英语二年级单词表_百度知道
英语二年级单词表
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a big mouth一个大嘴巴40fruit shop水果店2a bottle of jam一瓶果酱41go shopping去购物3a red nose一个红鼻子42go to school去学校4a white bird白鸟43hands down放下手5an apple pie一个苹果派44hands up举起手6ask and answer问和答45have a breakfast吃早饭7at the circus在马戏团46have a picnic举行一个野餐8big and blue大又蓝47have dinner吃晚餐9big and tall又大又高48have lunch吃午饭10birds nest鸟巢49here we are我们在那儿11black hair黑头发50in her cage在她的笼子里12blow a kiss 飞吻51in the classroom在教室13boiled dumpling水饺52in the country park在乡村公园14bread with jam面包夹果酱53in the garden在花园15break time休息时间54in the rain在雨中16brush your teeth刷牙55in the street在街道17chicken wing鸡翅56in the tree在树上18chicks with the hen小鸡跟着母鸡57join the cups把杯子连起来19clap your hands拍手58jump high跳高20class time上课时间59jumping up and down跳跃21clean the blackboard擦黑板60learn the letter学字母22clean the table整理桌子61little finger / small babies小拇指23close the window关窗户62look at the animals看这些动物24close your books关上书本63look in the mirror照镜子25colour picture涂颜色64make a shopping list做一个购物单26colour the circle给圆圈上色65make a telephone做个电话机27come here过来66make two holes做两个洞28count the chicks数小鸡67middle finger中指29cut down砍倒68nep time小睡时间30cut the circle剪下圆圈69on the road在马路上31dance with me和我跳舞70on the roof在屋顶32day and night日日夜夜71open the door打开门33draw a circle画个圆圈72open the gate打开大门34draw two dots画两个点73open the window打开窗35excuse me对不起74open your books打开书本36fall down倒下75play soccer玩足球37fly a kite放风筝76play with yo-yo玩溜溜球38fore finger / thin girls食指77primary school小学39fried bread stick油条78put on my shoes穿上我的鞋子79put on your coat穿上外套118wave youe hands挥挥手80read a book读书119with a string用一根细绳81read and tick读和打勾120writer a letter写信82red flowers红花121your camera你的照相机83ring finger / short mothers无名指84run after追赶85say a poem讲个小故事86say a rhyme读小诗87school bus校车88sing a song 唱歌89sit down坐下90six legs六条腿91six sandwichs六个三明治92smell the air闻空气93smile at me对我微笑94snack time点心时间95soybean milk豆浆96spider web蜘蛛网97spin the spinner转陀螺98spring rolls春卷99stand up起立100steamed buns馒头101super market超市102super show超级演出103sweet sweets甜的糖104table tennis乒乓球105take off your shoes脱下鞋子106the Great Wall长城107the natural word自然界108the red sun红太阳109thumb / fat boys大拇指110turn off the tap关闭龙头111turn on the tap打开龙头112two brown eyes两只棕色的眼睛113two crayons两只蜡笔114two long ears两只长耳朵115two paper cups两个纸杯116up and down上上下下117wash your towel洗毛巾
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出门在外也不愁408 Request Time-out
Your browser didn't send a complete request in time.小清新又袭来：TwoDots领跑多国排行榜
　　各位客官对去年风靡一时的小清新解谜游戏 Dots《点点连线》还有印象么？Dots《点点连线》是一款轻松简单，风格清新简约的休闲游戏。它的玩法很简单，只要将点与点连起来，争取一次性消除最多同色的点，而在消除的同时，游戏音乐也会发生变化，比较特别。
　　游戏截图
　　而 Dots 的续作 TwoDots，也就是本文的主人公，也于5月29日上架。该作在原作基础上并没有做太多的改动，在风格上依旧沿袭着以前的小清新风，而且在圆点之间加入了一个小剧情，孤独的圆点找到了另一个圆点朋友，于是 TwoDots 就诞生了。
　　游戏截图
　　咔!下面当然不是玩法介绍，我们聊一聊排行榜的事情。5月29日上架，截止到近日，该作在AppStore中已经斩获了57个国家的免费榜首位，包括北美在内。在127国家内获得了最佳新游的推荐。
　　排名情况图
　　首位统计情况图
　　畅销榜上，该作也表现给力。在美国已经杀入了Top30，在其他国家内也能看出明显的上升趋势。
　　畅销排行榜情况图
　　综上，可以看出这个叫做点点的游戏已经有了一点流行的趋势。这份小清新，让我第一个联想的就是《Threes!》，同样的清新别致，玩法简单简洁明了。当然，一说到这个我就十分好奇我过几天会不会看到ThreeDots、FourDots什么的。
　　当然，由于这个游戏才出来几天，除了表现抢眼一点确实没有太多可以八卦的地方，那最后就转一下话题，看下它以前的那些先辈们是什么样的状况，聊以参考，展望一下该作的未来之路。
　　历史排名情况图
　　之前火极一时的Smash Hit，还有印象么?
　　历史排名情况图
　　相比来说，小怨妇《Threes!》比楼上混得好一点。
　　《2048》历史排名情况图
　　当然，那是不和三楼比的情况下。
责任编辑：陶国琪(QN056)
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科技万花筒
•&•&•&•&•&The Notation in Principia Mathematica (Stanford Encyclopedia of Philosophy)
The Notation in Principia Mathematica
Principia Mathematica by A.N. Whitehead and Bertrand
Russell, published
in three volumes by Cambridge University
Press, contains a derivation of large portions of mathematics using
notions and principles of symbolic logic.
The notation in that work
has been superseded by the subsequent development of logic during the
20th century, to the extent that the beginner has trouble reading PM
at all. This article provides an introduction to the symbolism of PM,
showing how that symbolism can be translated into a more contemporary
notation which should be familiar to anyone who has had a first course
in symbolic logic. This translation is offered as an aid to learning
the original notation, which itself is a subject of scholarly dispute,
and embodies substantive logical doctrines so that it cannot simply be
replaced by contemporary symbolism. Learning the notation, then, is a
first step to learning the distinctive logical doctrines
of Principia Mathematica.
Principia Mathematica [PM] was written jointly by Alfred
North Whitehead and Bertrand Russell over several years, and published
in three volumes, which appeared between 1910 and 1913. It presents a
system of symbolic logic and then turns to the foundations of
mathematics to carry out the logicist project of defining mathematical
notions in terms of logical notions and proving the fundamental axioms
of mathematics as theorems of logic. While hugely important in the
development of logic, philosophy of mathematics and more broadly of
&Early Analytic Philosophy&, the work itself is no longer studied for
these topics. As a result the very notation of the work has become
alien to contemporary students of logic, and that has become a barrier
to the study of Principia Mathematica.
This entry is intended to assist the student of PM in
reading the symbolic portion of the work. What follows is a partial
translation of the symbolism into a more contemporary notation, which
should be familiar from other articles in this Encyclopedia, and which
is quite standard in contemporary textbooks of symbolic logic. No
complete algorithm is supplied, rather various suggestions are
intended to help the reader learn the symbolism of PM. Many issues of
interpretation would be prejudged by only using contemporary notation,
and many details that are unique to PM depend on that
notation. It will be seen below, with some of the more contentious
aspects of the notation, that doctrines of substance are built into the
notation of PM. Replacing the notation with a more modern symbolism
would drastically alter the very content of the book.
Below the reader will find, in the order in which they are introduced
in PM, the following symbols, which are briefly described. More detail
is provided in what follows:
pronounced &star&; indicates a number, or chapter, as
a centered dot (an old British decimal point); indicates a numbered sentence in the
order by first digit (all the 0s preceding all the 1's etc.), then
second digit, and so on. The first definitions and propositions of
*1 illustrate this
&lexicographical& ordering: 1&01,
1&1, 1&11,
1&71, 1&72.
the assertion- indicates an assertion,
either an axiom (i.e., a primitive proposition,
which are also annotated &Pp&) or a theorem.
follows a definition.
are dots used for de in
contemporary logic, we use (, ), [, ], {, }, etc.
are propositional variables.
are the familiar sentential connectives,
corresponding to &or&&,
&if-then&&,
&not&&, &if and only
if&& and &and&&,
respectively.
[In the Second Edition of PM, 1925&27, the Sheffer
Stroke &|& is the one primitive
connective.
It means &not both & and
y, z, etc.
are individual variables, which are to be
read with &typical ambiguity&, i.e., with their logical
types to be filled in (see below).
b, c, etc.
are individual constants, and stand for
individuals (of the lowest type).
These occur only in the Introduction
to PM, and not in the official system.
R(x), etc.
are atomic predications, in which the
objects named by the variables or constants stand in the relation
R or have the property
R. These occur only in the Introduction.
&a& and &b& occur as
constants only in the Second Edition.
The predications
etc., are used only in the Second Edition.
are variables which range over propositional
functions, no matter whether those functions are simple or
&(x,y), etc.
open atomic formulas in which both
&&& are free. [An alternative
interpretation is to view
&&x& as a schematic letter
standing for a formula in which the variable
&x& is free.]
when placed over a variable in
an open formula (as in
results in a term for a function. [This matter is controversial. See Landini 1998.]
When the circumflected variable precedes
a complex variable, the result indicates a class, as in
&(x&,z&), etc.
Terms for propositional functions. Here are examples
of such terms which are constants:
happy&, &x&
is bald and x& is
happy&, &4 &
If we apply, for example, the function
x& is bald and
x& is happy to the
particular individual b, the result is the
proposition
b is bald and b is
are the quantifiers &there
exists& and &for all& (&every&),
respectively.
For example, where &x is a
simple or complex open formula,
&there exists an x such that
&there exists a propositional function
& such that &x&
&every x is such that
&every propositional function & is such
[These were used by Peano.
More recently, & has been added
for symmetry with &.
Some scholars see the
quantfiers (&) and
(&&) as substitutional.]
This is notation that is used to abbreviate
universally quantified variables.
In modern notation, these become
&x(&x & &x) and
&x(&x & &x),
respectively.
See the definitions for this notation at the end of
Section 3.2 below.
pronounced &shriek&; indicates that a function is
predicative, as in &!x or
See Section 7.
expresses identity,
which is a defined notion in PM, not primitive as in contemporary logic.
read as &the&; is the inverted
iota or description operator and is used in
expressions for definite descriptions, such as
(which is read: the x such that
a definite des this is a scope
indicator for definite descriptions.
is defined at
*14&02, in the context
to mean that the description
is proper, i.e., there is exactly one &.
is defined at
*24&03, in the context
&!&, to mean that the class
& is non-empty, i.e., has a
An immediate obstacle to reading PM is the unfamiliar use of dots
for punctuation, instead of the more common parentheses and
brackets. The system is precise, and can be learned with just a little
practice. The use of dots for punctuation is not unique to
PM. Originating with Peano, it was later used in works by Alonzo
Church, W.V.O.Quine, and others, but it has now largely
disappeared. The best way to learn to use it is to look at a few
samples which are translated to formulae using parentheses, and thus
to get the feel for it. What follows is an explanation as presented in
PM, pages 9&10, followed by a number of examples which illustrate each
of its clauses:
The use of dots. Dots on the line of the
symbols have two uses, one to bracket off propositions, the other to
indicate the logical product of two propositions. Dots immediately
preceded or followed by &v& or
&&& or &&& or
&⊢&,
or by &(x)&,
&(x,y)&, &(x,y,z)&
& or &(&x)&,
&(&x,y,z)& & or
&[(x)(&x)]& or
&[R&y]& or analogous expressions,
serve to brack dots occurring otherwise serve to
mark a logical product. The general principle is that a larger number
of dots indicates an outside bracket, a smaller number indicates an
inside bracket. The exact rule as to the scope of the bracket
indicated by dots is arrived at by dividing the occurrences of dots
into three groups which we will name I, II, and III. Group I consists
of dots adjoining a sign of implication (&) or
equivalence (&) or of disjunction (v) or of
equality by definition (=Df).
Group II consists of
dots following brackets indicative of an apparent variable, such as
(x) or (x,y) or
[(x)(&x)] or analogous
expressions. Group III consists of dots which stand between
propositions in order to indicate a logical product.
Group I is of
greater force than Group II, and Group II than Group III.
of the bracket indicated by any collection of dots extends backwards
or forwards beyond any smaller number of dots, or any equal number
from a group of less force, until we reach either the end of the
asserted proposition or a greater number of dots or an equal number
belonging to a group of equal or superior force. Dots indicating a
logical product have a scope which works both ba
other dots only work away from the adjacent sign of disjunction,
implication, or equivalence, or forward from the adjacent symbol of
one of the other kinds enumerated in Group II. Some examples will
serve to illustrate the use of dots. (PM, 9&10)
3.1 Some Basic Examples
Consider the following series of extended examples, in which we
examine propositions in PM and then discuss how to translate them step
by step into modern notation. (Symbols below are sometimes used as
names for themselves, thus avoiding some otherwise needed quotation
marks. Russell is often accused of confusing use and mention, so there
may well be some danger in this practice.)
In all cases, we use
boldface italics for the notation in PM, and use normal italics for
modern notation (or hybrid notation).
p v p . & .
This is the second assertion of &star& 1.
It is in fact an axiom or
&Primitive Proposition& as indicated by the 'Pp'.
That this is an assertion (axiom or theorem) and not a definition is
indicated by the use of
&⊢&.
(By contrast, a definition would omit the assertion sign but conclude
with a 'Df' sign.) Now the first step in the process of
translating *1&2 into
modern notation is to note the colon.
Recall, from the above quoted
passage, that &a larger number of dots indicates an outside bracket, a
smaller number indicates an inside bracket&.
Thus, the colon here
(which consists of a larger number of dots than the single
dots occurring on the line in
*1&2) represents an
outside bracket.
So, the first step is to translate
⊢ [p v p . & .
So the brackets &[& and &]& represent the colon in
The scope of the
colon thus extends past any smaller number of dots (i.e., one dot) to
the end of the formula.
Next, the dots around the '&' are represented in modern notation
by the parenthesis around the antecedent and consequent.
Recall, in
the above passage, we find && dots only work away from the
adjacent sign of disjunction, implication, or equivalence &&.
Thus, the next step in the translation process is to move to the
[(p v p) & (p)]
Finally, standard modern conventions allow us to delete the outer
brackets and the parentheses around single letters, yielding:
⊢ (p v p) & p
Our next example involves conjunction, which is indicated by simple
juxtaposition of atomic sentences, or with a dot when a substitution
instance might be considered, as in the definition of conjunction in
the following:
= . ~(~p v ~q)
Here we have a case in which dots occur indicate both a
&logical product& (i.e., conjunction) and delimiting
As a first step in translating
*3&01 into modern
notation, we replace the first dot by an ampersand (and its
corresponding scope delimiters) and replace
by &=df&&, to yield:
(p & q) & =df &
[~(~p v ~q)]
The above step clearly illustrates how a &dot indicating a logical
product has a scope which works both backwards and forwards&.
that the first dot in
*3&01, i.e., between the
p and q, is really
optional, given the above quotation from PM.
However, since we may
sometimes want to substitute entire formulas for
p and q, the dot
indicates the extent of the substituted formulas. Thus, we might have,
as a substitution instance: r v s
. q & s (in PM notation) or
(r v s) & (q &
s) (in contemporary symbols).
Finally, our modern conventions allow us to eliminate the outer
parentheses from the definiendum and the brackets &[& and &]& from the
definiens, yielding:
p & q & =df &
~(~p v ~q)
Notice that the scope of the negation
sign &~& in *3&01 is not
indicated with dots, even in the PM system, but rather requires
parentheses.
(&x) . ~&x
If we apply the rule &dots only work away from the adjacent sign of
disjunction, implication, or equivalence, or forward from the adjacent
symbol of one of the other kinds enumerated in Group II& (where Group
II includes &(&x)&), then the modern equivalent would
~(x)&x & =df &
~&x&x & =df &
3.2 The Force of Connectives
The ranking of connectives in terms of relative &force&, or
scope, is a standard convention in contemporary logic. If
there are no explicit parentheses to indicate the scope of a
connective those which have precedence in the ranking are presumed to be
the principal connective, and so on for subformulas. Thus, instead
formulating the following DeMorgan's law as the cumbersome:
[(~p) v (~q)] & [~(p
we nowadays write it as:
~p v ~q & ~(p & q)
This simpler formulation is natural because & takes precedence
over (has wider &scope& than) v and &, and the latter take
precedence over ~.
Indeed parentheses are often unneeded around
&, given a further convention on which & takes precedence
Thus, the formula p & q &
~p v q becomes unambiguous.
We might represent
these conventions by listing the connectives in groups with those with
widest scope at the top:
For Whitehead and Russell, however, the symbols
and &=&Df, in Group I, are of equal
force. Group II consists of the variable binding expressions,
quantifiers and scope indicators for definite descriptions, and Group
III consists of conjunctions. Negation is below all of these. So the
ranking in PM would be:
(x), (x,y)
p . q & & (conjunction)
This is what Whitehead and Russell seem to mean when they say
&Group I is of greater force than Group II, and Group II than Group
Consider the following:
⊢ : ~p
This theorem illustrates how to read multiple uses of the same number
of dots within one formula. The first two dots around the
v simply &work away& from the connective. The second
&extends& until it meets with the next of the same number (the third
single dot). That third dot, and the fourth &work away& from the
second v, and the final dot indicates a conjunction
with narrowest scope. The result, formulated with all possible
punctuation for maximum explicitness, is:
{[(~p) v (~q)] v
If we employ all the standard conventions for dropping parentheses,
this becomes:
(~p v ~ q) v (p & q)
This illustrates the passage in the above quotation which says &The
scope of the bracket indicated by any collection of dots extends
backwards or forwards beyond any smaller number of dots, or any equal
number from a group of less force, until we reach either the end of
the asserted proposition or a greater number of dots or an equal
number belonging to a group of equal or superior force.&
Before we look at a wider range of examples, a detailed example
involving quantified variables will prove to be instructive. Whitehead
and Russell follow Peano's practice of expressing universally
quantified conditionals (such as &All &s are &s&) with the
bound variable subscripted under the conditional sign. Similarly with
universally quantified biconditionals (&All and only &s are
That is, the expressions
&&x&&x&&x&
&&x&&x&&x&
are defined as follows:
. = . (x). &x &
&x & & & Df
. = . (x). &x &
&x & & & Df
and correspond to the following more modern formulas, respectively:
&x(&x & &x)
&x(&x & &x)
As an exercise the reader might be inclined to formulate a rigorous
algorithm for converting PM into a particular contemporary symbolism
(with conventions for dropping parentheses), but the best way to learn
the system is to look over a few more examples of translations, and
then simply begin to read formulae directly.
3.3 More Examples
In the examples below, each formula number is followed
first by Principia notation and then its
modern translation. Notice that in *1&5
parentheses are used for punctuation in addition to dots.
(Primitive Propositions *1&2,
*1&6 together constitute the axioms
for propositional logic in PM. )
Proposition *1&5 was shown to be
redundant by Paul Bernays in 1926. It can be derived from appropriate
instances of the others and the rule of modus ponens.
⊢ :& q .
⊢ :& p v q . & . q v p
⊢ :& p
r ) &. & .
q v (p v r ) &&& Pp
q v (p v r )
⊢ :&. q
& r . & : p v q . & .
p v r &&& Pp
(q & r ) & (p v q
⊢ : p
& ~q . & . q & ~p
(p & ~q) & (q &
⊢ :&. p . q
. & . r : & : p . & . q
[(p & q) & r] &
[p & (q & r)]
⊢ :&. p
. q . & . ~r : & : q .
r . & . ~p
p & q & ~r &
q & r & ~p
⊢ :&. q
& ~r . & : p v q . r
. & . p . r
(q & ~r) &
[(p v q) & r & p
p . v . (x). &x : =
. (x). &x v p
=df & &x(&x v
⊢ :&:
. & . q : & :&.
(&x). &x . v .
r : & . q v r
[(&x&x) & q] &
[((&x&x) v r) &
⊢ :&.
: &x &x &x
: &x &x &x
&x(&x & &x) &
&x(&x & &x)
&x(&x & &x)
There are two kinds of functions in PM. Propositional
functions such as x& is a
natural number are to be distinguished from the more familiar
mathematical functions, which are called &descriptive functions& (PM,
Descriptive functions are defined using relations and definite
descriptions.
Examples of descriptive functions are
x + y and the successor of
Focusing on propositional functions, Whitehead and Russell
distinguish between expressions with a free variable (such as
&x is hurt&) and names of
functions (such as &x&
is hurt&) (PM, 14&15).
The propositions which
result from the formula by assigning allowable values to the free
variable &x& are said to be the &ambiguous
values& of the function.
Expressions using the circumflex
notation, such as
only occur in the introductory material in the technical sections of
PM and not in the technical sections themselves (with the exception of
the sections on the theory of classes), prompting some scholars to say
that such expressions do not really occur in the formal system of PM.
This issue is distinct from that surrounding the interpretation of
such symbols. Are they &term-forming operators& which turn an open
formula into a name for a function, or simply a syntactic device, a
placeholder, for indicating the variable for which a substitution can
made in an open formula? If they are to be treated as term-forming
operators, the modern notation for
would be &&x&&x&.
&-notation has the advantage of clearly revealing that the
variable x is bound by the term-forming operator
&, which takes a predicate & and yields a term
&x&&x (which in some logics is a
singular term that can occur in the subject position of a sentence,
while in other logics is a complex predicative expression). Unlike
&-notation, the PM notation using the circumflex cannot
indicate scope. The function expression
&&(x&,z&)&
is ambiguous between
&&x&y&&xy&
&&y&x&&xy&,
without some further convention.
Indeed, Whitehead and Russell
specified this convention for relations in extension (on p. 200 in the introductory
material of *21, in terms of the order of
the variables), but the ambiguity it brought out most clearly by using
& notation: the first denotes the relation of being an
x and y such that &xy and the second
denotes the converse relation of being a y and x
such that &xy.
This section explains notation that is not in Principia
Mathematica. Except for some notation for &relative& types in
Volume II, there are famously no symbols for types in Principia
Mathematica! Sentences are generally to be taken as &typically
ambiguous& and so standing for expressions of a whole range of types
and so just as there are no individual or predicate constants, there
are no particular functions of any specific type. So not only does one
not see how to symbolize the argument:
All men are mortal
Socrates is a man
Therefore, Socrates is mortal
but also there is no indication of the logical type of the function
x& is mortal.
project of PM is to reduce mathematics to logic, and part of the view
of logic behind this project is that logical truths are all completely
general. The derivation of truths of mathematics from definitions and
truths of logic will thus not involve any particular constants other
than those introduced by definition from purely logical notion. As a
result no notation is included in PM for describing those types. Those
of us who wish to consider PM as a logic which can be applied, must
supplement it with some indication of types.
Readers should note that the explanation of types outlined below is
not going to correspond with the statements about types in the text of
Alonzo Church [1976] developed a simple, rational reconstruction
of the notation for both the simple and ramified theory of types as
implied by the text of PM. (There are alternative, equivalent
notations for the theory of types.) The full theory can be seen as a
development of the simple theory of types.
A definition of the simple types can be given as follows:
& (Greek iota) is the type for an individual.
Where &1,&,&n are
any types, then
&(&1,&,&n)&
is the type of a propositional function whose arguments are of types
&1,&,&n, respectively.
is the type of propositions.
Here are some intuitive ways to understand the definition of type.
Suppose that 'Socrates' names an individual. (We are here ignoring
Russell's considered opinion that such ordinary individuals are in
fact classes of classes of sense data, and so of a much higher type.)
Then the individual constant 'Socrates' would be of type &.
monadic propositional function which takes individuals as arguments is
of type (&).
Suppose that &is mortal& is a predicate
expressing such a function. The function
x& is mortal will also
be of type (&).
A two-place or binary relation between
individuals is of type (&,&). Thus, a relation expression
like &parent of& and the function
x& is a parent of
z& will be of type (&,&).
Propositional functions of type (&) are often called &first
order&; hence the name &first order logic& for the familiar logic
where the variables only range over arguments of first order
functions.
A monadic function of arguments of type & are of type
(&) and so functions of such functions are of type
((&)). &Second order logic& will have variables for the arguments
of such functions (as well as variables for individuals). Binary
relations between functions of type & are of type (&,&),
and so on, for relations of having more than 2 arguments. Mixed types
are defined by the above. A relation between an individual and a
proposition (such as
x& believes
that P&) will be of type
To construct a notation for the full ramified theory of types of PM,
another piece of information must be encoded in the symbols. Church
calls the resulting system one of r-types.
The key idea of
ramified types is that any function defined using quantification over
functions of some given type has to be of a higher &order&
than those functions. To use Russell's example:
x& has all
the qualities that great generals have
is a function true of persons (i.e., individuals), and from the point
of view of simple type theory, it has the same simple logical
type as particular qualities of individuals (such as bravery and
decisiveness).
However, in ramified type theory, the above function
will be of a higher order than those particular qualities
of individuals, since unlike those particular qualities, it involves
a quantification over those qualities.
So, whereas the expression
brave& denotes a function of r-type (&)/1, the
expression &x& has all
the qualities that great generals have& will have
r-type (&)/2.
In these r-types, the number after the
&/& indicates the level of the function.
order of the functions will be defined and computed given the
following definitions.
Church defines the r-types as follows:
& (Greek iota) is the r-type for an individual.
Where &1,&,&m are
any r-types,
&(&1,&,&m)/n&
is an r- this is the r-type of a m-ary propositional
function of level n, which has arguments of
r-types &1,&,&m.
The order of an entity is defined as follows (here we
no longer follow Church, for he defines orders for variables, i.e.,
expressions, instead of orders for the things the variables range
the order of an individual (of r-type &) is 0,
the order of a function of r-type
(&1,&,&m)/n is
n+N, where N is the greatest of the order
of the arguments
These two definitions are supplemented with a principle which
identifies the levels of particular defined functions, namely, that
the level of a defined function should be one higher than the highest
order entity having a name or variable that appears in the definition
of that function.
To see how these definitions and principles can be used to compute
the order of the function
has all the qualities that great generals have, note
that the function can be represented as follows, where
are variables ranging over individuals of r-type & (order 0),
&GreatGeneral(y)& is a predicate
denoting a propositional function of r-type (&)/1 (and so of
order 1), and &&& is a variable
ranging over propositional functions of r-type (&)/1 (and so of
order 1) such as great general, bravery,
leadership, skill, foresight, etc.:
(&){[(y)(GreatGeneral(y) & &(y)]
We first note that given the above principle, the r-type of this
function is (&)/2; the level is 2 because the level of the r-type
of this function has to be one higher than the highest order of any
entity named (or in the range of a variable used) in the definition.
In this case, the denotation of GreatGeneral, and the
range of the variable &&&, is of
order 1, and no other expression names or ranges over an entity of
higher order.
Thus, the level of the function named above is defined
to be 2. Finally, we compute the order of the function denoted above
as it was defined: the sum of the level plus the greatest of the
orders of the arguments of the above function.
Since the only
arguments in the above function are individuals (of order 0), the
order of our function is just 2.
Quantifying over functions of r-type
(&)/n of order k in a definition of a new
function yields a function of r-type
(&)/n+1, and so a function of order one higher,
k+1. Two kinds of functions, then, can be of the second
order: (1) functions of first-order functions of individuals, of
r-type ((&)/1)/1, and
(2) functions of r-type (&)/2, such as our example
x& has all the qualities that
great generals have. This latter will be a function true of
individuals such as Napoleon, but of a higher order than simple
functions such as x& is
brave, which are of r-type (&)/1.
Logicians today use a different notion of &order&.
Today, first-order logic is a logic with only variables for
individuals.
Second order logic is a logic with variables for both
individuals and properties of individuals.
Third-order logic is a
logic with variables for individuals, properties of individuals, and
properties of properties of individuals.
And so forth.
By contrast,
Church would call these logics, respectively, the logic of functions
of the types (&)/1 and (&,&,&)/1, the logic of
functions of the types
((&)/1)/1 and ((&,&,&)/1,&,(&,&,&)/1)/1,
and the logic of functions of the types (((&)/1)/1)/1
etc. (i.e., the level-one functions of the functions of the preceding
type). Given Church's definitions, these are logics of first-, second-
and third-order functions, respectively, thus coinciding with the
modern terminology of &nth-order
As mentioned previously, there are no individual or predicate
constants in the formal system of PM, only variables. The
Introduction, however, makes use of the example
&a standing in the relation
R to b& in a
discussion of atomic facts (PM, 43). Although
&R& is later used as a variable
that ranges over relations in extension, and
are individual variables, let us temporarily add them to the system as
predicate and individual constants, respectively, in order to discuss
the use of variables in PM.
PM makes special use of the distinction between &real&,
or free, variables and &apparent&, or bound,
variables. Since &x& is a
variable, &xRy& will be an
atomic formula in our extended language, with
&y& real variables. When such
formulae are combined with the propositional connectives
~ , v, etc., the result is a
matrix. For example, &aRx . v
would be a matrix.
we saw earlier, there
are also variables which range over functions:
&&, &, & ,
The expression
thus contains two variables and stands for a proposition, in
particular, the result of applying the function
& to the individual x.
Theorems are stated with real variables, which gives them a
special significance with regard to the theory. For example,
: (x) . &x . & .
is a fundamental axiom of the quantificational theory of PM. In this
Primitive Proposition the variables
&y& are real (free), and the
&x&& is apparent
(bound). As there are no constants in the system, this is the closest
that PM comes to a rule of universal instantiation.
Whitehead and Russell interpret
&(x) . & x&& as
&the proposition
which asserts all the values for
&x&& (PM 41). The
use of the word &all& has special significance within the theory of
types. They present the &vicious circle principle&, which underlies the
theory of types, as asserting that
& generally, given any set of objects such that, if we
suppose the set to have a total, it will contain members which
presuppose this total, then such as set cannot have a total. By saying
that the set has &no total&, we mean, primarily, that no significant
statement can be made about &all its members&. (PM, 37)
Specifically, then, a quantified expression, since it talks about
&all& the members of a totality, must range over a
specific logical type in order to observe the vicious circle
principle. Thus, when interpreting a bound variable, we must assume
that it ranges over a specific type of entity, and so types must be
assigned to the other entities represented by expressions in the
formula, in observance with the theory of types.
A question arises, however, once one realizes that the statements of
primitive propositions and theorems in PM such as
*10&1 are taken to be
&typically ambiguous& (i.e., ambiguous with respect to
These statements are actually schematic and represent all the
possible specific assertions which can be derived from them by
interpreting types appropriately.
But if statements like
*10&1 are schemata and
yet have bound variables, how do we assign types to the entities over
which the bound variables range?
The answer is to first decide which
type of thing the free variables in the statement range over.
example, assuming that the variable y in
*10&1 ranges over
individuals (of type &), then the variable &
must range over functions of type (&)/n, for some
Then the bound variable x will
also range over individuals.
If, however, we assume that the variable
ranges over functions of type (&)/1, then the variable
& must range over functions of type
((&)/1)/m, for some m.
In this case, the bound
variable x will range over functions of type
So y and & are
called &real& variables
because they are free but also because they can range over any type.
Whitehead and Russell frequently say that real variables are taken
to ambiguously denote &any&
instances, while bound variables (which also ambiguously
denote) range over &all& of their instances (within a legitimate
totality, i.e. type).
The exclamation mark &!& following
a variable for a function and preceding
the argument, as in
&f!x&&, &&!x&,
indicates that the function is predicative, that is, of the
lowest order which can apply to its arguments. In Church's notation,
this means that predicative functions are all of the first level, with
types of the form (&)/1.
As a result, predicative functions
will be of order one more than the highest order of any of their
arguments. This analysis is based on quotations like the following, in
the Introduction to PM:
We will define a function of one variable as predicative when it is of the
next order above that of its argument, i.e., of the lowest order
compatible with its having that argument. [PM, 53]
Unfortunately in the summary of *12,
we find &A predicative function is one which contains no apparent
variables, i.e., is a matrix& [PM, 167]. Reconciling this
statement with that definition in the Introduction
is a problem for scholars.
To see the shriek notation in action, consider the following definition of
x = y . = : (&) : &!x
. & . &!y & & &
That is, x is identical with y if and only if
y has every predicative function & which is possessed by
(Of course the second occurrence of
&=& indicates a definition, and does not
independently have meaning. It is the first occurrence, relating
individuals x and y, which is defined.)
To see how this definition reduces to the more familiar definition
of identity (on which objects are identical iff they share the same
properties), we need the Axiom of Reducibility.
The Axiom of
Reducibility states that for any function there is an equivalent
function (i.e., one true of all the same arguments) which is
predicative:
Axiom of Reducibility:
: &x . &x .
f!x & & Pp
To see how this axiom implies the more familiar definition of identity,
note that the more familiar definition of identity is:
x = y . = : (&) : &x
. & . &y & & &
for & of &any& type. (Note that this
differs from *13&01 in
that the shriek no longer appears.)
Now to prove this, assume both
*13&01 and the Axiom of
Reducibility, and suppose, for proof by reductio, that
x = y, and
&x, and not
&y, for some function
& of arbitrary type. Then, the Axiom of
Reducibility *12&1
guarantees that there will be a predicative function
&!, which is coextensive with
& such that &!x but
not &!y, which contradicts
The inverted Greek letter iota
is used in PM, always followed by
a variable, to begin a definite description.
(x)&x is read as &the x such that
x is &&, or
more simply, as &the &&. Such
expressions may occur in subject position, as in
&(x)&x, read as
&the & is &&.
The formal part of Russell's famous &theory of definite
descriptions& consists of a definition of all formulas
&&&(x)&x&&
in which a description occurs. To distinguish the portion & from the rest of a larger sentence (indicated by the ellipses above)
in which the expression &(x)&x
the scope of the description is indicated by repeating the definite description within brackets:
[(x)&x] . &(x)&x
The notion of scope is meant to explain a distinction which Russell
famously discusses in &On Denoting& (1905).
Russell says that the
sentence &The present King of France is not bald& is
ambiguous between two readings: (1) the reading where it says of the
present King of France that he is not bald, and (2) the reading on
which denies that the present King of France is bald.
The former
reading requires that there be a unique King of France on the list of
things that are not bald, whereas the latter simply says that there is
not a unique King of France that appears on the list of bald things.
Russell says the latter, but not the former, can be true in a
circumstance in which there is no King of France.
Russell analyzes
this difference as a matter of the scope of the definite description,
though as we shall see, some modern logicians tend to think of this
situation as a matter of the scope of the negation sign.
Russell introduces a method for indicating the scope of the definite
description.
To see how Russell's method of scope works for this case, we must understand
the definition which introduces definite descriptions (i.e., the inverted iota operator).
Whitehead and Russell define:
. = : (&b) :
x=b : &b & &
This kind of definition is called a contextual
definition, which are to be contrasted with explicit definitions.
An explicit definition of the definition description would have to look
something like the following:
which would allow the definite description to be
replaced in any context by whichever defining expression fills in
the ellipsis. By contrast, *14&01
shows how a sentence, in which there is occurrence of a description
in a context &, can be replaced by some other
sentence (involving & and &)
which is equivalent.
To develop an instance of this definition, start
with the following example:
The present King of France is bald.
Using PKFx to represent the propositional
function of being a present King of France and
B to represent the propositional function of
being bald, Whitehead and Russell would represent the above claim
[(x)(PKFx)].
B(x)(PKFx)
PKFx . &x .
In words, there is one and only one b which is a present
King of France and which is bald.
In modern symbols, using b
non-standardly, as a variable, this becomes:
(&b)[&x(PKFx &
x=b) & Bb]
Now we return to the example which shows how the scope of the description
makes a difference:
The present King of France is not bald.
There are two options for representing this sentence.
[(x)(Kx)] .
~[(x)(Kx)] .
In the first, the description has &wide& scope, and in the second,
the description has &narrow& scope.
Russell says that the description
has &primary occurrence& in the former, and &secondary occurrence& in
the latter.
Given the definition
*14&01, the two PM
formulas immediately above become expanded into primitive notation
In modern notation these become:
&x[&y(PKFy
& y=x) & ~Bx]
~&x[&y(PKFy
& y=x) & Bx]
The former says that there is one and only one object which is a
present King of France a i.e., there is exactly
one present King of France and he is not bald.
This reading is false,
given that there is no present King of France. The latter says it is
not the case that there is exactly one present King of France which is
This reading is true.
Although Whitehead and Russell take the descriptions in these
examples to be the expressions which have scope, the above readings in
both expanded PM notation and in modern notation suggest why some
modern logicians take the difference in readings here to be a matter
of the scope of the negation sign.
The circumflex &&& over a variable preceding a
formula is used to indicate a class, thus
is the class of things x which are such
In modern notation we
represent this class as {x | &x}, which is read:
the class of x which are such that x has
&. Recall that
with the circumflex over a variable after the predicate variable,
expresses the propositional function of being an
x such that
&x. In the type theory of PM, the class
the same logical type as the function
This makes it appropriate to use the following contextual
definition, which allows one to eliminate the class term
from occurrences in the context f&:
&!x . &x .
&x : f{&!z&}
or in modern notation:
f{z | &z} &
&&[&x(&x & &x)
f(&x &x)],
where & is a predicative function of x
Note that f has to be interpreted as a higher-order function which
is predicated of the function &!z&.
In the modern notation used above, the language has to be a typed
language in which & expressions are allowed in argument
As was pointed out later (Chwistek 1924, G&del 1944,
and Carnap 1947) there should be scope indicators for class
expressions just as there are for definite descriptions. Chwistek, for
example, proposed copying the notation for definite descriptions, thus
replacing *20&01 with:
[z&(&z)] .
. = : (&&) :
&!x . &x .
&x : f{&!z&}
Contemporary formalizations of set theory make use of something like
these contextual definitions, when they require an
&existence& theorem of the form
&x&y(y & x
& &y&), in order to justify the
introduction of a singular term {y |
(Given the law of extensionality, it
follows from &x&y(y &
x & &y&) that there is a unique
such set.)
The relation of membership in classes & is defined in
PM by first defining a similar relationship between objects and
propositional functions:
*20&02. & &
x & (&!z&)
. = . &!x & & Df
or, in modern notation:
x & &z&z &
*20&01 and
*20&02 together are then
used to define the more familiar notion of membership in a class.
formal expression &y &
can now been seen as a context in which t it is
then eliminated by the contextual definition
*20&01. (Exercise)
PM also has Greek letters for classes: &, &, &,
etc. These will appear as bound (real) variables, apparent (free)
variables and in abstracts for propositional functions true of
classes, as in
Only definitions of the bound Greek variables appear in the body of
the text, the others are informally defined in the
Introduction:
. = . (&) .
f{z&(&!z)}
or, in modern notation,
&&f{z|&z}, where
& is a predicative function
Thus universally quantified class variables are defined in terms of
quantifiers ranging over predicative functions. Likewise for
existential quantification:
*20&071. &
. = . (&&) .
f{z&(&!z)}
or, in modern notation,
&&f{z|&z}, where
& is a predicative function
Expressions with a Greek variable to the left of
& are defined:
*20&081. & &
. = . &!& & & Df
These definitions do not cover all possible occurrences of Greek
variables. In the Introduction to PM, further definitions of are
proposed, but it is remarked that the definitions are in some way
peculiar and they do not appear in the body of the work. The
definition considered for
or, in modern notation,
&&f& =df &
&& f{x | &x}
is an expression naming the function which takes a function &
to a proposition which asserts f of the class of
(The modern notation shows that in the proposed
definition of
in PM notation, we shouldn't expect & in the
definiens, since it is really a bound variable in
f&&; similarly,
we shouldn't expect & in the definiendum because it is
a bound variable in the definiens.)
One might also expect definitions like
*20&07 and
to hold for cases in which the Roman letter
&z& is replaced
by a Greek letter. The definitions in PM are thus not complete, but it is possible
to guess at how they would be extended to cover all occurrences of
Greek letters.
This would complete the project of the &no-classes&
theory of classes by showing how all talk of classes can be reduced to
the theory of propositional functions.
Although students of philosophy usually read no further than
*20 in PM, this is in fact the point where
the &construction& of mathematics really
begins. *21 presents the &General Theory
of Relations& (the theory of re in contemporary
logic these are treated as sets of ordered pairs, following Wiener).
x& y& & (x , y)
is the relation between x and
which obtains when
In modern notation we
represent this as as the set of ordered pairs { & x , y &
| &( x , y ) }, which is read:
the set of ordered pairs
which are such that x bears the relation
The following contextual
definition allows one to eliminate the relation term
y& & (x , y)
from occurrences in the context f&:
&! ( x , y ) . &x,y .
& ( x , x ) : f {&! (
or in modern notation:
f {& x , y
& | & ( x , y )} &
&&[&xy (&(x , y) & & ( x , y) )
where & is a predicative function of u and v.
Principia does not analyze relations (or mathematical
functions) in terms of sets of ordered pairs, but rather takes the
notion of propositional function as primitive and defines relations
and functions in terms of them. The upper case
letters R, S and T,
etc., are used after *21 to stand for these
&relations in extension&, and are distinguished from
propositional functions by being written between the arguments. Thus
with arguments
after the propositional function symbol, but
xRy . From *21
functions & and &, etc., disappear and only
relations in extension, R, S
and T, etc., appear in the pages of
Principia . While propositional functions might be true of the
same objects yet not be identical, no two relations in extension are
true of the same objects. The logic of Principia is thus
&extensional&, from page 200 in volume I, through to the end
in Volume III.
*22 on the &Calculus of Classes&
presents the elementary set theory of intersections, unions and the
empty set which is often all the set theory used in elementary
mathematics of other sorts.
The student looking for the set theory
of Principia to compare it with, say the Zermelo-Fraenkel
system, will have to look at various numbers later in the text. The
Axiom of Choice is defined at *88 as the
&Multiplicative Axiom& and a version of the Axiom of
Infinity appears at *120 in Volume II as
&Infin ax&. The set theory of Principia comes
closest to Zermelo's axioms of 1908 among the various familiar axiom
systems, which means that it lacks the Axiom of Foundation and Axiom
of Replacement of the now standard Zermelo-Fraenkel axioms of set
*30 on &Descriptive Functions&
provides Whitehead and Russell's analysis of mathematical functions in
terms of relations and definite descriptions.
Frege had used the
notion of function, in the mathematical sense, as a basic notion in
his logical system. Thus a Fregean &concept& is a function from
objects as arguments to one of the two &truth values& as its
values. A concept yields the value &True& for each object to
which the concept applies, and &False& for all
others. Russell, from 1904, well before the writing
of Principia had preferred to analyze functions in terms of
the relation between each argument and value, and the notion of
&uniqueness&. With modern symbolism, his view would be
expressed as follows.
For each function
&x f(x), there will be some relation
(in extension) R, such that the value of the function for an
argument a, that is f(a), will be the
unique individual which bears the relation R
to a. (Nowadays we reduce functions to a binary relation
between the argument in the first place and value in the second
The result is that there are no function symbols
in Principia. As Whitehead and Russell say, the familiar
mathematical expressions such as &sin
will be analyzed with a relation and a definite description, as a
&descriptive function&. The &descriptive
function&, R&y
(the R of y), is defined as follows:
= (x)xRy & & & Df
With the exception of an incomplete notation for relative
types in Volume II, the reader should be able to work out all of the
rest of the notation in PM using the explanations above and the
definitions in the lists at the end of volume I of PM. We conclude
by presenting a number of prominent examples from these later numbers
below, with their intuitive meaning, location in PM, definition in PM,
and a modern equivalent. (Some of these numbers are theorems rather
than definitions.)
Note, however, that the modern equivalent will
sometimes logically differ from the original version in PM, such as by
treating relations as sets of ordered pairs, etc.
For each formula number, we present the information in the
following format:
PM Symbol (Intuitive Meaning) [Location]
PM Definition
Modern Equivalent
(& is a subset of &)
x&& . &x .
(the intersection of & and &)
(the union of & and &)
(the complement of &)
[i.e., x&~(x&&)
by *20&06]
{x | x & & }
(& minus &)
{x | x&& & x&&}
(the universal class)
{x | x = x}
(the empty class)
(the R of y) (a descriptive function)
f&1(y), where f =
{&x,y& | Rxy}
(the converse of R)
{&x,z& | Rzx}
(the R-predecessors of y)
(the R-successors of x)
(the domain of R)
{x | &yRxy}
(the range of R)
{z | &xRxz}
(the field of R)
xRy . v . yRx}
{x | &y(xRy v yRx)}
(the relative product of R and S)
x&z&{(&y). xRy
{&x,z& | &y(xRy
(the restriction of R to &)
{&x,z& | z&& &
(the Cartesian product of & and &)
&X&, or {&x,z& |
x&& & z&&}
(the projection of & by R)
{x | &y(y&& & Rxy)}
(singleton of x)
(the ancestral of R)
Frege's definition: y is in all the R-hereditary
classes x is in.
Carnap, R., 1947, Meaning and Necessity, Chicago:
University of Chicago Press.
Church, A. , 1976, &Comparison of Russell's Resolution of the
Semantical Antinomies with That of Tarski&, Journal of
Symbolic Logic, 41: 747&60.
Chwistek, L., 1924, &The Theory of Constructive Types&,
Annales de la Soci&t& Polonaise de
Math&matique (Rocznik Polskiego Towarzystwa
Matematycznego), II: 9&48.
Feys, R. and Fitch, F.B., 1969, Dictionary of Symbols of
Mathematical Logic, Amsterdam: North Holland.
G&del, K., 1944, &Russell's Mathematical Logic&, in
P.A. Schilpp, ed., The Philosophy of Bertrand Russell,
LaSalle: Open Court, 125&153.
Landini, G., 1998, Russell's Hidden Substitutional
Theory, New York and Oxford: Oxford University Press.
Linsky, B., 1999, Russell's Metaphysical Logic,
Stanford: CSLI Publications.
Linsky, B., 2009, &From Descriptive Functions
to Sets of Ordered Pairs&, in
Reduction & Abstraction & Analysis, A. Hieke and
H. Leitgeb (eds.), Ontos: Munich, 259&272.
Linsky, B., 2011, The Evolution of Principia Mathematica:
Bertrand Russell's Manuscripts and Notes for the Second Edition,
Cambridge: Cambridge University Press.
Russell, B., 1905, &On Denoting&, Mind
(N.S.), 14: 530&538.
Whitehead, A.N., and Russell, B. [PM], Principia
Mathematica, Cambridge: Cambridge University Press, 1910&13,
second edition, 1925&27.
at , with links to its database.
reproduced in the University of Michigan Historical Math Collection.
from the reprint in Logic and Knowledge (R. Marsh, ed., 1956) of
the original article in Mind 1905, typed into HTML by
Cosma Shalizi (Center for the Study of Complex Systems, U. Michigan)
Acknowledgments
The author would like to thank: Gregory Landini, Dick Schmitt, Franz Fritsche, Rafal Urbaniak, Adam Trybus, Pawel Manczyk and Kenneth Blackwell for corrections to this entry .
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