Encyclopedia Dictionary of Mathematics (Section D)

604 Pages·1987·43.425 MB·English

1549 410 B
Surfaces
41O(Vl.21) surface, one-sided. A nonorientable closed
Surfaces surface without boundary cannot be embed-
ded in the Euclidean space E3 (- 56 Charac-
teristic Classes, 114 Differential Topology).
A. The Notion of a Surface The first example of a nonorientable surface
(with boundary) is the so-called Miihius strip
The notion of a surface may be roughly ex- or Miihius hand, constructed as an tidenti-
pressed by saying that by moving a curve we fication space from a rectangle by twisting
get a surface or that the boundary of a solid through 180” and identifying the opposite
body is a surface. But these propositions can- edges with one another (Fig. 1).
not be considered mathematical definitions of
a surface. We also make a distinction between A1
surfaces and planes in ordinary language,
where we mean by surfaces only those that are
not planes. In mathematical language, how- B C
ever, planes are usually included among the
4!i!EQ
surfaces.
A
A surface can be defined as a 2-dimensional i
+continuum, in accordance with the definition
of a curve as a l-dimensional continuum. DB
However, while we have a theory of curves
Fig. 1
based on this definition, we do not have a
similar theory of surfaces thus defined (- 93
As illustrated in Fig. 2, from a rectangle
Curves).
ABCD we can obtain a closed surface homeo-
What is called a surface or a curved surface
morphic to the product space S’ x S’ by
is usually a 2-dimensional ttopological mani-
identifying the opposite edges AB with DC
fold, that is, a topological space that satisfies
and BC with AD. This surface is the so-called
the tsecond countability axiom and of which
2-dimensional torus (or anchor ring). In this
every point has a neighborhood thomeomor-
case, the four vertices A, B, C, D of the rec-
phic to the interior of a circular disk in a
tangle correspond to one point p on the sur-
2-dimensional Euclidean space. In the follow-
face, and the pairs of edges AB, DC and BC,
ing sections, we mean by a surface such a 2-
AD correspond to closed curves a’ and h’ on
dimensional topological manifold.
the surface. We use the notation aba-‘bm’ to
represent a torus. This refers to the fact that
the torus is obtained from an oriented four-
B. Examples and Classification
sided polygon by identifying the first side and
the third (with reversed orientation), the sec-
The simplest examples of surfaces are the 2-
ond side and the fourth (with reversed orienta-
dimensional tsimplex and the 2-dimensional
tion). Similarly, aa m1 represents a sphere (Fig.
isphere. Surfaces are generally +simplicially
3), and a,b,a;lb;‘a,b,a;lb;l represents the
decomposable (or triangulable) and hence
closed surface shown in Fig. 4.
homeomorphic to 2-dimensional polyhedra (T.
Rad6, Acta Sci. Math. Szeged. (1925)). A +com-
pact surface is called a closed surface, and a
noncompact surface is called an open surface.
A closed surface is decomposable into a finite
B b C
number of 2-simplexes and so can be inter-
preted as a tcombinatorial manifold. A 2-
dimensional topological manifold having a
boundary is called a surface with boundary. A
2-simplex is an example of a surface with
boundary, and a sphere is an example of a
Fig. 2
closed surface without boundary.
Surfaces are classified as torientable and
tnonorientable. In the special case when a sur-
face is +embedded in a 3-dimensional Euclid-
ean space E3, whether the surface is orien-
table or not depends on its having two sides
(the “surface” and “back”) or only one side.
Therefore, in this special case, an orientable
surface is called two-sided, and a nonorientable Fig. 3
410 B 1550
Surfaces
The l-dimensional Betti number of this
surface is q - 1, the O-dimensional and 2-
dimensional Betti numbers are 1 and 0, re-
spectively, the l-dimensional torston coeffi-
cient is 2, the O-dimensional and 2-dimensional
torsion coefficients are 0, and q is called the
genus of the surface. A closed nonorientable
surface of genus q with boundaries c, , , ck
is represented by
-1
w,c,w, . ..WkCkWk -‘alal . ..uquy. (4)
Each of forms (l))(4) is called the normal form
of the respective surface, and-the curves q, b,,
wk are called the normal sections of the surface.
To explain the notation in (3), we first take the
simplest case, aa. In this case, the surface is
obtained from a disk by identifying each pair
of points on the circumference that are end-
points of a diameter (Fig. 6). The :surface au is
then homeomorphic to a iproject-lve plane of
which a decomposition into a complex of
triangles is illustrated in Fig. 7. On the other
hand, aabb represents a surface like that
shown in Fig. 8, called the Klein bottle. Fig. 9
shows a handle, and Fig. 10 shows a cross cap.
Fig. 4
All closed surfaces without boundary are
constructed by identifying suitable pairs of
sides of a 2n-sided polygon in a Euclidean
plane E*. Furthermore, a closed orientable
surface without boundary is homeomorphic to
Fig. 6
the surface represented by au-’ or
u,h,a;‘b,‘...a,b,a,‘b,‘. (1) .A
The 1 -dimensional +Betti number of this B c
surface is 2p, the O-dimensional and 2-dimen-
F .E
sional +Betti numbers are 1, the ttorsion coefi-
cients are all 0, and p is called the genus of the
C’ I) B
surface. Also, a closed orientable surface of
genus p with boundaries ci , . , ck is repre- @
A
sented by
Fig. I
w,c, w;’ w,c,w,‘a,b,a;‘b,’ . ..a.b,a,‘b,’
(2)
b
(Fig. 5). A closed nonorientable surface with-
out boundary is represented by n 6 =
(3) tl
Fig. 8
Fig. 5 Fig. 9
1551 411 B
Symbolic Logic
[4] D. Hilbert and S. Cohn-Vossen, Anschau-
fiche Geometrie, Springer, 1932; English
translation, Geometry and the imagination,
Chelsea, 1952.
[S] W. S. Massey, Algebraic topology: An
introduction, Springer, 1967.
[6] E. E. Moise, Geometric topology in dimen-
sions 2 and 3, Springer, 1977.
Fig. 10
411 (1.4)
The last two surfaces have boundaries; a Symbolic Logic
handle is orientable, while a cross cap is non-
orientable and homeomorphic to the Mobius
strip. If we delete p disks from a sphere and A. General Remarks
replace them with an equal number of handles,
then we obtain a surface homeomorphic to Symbolic logic (or mathematical logic) is a field
the surface represented in (1) while if we of logic in which logical inferences commonly
replace the disks by cross caps instead of by used in mathematics are investigated by use of
handles, then the surface thus obtained is mathematical symbols.
homeomorphic to that represented in (3). The algebra of logic originally set forth by
Now we decompose the surfaces (1) and (3) G. Boole [l] and A. de Morgan [2] is actually
into triangles and denote the number of i- an algebra of sets or relations; it did not reach
dimensional simplexes by si (i = 0, 1,2). Then in the same level as the symbolic logic of today.
view of the tEuler-Poincare formula, the sur- G. Frege, who dealt not only with the logic
faces (1) and (3) satisfy the respective formulas of propositions but also with the first-order
predicate logic using quantifiers (- Sections
C and K), should be regarded as the real
a,-q+a,=2-q. originator of symbolic logic. Frege’s work,
however, was not recognized for some time.
The tRiemann surfaces of talgebraic func-
Logical studies by C. S. Peirce, E. Schroder,
tions of one complex variable are always sur-
and G. Peano appeared soon after Frege, but
faces of type (1) and their genera p coincide they were limited mostly to propositions and
with those of algebraic functions. did not develop Frege’s work. An essential
All closed surfaces are homeomorphic to development of Frege’s method was brought
surfaces of types (I), (2), (3), or (4). A necessary about by B. Russell, who, with the collabor-
and sufficient condition for two surfaces to be ation of A. N. Whitehead, summarized his
homeomorphic to each other is coincidence of results in Principia mathematics [4], which
the numbers of their boundaries, their orienta- seemed to have completed the theory of sym-
bility or nonorientability, and their genera (or
bolic logic at the time of its appearance.
+Euler characteristic a0 -u’ + 3’). This propo-
sition is called the fundamental theorem of the
topology of surfaces. The thomeomorphism B. Logical Symbols
problem of closed surfaces is completely solved
by this theorem. The same problem for n If A and B are propositions, the propositions
(n > 3) manifolds, even if they are compact, (A and B), (A or B), (A implies B), and (not A)
remains open. (For surface area - 246 Length are denoted by
and Area. For the differential geometry of
A A B, AvB, A-tB, lA,
surfaces - 111 Differential Geometry of
Curves and Surfaces.) respectively. We call 1 A the negation of A,
A A B the conjunction (or logical product),
A v B the disjunction (or logical sum), and
References A + B the implication (or B by A). The propo-
sition (A+B)r\(B+A) is denoted by AttB
[l] B. Kerekjarto, Vorlesungen iiber Topo- and is read “A and B are equivalent.” A v B
logie, Springer, 1923. means that at least one of A and B holds. The
[2] H. Seifert and W. Threlfall, Lehrbuch der propositions (For all x, the proposition F(x)
Topologie, Teubner, 1934 (Chelsea, 1945). holds) and (There exists an x such that F(x)
[3] S. Lefschetz, Introduction to topology, holds) are denoted by VxF(x) and 3xF(x),
Princeton Univ. Press, 1949. respectively. A proposition of the form V.xF(x)
411 c 1552
Symbolic Logic
is called a universal proposition, and one of the E. Propositional Logic
form &F(x), an existential proposition. The
symbols A, v , -+, c--), 1, V, 3 are called log- Propositional logic is the field in symbolic
ical symbols. logic in which we study relations between
There are various other ways to denote propositions exclusively in connection with the
logical symbols, including: four logical symbols A, v , +, and 1, called
propositional connectives.
AAB: A&B, A.B,
In propositional logic, we deal only with
AvB: A+B, operations of logical operators denoted by
propositional connectives, regarding the vari-
A+B: AxB, A-B,
ables for denoting propositions, called propo-
AttB: APB, A-B, A-B, AIcB, A-B, sition variables, only as prime formulas. We
examine problems such as: What kinds of
1A: -A, A;
formulas are identically true when their propo-
sition variables are replaced by any propo-
VxF(x): (x)F(x), rIxF(x), &Jw,
sitions, and what kinds of formulas can some-
3xF(x): (Ex)F(x), CxF(x), VxF(x). times be true?
Consider the two symbols v and A,
read true and false, respectively, and let A =
C. Free and Bound Variables {V, A}. A univalent function frotn A, or
more generally from a Cartesian product
Any function whose values are propositions is A x . x A, into A is called a truth function.
called a propositional function. Vx and 3x can We can regard A, v, +, 1 as the following
be regarded as operators that transform any truth functions: (1) A A B= Y for 4 = B= v,
propositional function F(x) into the propo- and AA B= h otherwise; (2) A vB= h for
sitions VxF(x) and 3xF(x), respectively. Vx and A=B=h,andAvB= Votherwise;(3)
3x are called quantifiers; the former is called A-B= h for A= Y and B= h, and
the universal quantifier and the latter the A+B= v otherwise; (4) lA= h for A= v,
existential quantifier. F(x) is transformed and lA=Y for A= h.
into VxF(x) or 3xF(x) just as a function f(x) If we regard proposition variabmles as vari-
is transformed into the definite integral ables whose domain is A, then each formula
Jd f(x)dx; the resultant propositions VxF(x) represents a truth function. Conversely, any
and 3xF(x) are no longer functions of x. The truth function (of a finite number of indepen-
variable x in VxF(x) and in 3xF(x) is called a dent variables) can be expressed by an appro-
bound variable, and the variable x in F(x), priate formula, although such a formula is not
when it is not bound by Vx or 3x, is called a uniquely determined. If a formula is regarded
free variable. Some people employ different as a truth function, the value of thle function
kinds of symbols for free variables and bound determined by a combination of values of the
variables to avoid confusion. independent variables involved in the formula
is called the truth value of the formula.
A formula corresponding to a truth function
D. Formal Expressions of Propositions that takes only v as its value is called a tau-
tology. For example, %v 12I and ((‘X-B)
A formal expression of a proposition in terms +5X)+ 9I are tautologies. Since a truth func-
of logical symbols is called a formula. More tion with n independent variables takes values
precisely, formulas are constructed by the corresponding to 2” combinations of truth
following formation rules: (1) If VI is a formula, values of its variables, we can determine in a
1% is also a formula. If 9I and 8 are for- finite number of steps whether a given formula
mulas, 9I A %, Cu v 6, % --) b are all formulas. is a tautology. If a-23 is a tautology (that is,
(2) If 8(a) is a formula and a is a free variable, Cu and !.I3 correspond to the same truth func-
then Vxg(x) and 3x5(x) are formulas, where x tion), then the formulas QI and 23 .are said to be
is an arbitrary bound variable not contained equivalent.
in z(a) and 8(x) is the result of substituting x
for a throughout s(a).
We use formulas of various scope accord- F. Propositional Calculus
ing to different purposes. To indicate the scope
of formulas, we fix a set of formulas, each It is possible to choose some specific tau-
element of which is called a prime formula (or tologies, designate them as axioms, and derive
atomic formula). The scope of formulas is the all tautologies from them by appropriately
set of formulas obtained from the prime for- given rules of inference. Such a system is called
mulas by formation rules (1) and (2). a propositional calculus. There are many ways
1553 411 H
Symbolic Logic
to stipulate axioms and rules of inference for field are predicate variables defined over a
a propositional calculus. certain common domain and object variables
The abovementioned propositional calculus running over the domain. Propositional vari-
corresponds to the so-called classical propo- ables are regarded as predicates of no vari-
sitional logic (- Section L). By choosing ap- ables. Each expression F(a,, . . , a,) for any
propriate axioms and rules of inference we can predicate variable F of n variables a,, , a,
also formally construct intuitionistic or other (object variables designated as free) is regarded
propositional logics. In intuitionistic logic the as a prime formula (n = 0, 1,2, ), and we deal
law of the texcluded middle is not accepted, exclusively with formulas generated by these
and hence it is impossible to formalize intui- prime formulas, where bound variables are
tionistic propositional logic by the notion of also restricted to object variables that have a
tautology. We therefore usually adopt the common domain. We give no specification for
method of propositional calculus, instead of the range of objects except that it be the com-
using the notion of tautology, to formalize mon domain of the object variables.
intuitionistic propositional logic. For example, By designating an object domain and sub-
V. I. Glivenko’s theorem [S], that if a formula stituting a predicate defined over the domain
‘91 can be proved in classical logic, then 1 1 CL1 for each predicate variable in a formula, we
can be proved in intuitionistic logic, was ob- obtain a proposition. By substituting further
tained by such formalistic considerations. A an object (object constant) belonging to the
method of extending the classical concepts of object domain for each object variable in a
truth value and tautology to intuitionistic proposition, we obtain a proposition having a
and other logics has been obtained by S. A. definite truth value. When we designate an
Kripke. There are also studies of logics inter- object domain and further associate with each
mediate between intuitionistic and classical predicate variable as well as with each object
logic (T. Umezawa). variable a predicate or an object to be sub-
stituted for it, we call the pair consisting of the
object domain and the association a model.
G. Predicate Logic Any formula that is true for every model is
called an identically true formula or valid
Predicate logic is the area of symbolic logic in formula. The study of identically true formu-
which we take quantifiers in account. Mainly las is one of the most important problems in
propositional functions are discussed in predi- predicate logic.
cate logic. In the strict sense only single-
variable propositional functions are called
predicates, but the phrase predicate of n argu- H. Formal Representations of Mathematical
ments (or wary predicate) denoting an n- Propositions
variable propositional function is also em-
ployed. Single-variable (or unary) predicates To obtain a formal representation of a math-
are also called properties. We say that u has ematical theory by predicate logic, we must
the property F if the proposition F(a) formed first specify its object domain, which is a non-
by the property F is true. Predicates of two empty set whose elements are called individ-
arguments are called binary relations. The uals; accordingly the object domain is called
proposition R(a, b) formed by the binary re- the individual domain, and object variables are
lation R is occasionally expressed in the form called individual variables. Secondly we must
aRb. Generally, predicates of n arguments are specify individual symbols, function symbols,
called n-ary relations. The domain of defini- and predicate symbols, signifying specific indi-
tion of a unary predicate is called the object viduals, functions, and tpredicates, respectively.
domain, elements of the object domain are Here a function of n arguments is a univa-
called objects, and any variable running over lent mapping from the Cartesian product
the object domain is called an object variable. D x x D of n copies of the given set to D.
We assume here that the object domain is not Then we define the notion of term as in the
empty. When we deal with a number of predi- next paragraph to represent each individual
cates simultaneously (with different numbers of formally. Finally we express propositions for-
variables), it is usual to arrange things so that mally by formulas.
all the independent variables have the same Definition of terms (formation rule for terms):
object domain by suitably extending their (1) Each individual symbol is a term. (2) Each
object domains. free variable is a term. (3) f(tt , , t,) is a term
Predicate logic in its purest sense deals if t, , , t, are terms and ,f is a function symbol
exclusively with the general properties of of n arguments. (4) The only terms are those
quantifiers in connection with propositional given by (l)-(3).
connectives. The only objects dealt with in this As a prime formula in this case we use any
411 I 1554
Symbolic Logic
formula of the form F(t,, , t,), where F is a contradiction. The validity of a proof by
predicate symbol of n arguments and t,, , t, reductio ad absurdum lies in the f.act that
are arbitrary terms. To define the notions of
((Il-r(BA liB))-1%
term and formula, we need logical symbols,
free and bound individual variables, and also a is a tautology. An affirmative proposition
list of individual symbols, function symbols, (formula) may be obtained by reductio ad
and predicate symbols. absurdum since the formula (of flropositional
In pure predicate logic, the individual logic) representing the discharge of double
domain is not concrete, and we study only negation
general forms of propositions. Hence, in this
1 lT!+'U
case, predicate or function symbols are not
representations of concrete predicates or func- is a tautology.
tions but are predicate variables and function
variables. We also use free individual variables
instead of individual symbols. In fact, it is now J. Predicate Calculus
most common that function variables are
dispensed with, and only free individual vari- If a formula has no free individual variable, we
ables are used as terms. call it a closed formula. Now we consider a
formal system S whose mathematical axioms
are closed. A formula 91 is provable in S if
I. Formulation of Mathematical Theories and only if there exist suitable m.athematical
axioms E,, ,E, such that the formula
To formalize a theory we need axioms and
rules of inference. Axioms constitute a certain
specific set of formulas, and a rule of inference is provable without the use of mathematical
is a rule for deducing a formula from other axioms. Since any axiom system can be re-
formulas. A formula is said to be provable if it placed by an equivalent axiom system contain-
can be deduced from the axioms by repeated ing only closed formulas, the study of a formal
application of rules of inference. Axioms are system can be reduced to the study of pure
divided into two types: logical axioms, which logic.
are common to all theories, and mathematical In the following we take no individual sym-
axioms, which are peculiar to each individual bols or function symbols into consideration
theory. The set of mathematical axioms is and we use predicate variables as predicate
called the axiom system of the theory. symbols in accordance with the commonly
(I) Logical axioms: (1) A formula that is the accepted method of stating properties of the
result of substituting arbitrary formulas for the pure predicate logic; but only in the case of
proposition variables in a tautology is an predicate logic with equality will ‘we use predi-
axiom. (2) Any formula of the form cate variables and the equality predicate = as
a predicate symbol. However, we can safely
state that we use function variables as function
is an axiom, where 3(t) is the result of sub- symbols.
stituting an arbitrary term t for x in 3(x). The formal system with no mathematical
(II) Rules of inference: (I) We can deduce a axioms is called the predicate calculus. The
formula 23 from two formulas (rl and ‘U-8 formal system whose mathematical axioms are
(modus ponens). (2) We can deduce C(I+VX~(X) the equality axioms
from a formula %+3(a) and 3x3(x)+%
u=u, u=/J + m4+im))
from ~(a)+%, where u is a free individual
variable contained in neither ‘11 nor s(x) and is called the predicate calculus with equality.
%(a) is the result of substituting u for x in g(x). In the following, by being provable we mean
If an axiom system is added to these logical being provable in the predicate calculus.
axioms and rules of inference, we say that a (1) Every provable formula is valid.
formal system is given. (2) Conversely, any valid formula is prov-
A formal system S or its axiom system is able (K. Code1 [6]). This fact is called the
said to be contradictory or to contain a con- completeness of the predicate calculus. In fact,
tradiction if a formula VI and its negation 1 CLI by Godel’s proof, a formula (rI is provable if
are provable; otherwise it is said to be consis- 9I is always true in every interpretation whose
tent. Since individual domain is of tcountable cardinality.
In another formulation, if 1 VI is not provable,
the formula 3 is a true proposition in some
is a tautology, we can show that any formula interpretation (and the individual domain in
is provable in a formal system containing a this case is of countable cardinality). We can
1555 411 K
Symbolic Logic
extend this result as follows: If an axiom sys- a normal form 9I’ satisfying the condition: YI’
tem generated by countably many closed has the form
formulas is consistent, then its mathematical
Q,-xl . . . Q.x,W,, . . ..x.),
axioms can be considered true propositions
by a common interpretation. In this sense, where Qx means a quantifier Vx or 3x, and
Giidel’s completeness theorem gives another %(x,, , x,) contains no quantifier and has no
proof of the %kolem-Lowenheim theorem. predicate variables or free individual variables
(3) The predicate calculus is consistent. not contained in ‘Ll. A normal form of this
Although this result is obtained from (1) in this kind is called a prenex normal form.
section, it is not difftcult to show it directly (7) We have dealt with the classical first-
(D. Hilbert and W. Ackermann [7]). order predicate logic until now. For other
(4) There are many different ways of giving predicate logics (- Sections K and L) also, we
logical axioms and rules of inference for the can consider a predicate calculus or a formal
predicate calculus. G. Gentzen gave two types system by first defining suitable axioms or
of systems in [S]; one is a natural deduction rules of inference. Gentzen’s fundamental
system in which it is easy to reproduce formal theorem applies to the intuitionistic predicate
proofs directly from practical ones in math- calculus formulated by V. I. Glivenko, A.
ematics, and the other has a logically simpler Heyting, and others. Since Gentzen’s funda-
structure. Concerning the latter, Gentzen mental theorem holds not only in classical
proved Gentzen’s fundamental theorem, which logic and intuitionistic logic but also in several
shows that a formal proof of a formula may be systems of frst-order predicate logic or pro-
translated into a “direct” proof. The theorem positional logic, it is useful for getting results
itself and its idea were powerful tools for ob- in modal and other logics (M. Ohnishi, K.
taining consistency proofs. Matsumoto). Moreover, Glivenko’s theorem
(5) If the proposition 3x’.(x) is true, we in propositional logic [S] is also extended to
choose one of the individuals x satisfying the predicate calculus by using a rather weak
condition ‘LI(x), and denote it by 8x%(x). When representation (S. Kuroda [12]). G. Takeuti
3x91(x) is false, we let c-:x’lI(x) represent an expected that a theorem similar to Gentzen’s
arbitrary individual. Then fundamental theorem would hold in higher-
order predicate logic also, and showed that
3xQr(x)+‘x(ExcLr(x)) (1)
the consistency of analysis would follow if
is true. We consider EX to be an operator as- that conjecture could be verified [ 131. More-
sociating an individual sxqI(x) with a propo- over, in many important cases, he showed
sition 9I(x) containing the variable x. Hilbert constructively that the conjecture holds par-
called it the transfinite logical choice function; tially. The conjecture was finally proved by
today we call it Hilbert’s E-operator (or E- M. Takahashi [ 141 by a nonconstructive
quantifier), and the logical symbol E used in method. Concerning this, there are also con-
this sense Hilbert’s E-symbol. Using the E- tributions by S. Maehara, T. Simauti, M.
symbol, 3xX(x) and Vx’lI(x) are represented by Yasuhara. and W. Tait.
Bl(EXPI(X)), \Ll(cx 1 VI(x)),
respectively, for any N(x). The system of predi- K. Predicate Logics of Higher Order
cate calculus adding formulas of the form (1)
as axioms is essentially equivalent to the usual In ordinary predicate logic, the bound vari-
predicate calculus. This result, called the c- ables are restricted to individual variables. In
theorem, reads as follows: When a formula 6 is this sense, ordinary predicate logic is called
provable under the assumption that every first-order predicate logic, while predicate logic
formula of the form (1) is an axiom, we can dealing with quantifiers VP or 3P for a predi-
prove (5 using no axioms of the form (1) if Cr cate variable P is called second-order predicate
contains no logical symbol s (D. Hilbert and logic.
P. Bernays [9]). Moreover, a similar theorem Generalizing further, we can introduce the
holds when axioms of the form so-called third-order predicate logic. First we
fix the individual domain D,. Then, by intro-
vx(‘.x(x)~B(x))~EX%(X)=CX%(X) (2)
ducing the whole class 0; of predicates of n
are added (S. Maehara [lo]). variables, each running over the object domain
(6) For a given formula ‘U, call 21’ a normal D,, we can introduce predicates that have 0;
form of PI when the formula as their object domain. This kind of predicate
is called a second-order predicate with respect
YIttW
to the individual domain D,. Even when
is provable and ‘% satisfies a particular con- we restrict second-order predicates to one-
dition For example, for any formula YI there is variable predicates, they are divided into vari-
411 L 1556
Symbolic Logic
ous types, and the domains of independent sitional logic, predicate logic, and type theory
variables do not coincide in the case of more are developed from the standpoint of classical
than two variables. In contrast, predicates logic. Occasionally the reasoning of intuition-
having D, as their object domain are called istic mathematics is investigated using sym-
first-order predicates. The logic having quan- bolic logic, in which the law of the excluded
tifiers that admit first-order predicate variables middle is not admitted (- 156 Foundations of
is second-order predicate logic, and the logic Mathematics). Such logic is called intuitionistic
having quantifiers that admit up to second- logic. Logic is also subdivided into proposi-
order predicate variables is third-order predi- tional logic, predicate logic, etc., according to
cate logic. Similarly, we can define further the extent of the propositions (formulas) dealt
higher-order predicate logics. with.
Higher-order predicate logic is occasionally To express modal propositions stating possi-
called type theory, because variables arise that bility, necessity, etc., in symbolic logic, J. tu-
are classified into various types. Type theory is kaszewicz proposed a propositional logic called
divided into simple type theory and ramified three-valued logic, having a third truth value,
type theory. neither true nor false. More generally, many-
We confine ourselves to variables for single- valued logics with any number of truth values
variable predicates, and denote by P such a have been introduced; classical logic is one of
bound predicate variable. Then for any for- its special cases, two-valued logic with two
mula ;4(a) (with a a free individual variable), truth values, true and false. Actually, however,
the formula many-valued logics with more than three truth
values have not been studied mu’ch, while
various studies in modal logic based on classi-
is considered identically true. This is the point cal logic have been successfully carried out.
of view in simple type theory. For example, studies of strict implication
Russell asserted first that this formula can- belong to this field.
not be used reasonably if quantifiers with
respect to predicate variables occur in s(x).
This assertion is based on the point of view
References
that the formula in the previous paragraph
asserts that 5(x) is a first-order predicate,
whereas any quantifier with respect to first- [l] G. Boole, An investigation of the laws of
order predicate variables, whose definition thought, Walton and Maberly, 1:554.
assumes the totality of the first-order predi- [2] A. de Morgan, Formal logic, or the cal-
cates, should not be used to introduce the first- culus of inference, Taylor and Walton, 1847.
order predicate a(x). For this purpose, Russell [3] G. Frege, Begriffsschrift, eine der arith-
further classified the class of first-order predi- metischen nachgebildete Formalsprache des
cates by their rank and adopted the axiom reinen Denkens, Halle, 1879.
[4] A. N. Whitehead and B. Russell, Principia
mathematics I, II, Ill, Cambridgl: Univ. Press,
for the predicate variable Pk of rank k, where 1910-1913; second edition, 1925-1927.
the rank i of any free predicate variable occur- [S] V. Glivenko, Sur quelques points de la
ring in R(x) is dk, and the rank j of any logique de M. Brouwer, Acad. Roy. de Bel-
bound predicate variable occurring in g(x) is gique, Bulletin de la classe des sciences, (5) 15
<k. This is the point of view in ramified type (1929) 1833188.
theory, and we still must subdivide the types if [6] K. Godel, Die Vollstlndigkeit der Axiome
we deal with higher-order propositions or des logischen Funktionenkalkiils, Monatsh.
propositions of many variables. Even Russell, Math. Phys., 37 (1930) 3499360.
having started from his ramified type theory, [7] D. Hilbert and W. Ackermann, Grundziige
had to introduce the axiom of reducibility der theoretischen Logik, Springer, 1928, sixth
afterwards and reduce his theory to simple edition, 1972; English translation, Principles of
type theory. mathematical logic, Chelsea, 1950.
[S] G. Gentzen, Untersuchungen iiber das
logische Schliessen, Math. Z., 39 (1935) 1766
210,4055431.
L. Systems of Logic
[9] D. Hilbert and P. Bernays, Grundlagen
der Mathematik II, Springer, 1939; second
Logic in the ordinary sense, which is based on edition, 1970.
the law of the excluded middle asserting that [lo] S. Maehara, Equality axiom on Hilbert’s
every proposition is in principle either true or a-symbol, J. Fat. Sci. Univ. Tokyo, (I), 7 (1957)
false, is called classical logic. Usually, propo- 419-435.
1557 412 C
Symmetric Riemannian Spaces and Real Forms
[l 11 A. Heyting, Die formalen Regeln der space (with respect to 0) if there exists an in-
intuition&&hen Logik I, S.-B. Preuss. Akad. volutive automorphism (i.e., automorphism of
Wiss., 1930,42%56. order 2) 0 of G satisfying the condition Kt c
[ 121 S. Kuroda, Intuitionistische Untersu- Kc K,, where K, is the closed subgroup con-
chungen der formalist&hen Logik, Nagoya sisting of all elements of G left fixed by 0 and
Math. J., 2 (195 l), 35-47. K,” is the connected component of the iden-
[13] G. Take&, On a generalized logic cal- tity element of K,,. In this case, the mapping
culus, Japan. J. Math., 23 (1953), 39-96. aK+B(a)K (aEG) is a transformation of
[14] M. Takahashi, A proof of the cut- G/K having the point K as an isolated fixed
elimination theorem in simple type-theory, J. point; more generally, the mapping OoO:a K --)
Math. Sot. Japan, 19 (1967), 399-410. a,O(a,)-’ O(a)K is a transformation of G/K
[ 151 S. C. Kleene, Mathematical logic, Wiley, that has an arbitrary given point a, K of G/K
1967. as an isolated fixed point. If there exists a G-
[16] J. R. Shoeniield, Mathematical logic, invariant Riemannian metric on G/K, then
Addison-Wesley, 1967. G/K is a symmetric Riemannian space with
[17] R. M. Smullyan, First-order logic, Sprin- symmetries { QaO1 a ,, E G} and is called a sym-
ger, 1968. metric Riemannian homogeneous space. A
sufficient condition for a symmetric homoge-
neous space G/K to be a symmetric Riemann-
ian homogeneous space is that K be a com-
412 (IV.13) pact subgroup. Conversely, given a symmetric
Symmetric Riemannian Riemannian space M, let G be the connected
component of the identity element of the Lie
Spaces and Real Forms
group formed by all the isometries of M; then
M is represented as the symmetric Riemannian
A. Symmetric Riemannian Spaces homogeneous space M = G/K and K is a com-
pact group. In particular, a symmetric Rie-
Let M be a +Riemannian space. For each point mannian space can be regarded as a Riemann-
p of M we can define a mapping gp of a suit- ian space that is realizable as a symmetric
able neighborhood U, of p onto U, itself so Riemannian homogeneous space.
that a,(x,)=x-,, where x, (It/ <E,x~=P) is any The Riemannian connection of a symme-
tgeodesic passing through the point p. We call tric Riemannian homogeneous space G/K is
M a locally symmetric Riemannian space if for uniquely determined (independent of the
any point p of M we can choose a neighbor- choice of G-invariant Riemannian metric), and
hood U,, so that crp is an tisometry of U,,. In a geodesic xt( j tI < co, x0 = a, K) passing
order that a Riemannian space M be locally through a point a, K of G/K is of the form
symmetric it is necessary and sufficient that the x, = (exp tX)a, K. Here X is any element of the
tcovariant differential (with respect to the Lie algebra g of G such that O(X)= -X, where
+Riemannian connection) of the tcurvature 0 also denotes the automorphism of g induced
tensor of A4 be 0. A locally symmetric Riemann- by the automorphism 0 of G and exp tX is the
ian space is a +real analytic manifold. We say tone-parameter subgroup of G defined by the
that a Riemannian space M is a globally sym- element X. The covariant differential of any G-
metric Riemannian space (or simply symmetric invariant tensor field on G/K is 0, and any G-
Riemannian space) if M is connected and if for invariant idifferential form on G/K is a closed
each point p of M there exists an isometry cp differential form.
of M onto M itself that has p as an isolated
fixed point (i.e., has no fixed point except p in a
certain neighborhood of p) and such that 0; is C. Classification of Symmetric Riemannian
the identity transformation on M. In this case Spaces
ap is called the symmetry at p. A (globally)
symmetric Riemannian space is locally sym- The tsimply connected tcovering Riemannian
metric and is a tcomplete Riemannian space. space of a symmetric Riemannian space is also
Conversely, a tsimply connected complete a symmetric Riemannian space. Therefore the
locally symmetric Riemannian space is a (glob- problem of classifying symmetric Riemannian
ally) symmetric Riemannian space. spaces is reduced to classifying simply con-
nected symmetric Riemannian spaces M and
B. Symmetric Riemannian Homogeneous determining tdiscontinuous groups of iso-
metries of M. When we take the +de Rham
Spaces
decomposition of such a space M and repre-
A thomogeneous space GJK of a connected sent M as the product of a real Euclidean
+Lie group G is a symmetric homogeneous space and a number of simply connected irre-
412 D 1.558
Symmetric Riemannian Spaces and Real Forms
ducible Riemannian spaces, all the factors are symmetric Riemannian space defines one of
symmetric Riemannian spaces. We say that M (l)-(4) as its tuniversal covering manifold; if
is an irreducible symmetric Riemannian space if the covering manifold is of type (3) or (4), the
it is a symmetric Riemannian space and is original symmetric Riemannian space is neces-
irreducible as a Riemannian space. sarily simply connected.
A simply connected irreducible symmetric
Riemannian space is isomorphic to one of the
following four types of symmetric Riemannian
D. Symmetric Riemannian Homogeneous
homogeneous spaces (here Lie groups are
Spaces of Semisimple Lie Groups
always assumed to be connected):
(1) The symmetric Riemannian homoge-
neous space (G x G)/{ (a, a) 1a E G) of the direct In Section C we saw that any irreducible sym-
product G x G, where G is a simply connected metric Riemannian space is representable as a
compact isimple Lie group and the involutive symmetric Riemannian homogeneous space
automorphism of G x G is given by (a, h)d(h, a) G/K on which a connected semisimple Lie
((a, h)~ G x G). This space is isomorphic, as a group G acts +almost effectively (-- 249 Lie
Riemannian space, to the space G obtained by Groups). Among symmetric Riemannian
introducing a two-sided invariant Riemannian spaces, such a space A4 = G/K is characterized
metric on the group G; the isomorphism is as one admitting no nonzero vector field that
induced from the mapping G x G ~(a, h)+ is tparallel with respect to the Riemannian
Ub-‘EG. connection. Furthermore, if G acis effectively
(2) A symmetric homogeneous space G/K, on M, G coincides with the connected compo-
of a simply connected compact simple Lie nent I(M)’ of the identity element of the Lie
group G with respect to an involutive auto- group formed by all the isometries of M.
morphism 0 of G. In this case, the closed sub- We let M = G/K be a symmetric Riemann-
group K, = {a E G) 0(u) = u} of G is connected. ian homogeneous space on which. a con-
We assume here that 0 is a member of the nected semisimple Lie group G acts almost
given complete system of representatives of the effectively. Then G is a Lie group that is tlocally
iconjugate classes formed by the elements of isomorphic to the group 1(M)‘, and therefore
order 2 in the automorphism group of the the Lie algebra of G is determined by M. Let g
group G. be the Lie algebra of G, f be the subalgebra of
(3) The homogeneous space G”/G, where GC g corresponding to K, and 0 be the involutive
is a complex simple Lie group whose tcenter automorphism of G defining the symmetric
reduces to the identity element and G is an homogeneous space G/K. The automorphism
arbitrary but fixed maximal compact subgroup of g defined by 6’ is also denoted by 0. Then f =
of CC. {XEgIQ(X)=X}. Puttingm={XEg/B(X)=
(4) The homogeneous space G,/K, where G, -X}, we have g = m + f (direct sum of linear
is a noncompact simple Lie group whose spaces), and nr can be identified in a natural
center reduces to the identity element and way with the tangent space at the point K of
which has no complex Lie group structure, G/K. The tadjoint representation of G gives
and K is a maximal compact subgroup of G. rise to a representation of K in g, which in-
In Section D we shall see that (3) and (4) are duces a linear representation Ad,,,(k) of K in m.
actually symmetric homogeneous spaces. All Then {Ad,,,(k) 1k E K} coincides wl th the +res-
four types of symmetric Riemannian spaces are tricted homogeneous holonomy group at the
actually irreducible symmetric Riemannian point K of the Riemannian space G/K.
spaces, and G-invariant Riemannian metrics Now let cp be the +Killing form of g. Then f
on each of them are uniquely determined up to and m are mutually orthogonal with respect to
multiplication by a positive number. On the cp, and denoting by qt and (P,” the restrictions
other hand, (1) and (2) are compact, while (3) of cp to f and m, respectively, qDti s a negative
and (4) are homeomorphic to Euclidean spaces definite quadratic form on f. If v,,~ is also a
and not compact. For spaces of types (1) and negative definite quadratic form on nt, g is a
(3) the problem of classifying simply connected compact real semisimple Lie algebra and G/K
irreducible symmetric Riemannian spaces is is a compact symmetric Riemannian space; in
reduced to classifying +compact real simple Lie this case we say that G/K is of compact type.
algebras and tcomplex simple Lie algebras, In the opposite case, where (pm is a tpositive
respectively, while for types (2) and (4) it is definite quadratic form, G/K is said to be of
reduced to the classification of noncompact noncompact type. In this latter case, G/K is
real simple Lie algebras (- Section D) (for the homeomo’rphic to a Euclidean space, and if
result of classification of these types - Ap- the center of G is finite, K is a maximal com-
pendix A, Table 5.11). On the other hand, any pact subgroup of G. Furthermore, the group
(not necessarily simply connected) irreducible of isometries I(G/K) of G/K is canonically