An Introduction to Symbolic Logic

✍ Scribed by Terence Parsons

• Year: 2013
• Tongue: English
• Leaves: 298
• Category: Library

✦ Synopsis

Version Aug 2013 of An Exposition of Symbolic Logic
is a lightly revised version of the August 2012 version of
An Introduction to Symbolic Logic (also known as Terry-Text).

The system of logic used here is essentially that of Kalish & Montague 1964 and Kalish,
Montague and Mar, Harcourt Brace Jovanovich, 1992. The principle difference is that written
justifications are required for boxing and canceling: 'dd' for a direct derivation, 'id' for an indirect
derivation, etc. This text is written to be used along with the UCLA Logic 2010 software program,
but that program is not mentioned, and the text can be used independently (although you would
want to supplement the exercises).
The system of notation is almost the same as KK&M; major differences are that the signs '∀' and
'∃' are used for the quantifiers, name and operation symbols are the small letters between ‘a’ and
‘h’, and variables are the small letters between ‘i’ and ‘z’.
The exercises are new.
Chapters 1-3 cover pretty much the same material as KM&M except that the rule allowing for the
use of previously proved theorems is now in chapter 2, immediately following the section on
theorems. (Previous versions of this text used the terminology ‘tautological implication’ in section
2.11. This has been changed to ‘tautological validity’ to agree with the logic program.)
Chapters 4-6 include invalidity problems with infinite universes, where one specifies the
interpretation of notation "by description"; e.g. "R(): ≤". These are discussed in the final
section of each chapter, so they may easily be avoided. (They are not currently implemented in
the logic program.)
Chapter 4 covers material from KK&M chapter IV, but without operation symbols. Chapter 4 also
includes material from KK&M chapter VII, namely interchange of equivalents, biconditional
derivations, monadic sentences without quantifier overlay, and prenex form.
Chapter 5 covers identity and operation symbols.
Chapter 6 covers Fregean definite descriptions, as in KK&M chapter VI.

✦ Table of Contents

Chapter Two
Sentential Logic with 'and', 'or', if-and-only-if'
1 SYMBOLIC NOTATION
2 ENGLISH EQUIVALENTS OF THE CONNECTIVES
3 COMPLEX SENTENCES
8 DERIVED RULES
9 OFFICIAL CONDITIONS FOR DERIVATIONS
10 TRUTH TABLES AND TAUTOLOGIES
11 TAUTOLOGICAL VALIDITY
Chapter Five
Identity and Operation Symbols
1 IDENTITY
2 AT LEAST AND AT MOST, EXACTLY, AND ONLY
3 DERIVATIONAL RULES FOR IDENTITY
44c23ac2-d876-40cb-9427-16641034ed00.pdf
Chapter Two
Sentential Logic with 'and', 'or', if-and-only-if'
1 SYMBOLIC NOTATION
P(Q ( R
P ( Q(R
P ( (Q(R)
2 |
T
2 2
T
2 2
EXERCISES
1. For each of the following state whether it is a sentence in official notation, or a sentence in informal notation, or not a sentence at all. If it is a sentence, parse it as indicated above.
2 ENGLISH EQUIVALENTS OF THE CONNECTIVES
The book is short, and it is interesting
The book is short, but it is interesting
The book is short; it is interesting
The game will be called off just in case it rains: Q ( R
EXERCISES
1. For each of the following sentences say which symbolic sentence it is equivalent to.
R ( P
R ( P
R ( P
W ( R
W ( R
R ( S
R ( S
Q ( R
Q ( R
2. Symbolize each of the following using this translation scheme:
S Sally will walk
V Veronica will give Sally a ride
R It will rain
Q Barbara will come with Quincy
T Barbara will come with Tom
3 COMPLEX SENTENCES
This is a complex sentence, with at least two different but equivalent symbolizations.
Neither Polk nor Quincy was president.
If neither Wilma nor Sally attends, either Robert or Peter will be bored.
If neither Wilma [attends] nor Sally attends, either Robert [will be bored] or Peter will be bored.
If neither W nor S, either R or P
~(W(S) ( (R(P)
If neither Wilma nor Sally attends, either Robert or Peter, but not Tom, will be bored.
If neither Wilma [attends] nor Sally attends, either Robert [will be bored] or Peter [will be bored], but Tom will not be bored.
If neither W nor S, either R or P, but not T
~(W(S) ( (R(P) & ~T
~(W(S) ( (R ( (P&~T))
(~(W(S) ( (R(P)) & ~T
Either Robert or Tom will attend, but not both
Either Robert [will attend] or Tom will attend, but not both [will attend]
Either R or T, but not R and T
(R(T) (~(R(T)
Robert will attend if Sally does, but she won't attend if neither Tom nor Wilma attend.
Robert will attend if Sally does [attend], but she won't attend if neither Tom [attends] nor Wilma attends.
R if S, but not S if neither T nor W
(S(R) ((~(T(W) ( ~S)
Neither Sally nor Robert will run, but if either Tom or Quincy run, Veronica will win.
Neither S nor R, but if either T or Q, V
~(S(R) ((T(Q ( V).
Given that Sally and Robert won't both run, Tom will run exactly if Q does.
Given that not both S and R, T exactly if Q.
~(S(R) ( (T(Q)
A variety of English expressions that we have not mentioned affect how a sentence is to be symbolized. Examples:
Quincy will whistle if Reggie sings without Susan singing or Susan sings without Reggie, but he won't whistle if they both sing
Q if R and not S or S and not R, but not Q if S and R
((R(~S) ( (S(~R) ( Q) ((S(R ( ~Q)
EXERCISES
This derivation illustrates how the conjunction rules are used:
P ( Q
( Q ( P
1. Show Q ( P
Addition indicates that from any sentence you may infer its disjunction with any other sentence.
For mtp you need the negation of a disjunct. In the case given, if 'T' and 'W' were both true, then the argument would have true premises and a false conclusion.
S ( P
P(Q ( ~R
R ( ~P
EXERCISES
These strategy hints will be put to use below, as we extend our list of Theorems from Chapter 1.
Notice that T26 and T4 from the previous chapter are both called "hypothetical syllogism".
1. Show (P(Q ( R) ( (P( (Q(R))
2. Show (P(Q ( R) ( (P( (Q(R))
4. Show P( (Q(R)
The rest of the work is filling in the remaining subderivations. It is often useful to develop a derivation as we did here by first sketching its overall structure, and then flesh it out with details afterwards.
EXERCISES
P
1. Show R
Here is a more highly abbreviated derivation.
P ( Q
R ( ~Q
Rule r also gives you a sentence -- the sentence on the line cited.
EXERCISES
1. Use the method of abbreviating derivations to produce shortened derivations for T38, T40-43.
Here are two arguments, and derivations, that use some theorems from Chapter 1 as rules.
S ( T
S ( Q
EXERCISES
Some additional theorems are given here for reference.
8 DERIVED RULES
EXERCISES
9 OFFICIAL CONDITIONS FOR DERIVATIONS
Let us summarize here what we can now use in constructing an unabbreviated derivation.
EXERCISES
10 TRUTH TABLES AND TAUTOLOGIES
EXERCISES
11 TAUTOLOGICAL VALIDITY
P ( P
EXERCISES