad.

2. Either the Cardinals or the Braves will win the pennant.

3. The President is Head of State; but, she is also Commander in Chief.

4. Abortion is exactly the same as murder.

            5. If Kathy and Dorothy and J. C. go on vacation, then Jerome will stay home.

6. The economy improves whenever auto sales increase.

7. Neither Steve nor Joshua are Hindu.

8. AIDS will decrease only if AZT becomes inexpensive and that won't happen.

9. If either the House and the Senate vote against it or the president vetoes it, the bill will fail.

10. If you are both unhappy and underpaid, you should quit. If you are happy and well paid, you should stay. (don't overlook the term compliments here--you should be using some ~'s)

ASSIGNMENT 2 (Two points each)

Which of the following are not well-formed formulas (WFF's)? For the statements below that are not WFFs, point out where the problem is. For those that are WFFs, write out how you would say the complex statement.

Model: ~(D É ~T) v [(S · R) É V]

"Either it is not the case that if D then not T or, if S and R then V"

1. ~L v ( É X · M)

2. (M v N) · v (P v ~Q)

3. (R É S) É (~T É ~V)

4. (X º Y) É ~[V · (L · M)]

5. (M v N v O) É P

6. ~A · (~B É C)

7. [(D · E) v F v G] · (H v ~I)

8. ~(R ~ S) É T

9. ~R v ~S º ~T

10. ~(C v D) É [(S º ~T) v (E É ~F)]

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