Table Of ContentMathematical logic and
formalized theories
A Survey of Basic Concepts and Results
ROBERT ROGERS
Professor of Philosophy, University of Colorado
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM . LONDON
AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
©NORTH-HOLLAND PUBLISHING COMPANY - 1971
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Library of Congress Catalog Card Number: 78-146195
ISBN North-Holland: 07204 2098 9
ISBN American Elsevier: 0 444 10083 0
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1st edition 1971
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PREFACE
This book is intended as a survey — primarily for people who
are not professional logicians — of the basic concepts and results
of mathematical logic and the study of formalized theories. It is
not a textbook, complete with proofs and exercises. A consider-
able number of concepts are defined in an exact way, and
numerous results and methods are carefully stated. Because it is
basically a survey rather than a textbook in the usual sense,
however, many important results are stated without proof, though
for many results proofs are given. This makes the book noticeably
easier to read (I hope) than an ordinary textbook on mathematical
logic. My priorities in writing the book have been readability and
precision. Holding to these priorities, I have attempted to give a
representative and reasonably up-to-date picture of the fundamen-
tals. Proofs which discourage all but the experts, however, are
generally omitted.
The book (and the subject!) demands a certain maturity in
symbolic thinking. No logic or mathematics is presupposed,
however. It is hoped that philosophers, as well as mathematicians,
who have a genuine interest in logic without being professional
logicians will be able to read it without too great difficulty. The
most difficult material has been put off until the last chapter.
Chapters I and VI are easiest to read; Chapters II, IV, V and VII
are somewhat more difficult; and Chapters HI and VIII will
probably strike the reader as the most difficult chapters. The
reader may find that he is unable to follow certain points on first
reading. Quite often, he should be able to proceed, however, and
can return to these points later.
Expressions which are being defined are put in italics. Also,
variables ar£ italicized, except for syntactical meta-variables, which
are put in boldface.
vii
viii PREFACE
I am especially indebted to the textbooks on mathematical logic
by E. Mendelson and A. Church. I have also profited from helpful
suggestions from North-Holland Publishing Company's reader,
from discussions with my colleague Professor Donald Monk of the
University of Colorado mathematics department, who read the
next-to-final draft of my manuscript; and from detailed comments
on an early draft of the manuscript, coming from my former
classmate Professor Richard Montague of the UCLA philosophy
department. To them, my thanks and gratitude. Many thanks are
also due to Mrs. Kathi George, for her help with the proofreading;
to my typist Mrs. Eloise Pearson, for her patience with the rather
difficult manuscript; and to my wife Marilyn for her patience with
me while I was writing it.
Robert Rogers
Boulder, Colorado
September, 1970
CHAPTER I
THE SENTENTIAL LOGIC
1.1. Introduction
Within the first four chapters of this book we shall be con-
cerned with a formal presentation of various branches of mathem-
atical logic. In this chapter we shall be concerned with the most
elementary branch of mathematical logic; viz., the sentential logic,
or the propositional calculus. This branch of logic has to do with
the logical properties of the various forms of sentential composi-
tion, by means of which sentences can be joined together so as to
result in compound sentences. We shall be especially concerned
here with the problem of distinguishing among sentences in
general those sentences which are true solely by virtue of the
logical properties of the sentential connectives; viz., the so-called
class of tautologies. These are the sentences which are true, as we
say, 'solely by virtue of the meanings of the sentential connectives
themselves'. These sentences form the most fundamental class of
logical truths.
Our approach to the sentential logic — and to the various more
advanced parts of logic taken up in Chapters II-IV - will be in
part syntactical, and in part semantical. Within syntax, we attend
only to various of the typographical, or structural, features of the
expressions with which we are concerned. Here no meaning or
interpretation is presupposed; symbols and expressions in general
are regarded as uninterpreted. Within semantics, however, we attend
not only to structural features of expressions, but also to inter-
pretations. Thus, within semantics symbols and expressions are
interpreted, and certain expressions are said to be true, and others
false, once given certain interpretations of those expressions.
ι
2 THE SENTENTIAL LOGIC
In our approach to each of the various areas of logic, and thus
to the sentential logic in particular, we shall proceed by developing
a certain formal system of logic (or a whole type of formal
systems of logic). This will be done in each case in a certain order.
First, we shall take up a certain part of the syntax of that system
of logic. Here we characterize principally the symbols and formu-
las of that system of logic in an exact way. In particular, in
characterizing, or distinguishing, certain of the expressions of that
system as formulas, no reference is made to any interpretation of
those expressions. The second step in setting forth a logical system
will consist in providing the semantics of that system. Here we
first specify in an exact fashion just how the expressions of that
system are to be interpreted; then define a number of important
semantic concepts, and establish a number of basic results con-
cerning those concepts. Most importantly, we here define the
fundamental concept of a logically valid formula within that
system. In the case of the sentential logic, the logically valid
formulas are the tautologies. Finally, we return to the syntactical
approach, and attempt to characterize syntactically this class of
valid formulas, which we have just defined semantically. We
attempt to do this by laying down certain formulas as axioms —
that is, as formulas accepted without proof. We then specify
certain rules of inference, and define as theorems those formulas
within the system which can be derived from those axioms by
means of those rules of inference. The attempt is to do all this in
such a way that the theorems of the particular system will
coincide with the valid formulas of that system. In the case of the
sentential logic this turns out to be possible. Here the class of valid
formulas can be successfully characterized by syntactical means.
This also remains true for that branch of logic taken up in
Chapters II and III; viz., the first-order predicate logic. It turns
out, however, no longer to be possible with respect to the logic of
Chapter IV; viz., the second-order predicate logic. Here the syntac-
tical approach falls short of the semantical approach; that is, here
the class of logically valid formulas can be characterized only
semantically.
The elements of the sentential logic were first studied by certain
SENTENTIAL CONNECTIVES 3
of the early Stoics of ancient times, and a number of minor
contributions to the sentential logic come from the medieval
period. Its study in a serious way, however, dates only from the
second half of the nineteenth century. Most important in the
whole history of this logic is Gottlob Frege (1848—1925), who has
been called the greatest logician of modern times. The first
formulation of the sentential logic as a formal system appeared in
Frege's Begriffsschrift of 1879. Other important figures in the
history of this logic include G. Boole (1815-1864), E. Schröder
(1841 — 1902), the American philosopher and logician CS. Peirce
(1939-1914), and E. Post.1
1.2. Sentential Connectives
Consider the sentence, Today is Monday, and tomorrow will be
Tuesday.' It is as obvious as can be that this sentence implies the
sentence, Today is Monday,' in the sense that it is impossible for
the former of these two sentences to be true without the latter
being true. We may say that this implication holds by virtue of the
very nature of sentential conjunction. Sentential conjunction is
one of the topics studied within the logic of sentences. It is there
assigned a precise analysis, as follows: A compound sentence of
the form
A and Β
is called a conjunction, with A and Β as its conjuncts. The
conjunction of A and Β is regarded as true just in that case when
the sentence A and the sentence Β are both true. That case is one
of a total of four possible cases: A and Β both true; A true, Β
false; A false, Β true; and A and Β both false. Only in the first of
these four cases is the conjunction
A and Β
1 For detailed notes on the history of the sentential logic, see A. Church 1956, sec-
tion 29.
4 THE SENTENTIAL LOGIC
true. All of this can be said very simply by making use of so-called
truth-tables, which are schematic diagrams of a sort. The truth-
table for the sentential connective 'and' is as follows, where the
letters T' and 'F' stand for the two truth-values, truth and falsity:
A Β A and Β
T T Τ
Τ F F
F T F
F F F
For example, consider the sentences 'Caesar was a Roman,'
'Shakespeare was an Englishman,' and 'Beethoven was an Italian.'
The first two of these sentences are true, and the third is false.
Therefore, the conjunction 'Caesar was a Roman and Shakespeare
was an Englishman' is a true sentence, while the conjunction
'Caesar was a Roman and Beethoven was an Italian' is a false
sentence.
This use of the connective 'and' accords reasonably well with
the way in which the word 'and' is used in ordinary informal
discourse. It differs from that usage principally in the fact that in
the logic of sentences any two sentences can be joined together by
that connective. In particular, it is not required that the conjuncts
A and Β be related to one another in what they are about; that is,
in their subject matter. Thus, for example, the two sentences
'Today is Monday' and '2 + 2 = 4' can be joined together to give
the compound sentence 'Today is Monday and 2 + 2 = 4.' In
ordinary discourse, this sentence would, perhaps, never be used,
since there is no "connection" between the subject matters of its
two conjuncts. But in sentential logic no such "connection" is
required, either here or in the case of any of the remaining
sentential connectives. By not requiring any such "connection",
the logic of sentences becomes much simpler by far than it would
otherwise be.
A second connective — a singulary connective, rather than a
binary one, as is the connective 'and' - is the connective for
negation', viz., 'not.' The truth-table for this connective is as
SENTENTIAL CONNECTIVES 5
follows:
A not A
Τ F
F Τ
Thus, the negation of a sentence counts as true when that sentence
itself is false, and as false when that sentence itself is true.
A compound sentence of the form
A or Β
is called a disjunction, with A and Β as its disjuncts. The
connective for disjunction is here understood in the so-called
inclusive sense: a disjunction counts as true not only in those cases
where one disjunct is false and the other true, but also in the case
where both disjuncts are true. A disjunction counts as false, then,
only when neither of its disjuncts is true. The truth-table for 'or,'
therefore, is as follows:
A Β A or Β
T T Τ
Τ F Τ
F T Τ
F F F
For example, the disjunction 'Caesar was a Roman or Shake-
speare was an Englishman' is a true sentence, as is the disjunction
'Caesar was a Roman or Beethoven was an Italian.' If we take the
sentence 'Beethoven was an Italian' for both the left and the right
disjuncts, however, we obtain a false sentence; viz., 'Beethoven
was an Italian or Beethoven was an Italian.'
We turn now to the connective for the conditional; viz., 'if ...
then.'
The logician's definition of this connective is admittedly a bit
peculiar, at variance with the ordinary usage, or usages, of the
expression 'if ... then.' Here, as with the case for 'and' and 'or,' it
is not required that the sentences joined by this connective have
anything in common in their subject matters. Any two sentences
