Natural Deduction
for Sentence Logic
Fundamentals
51. THE IDEA OF NATURAL DEDUCTION
I n chapter 4 you learned that saying an argument is valid means that any
case which makes all of the argument's premises true also makes its conclusion true. And you lea
More on Natural
Deduction for Predicate
Logic
61. MULTIPLE QUANTIFICATION AND HARDER PROBLEMS
I n chapter 5 I wanted you to focus on understanding the basic rules for
quantifiers. So there I avoided the complications that arise when we have
sentences, su
Predicate Logic
Semantics and Validity
21. INTERPRETATIONS
.
Recall that we used truth tables to give very precise definitions of the
meaning of '&','v' '', '3, nd 'BE'. We would like to do the same for the
a
meaning of quantifiers. But, as you will see
41.
Transcription
41. RESTRICTED QUANTIFIERS
For three chapters now I have been merrily transcribing '(3x)' both as
'something' and 'someone', and I have been transcribing '(Vx)' both as
'everything' and 'everyone.' I justified this by saying that when
Contents
A Modem
Formal Logic
Primer
Volume
LI
PREFACE TO VOLUMES I AND 11: A Guide to the
Primer
PART I: PREDICATE LOGIC
1.
PREDICATE LOGIC: Syntax
11.
12.
Quantifiers and Variables
13.
2.
We Need More Logical Form
T he Sentences of Predicate Logic
1
71. The Rule for Universal Owntifieation
Truth Trees for
Predicate Logic:
Fundamentals
71. THE RULE FOR UNIVERSAL QUANTIFICATION
You have already learned the truth tree method for sentence logic. And
now that you have a basic understanding of predicate
Index
Ambiguity:
lexical, 54
in quantified sentences, 4850
structural, 5354
Ancestor, 207
Arbitrary names, 7476
Argument:
of a function, 147
of a predicate, 3
Basis step, 170, 171
for strong induction, 17374
Binding, 7
Boldface:
for names, 8, 15859
f
More on Truth Trees
for Predicate Logic
81. CONTRADICTIONS, LOGICAL TRUTH, LOGICAL
EQUNALENCE, AND CONSISTENCY
In this section we are going to see how to apply the truth tree method to
hs il
test predicate logic sentences for some familiar properties. Ti
A n s w e r s t o E x e r c i s e s i n V o l u m e I1
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
. . 7 5
. . 7 6
. . 7 9
. . 8 2
. . 8 7
. . 9 9
. 1 1 8
. . 1 3 2
.
1 0 . .
Chapter 9
152
Chapter
172
a) CidistdLlc?rtba
A Modem
Formal Logic
Primer
Volume
II
Predicate Logic
Syntax
11. WE NEED M R L GC L FORM
OE O I A
In Volume I you gained a firm foundation in sentence logic. But there
must be more to logic, as you can see from the next examples. Consider
the following t
32. Quantifier Scope, Bound Variabks, and Free Variables
More about Quantifiers
3

49
Next, (5) says that everyone loves someone: Every person is such that
there is one person such that the first loves the second. In a world in
which (5) is true, each p
T ruth Trees for
Sentence Logic
what the rules are supposed to do. Our objective is to make line 1 true,
in all the minimally sufficient ways that this can be done. Because line 1
is a disjunction, we must do this by making each of the disjuncts true
alon
Transcrbtion between
English and Sentence
Logic
Z
21. TRANSCRIPTION VS. TRANSLATION
As we saw in chapter 1, for many English sentences we can find corresponding sentences of sentence logic. For example, if 'A' stands for the
sentence 'Adam loves Eve.' an
& I. valid*
Validity
and Conditionals
47
case 1
case 2
case 3
case 4
We know that cases 1 through 4 constitute all the ways in which any of
the sentences in the argument may turn out to be true or false. T i
hs
enables us to explain very exactly what we m
Logical Equivalence,
Logical Truths,
a nd Contradictions
3 1 LOGICAL EQUIVALENCE
.
I introduced logic as the science of arguments. But before turning to arguments, we need to extend and practice our understanding of logic's.
basic tools as I introduced t

A Modem
Formal Logic
Primer
Volume
Basic Ideas and Tools
I.

Sentence Logic
11. LOGIC AS THE SCIENCE OF ARGUMENT
Adam is happy, or so I tell you. If you don't believe me, I try to convince
i
s
you with an argument: Adam just got an 'A' o n h logic exa
Natural Deduction
for Sentence Logic
Strategies
61. CONSTRUCTING CORRECT DERIVATIONS
Knowing the rules for constructing derivations is one thing. Being able to
apply the rules successfully is another. There are no simple mechanical
guidelines to tell you
Natural Deduction
for Sentence Logic
We will pursue the obvious strategy of getting the conclusion by constructing a subderivation from the assumption of 'A' to 'B3C' as conclusion:
Derived Rules
and Derivations
without Premises
We have reduced our task t
Truth Trees for
Sentence Logic
Fundamentals
81. PROVING VALIDITY WITH TRUTH T E S
RE
You know, from the exercises of chapter 4, that you can use truth tables
to check the validity of any argument of sentence logic. But such truth
table checks for validit
Contents
A Modern
Formal Logic
Primer
Volume
I
Sentence Logic
PREFACE TO VOLUMES I AND 11:
1.
A Guide to the
Primer
BASIC IDEAS AND TOOIS
.1
11.
Logic As the Science of Argument
12.
Sentences and Connectives
13.
T ruth Tables and the Meaning of '', '&