神经活动中内在思想的逻辑演算

概述 Overview

神经活动中内在理念的逻辑演算

A LOGICAL CALCULUS OF THE IDEAS IMMANENT IN NERVOUS ACTIVITY

沃伦·S·麦卡洛克和沃尔特·皮茨，伊利诺伊大学医学院，伊利诺伊神经精神病学研究所精神病学系，芝加哥大学，芝加哥，美国

WARREN S. MCCULLOCH AND WALTER PITTS University of Illinois, College of Medicine, Department of Psychiatry at the Illinois Neuropsychiatric Institute, University of Chicago, Chicago, U.S.A.

由于神经活动的“全或无”特性，神经事件及其相互关系可以通过命题逻辑来处理。研究发现，每个网络的行为都可以用这些术语描述，对于包含回路的网络，需要添加更复杂的逻辑手段；并且对于任何满足特定条件的逻辑表达式，都可以找到一个行为与其描述一致的网络。研究表明，在可能的神经生理学假设中，许多特定选择是等价的，即对于在一种假设下运行的每个网络，都存在另一个在另一种假设下运行并给出相同结果（尽管可能不在同一时间）的网络。讨论了该演算的各种应用。

Because of the “all-or-none” character of nervous activity, neural events and the relations among them can be treated by means of propositional logic. It is found that the behavior of every net can be described in these terms, with the addition of more complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions, one can find a net behaving in the fashion it describes. It is shown that many particular choices among possible neurophysiological assumptions are equivalent, in the sense that for every net behaving under one assumption, there exists another net which behaves under the other and gives the same results, although perhaps not in the same time. Various applications of the calculus are discussed.

1. 引言。理论神经生理学基于某些基本假设。神经系统是一个由神经元组成的网络，每个神经元有一个胞体和轴突。它们的连接点，即突触，总是在一个神经元的轴突和另一个神经元的胞体之间。在任何时刻，神经元都有一个阈值，兴奋必须超过该阈值才能引发冲动。除了事实和发生的时间外，这由神经元而非兴奋决定。从兴奋点开始，冲动传播到神经元的各个部分。沿轴突的速度与其直径成正比，在通常较细的短轴突中速度低于 1 米/秒，在通常较粗的长轴突中速度超过 150 米/秒。因此，轴突传导时间在确定来自同一源点但距离不同的冲动到达时间时并不重要。跨突触的兴奋主要从轴突末梢到胞体。这是否取决于单个突触的不可逆性还是仅仅取决于普遍的解剖结构，仍是一个有争议的问题。假设后者不需要特别的假设，

1. Introduction. Theoretical neurophysiology rests on certain cardinal assumptions. The nervous system is a net of neurons, each having a soma and an axon. Their adjunctions, or synapses, are always between the axon of one neuron and the soma of another. At any instant a neuron has some threshold, which excitation must exceed to initiate an impulse. This, except for the fact and the time of its occurrence, is determined by the neuron, not by the excitation. From the point of excitation the impulse is propagated to all parts of the neuron. The velocity along the axon varies directly with its diameter, from < 1 ms^-1 in thin axons, which are usually short, to > 150 ms^-1 in thick axons, which are usually long. The time for axonal conduction is consequently of little importance in determining the time of arrival of impulses at points unequally remote from the same source. Excitation across synapses occurs predominantly from axonal terminations to somata. It is still a moot point whether this depends upon irreciprocity of individual synapses or merely upon prevalent anatomical configurations. To suppose the latter requires no hypothesis ad hoc

并且可以解释已知的例外，但任何关于原因的假设都与即将介绍的演算兼容。目前没有已知情况表明，通过单个突触的兴奋能在任何神经元中引发神经冲动，而任何神经元都可以被足够数量的邻近突触在潜伏叠加期（约 0.25 毫秒）内到达的冲动所兴奋。观察到在更大间隔内的冲动时间总和

and explains known exceptions, but any assumption as to cause is compatible with the calculus to come. No case is known in which excitation through a single synapse has elicited a nervous impulse in any neuron, whereas any neuron may be excited by impulses arriving at a sufficient number of neighboring synapses within the period of latent addition, which lasts ~0.25 ms. Observed temporal summation of impulses at greater intervals

《数学生物物理学》第 5 卷，第 115-133 页（1943 年）。W. S. 麦卡洛克和 W. 皮茨

of Mathematical Biophysics, Vol. 5, pp. 115-133 (1943). W. S. McCULLOCH AND W. PITTS

单个神经元不可能实现，经验上依赖于网络的结构特性。在 impulses 到达神经元与它自身产生传导 impulse 之间，存在一个 > 0.5 ms 的突触延迟。在神经 impulse 的第一部分，神经元对所有刺激绝对不应。此后其兴奋性迅速恢复，在某些情况下达到高于正常的水平，然后又降至低于正常，再缓慢回到正常。频繁的活动会加剧这种低常性。神经 impulse 所具有的这种特异性仅取决于它们的时间和位置，而不取决于神经能量的任何其他特异性。近来只有抑制被严肃地引用来反驳这一论点。抑制是由第二组神经元的同时或先前活动终止或阻止第一组神经元的活动。直到最近，这可以通过假设第二组神经元的先前活动可能提高中间神经元的阈值，以至于它们不再能被第一组神经元兴奋来解释，而第一组的 impulses 必须与这些中间神经元的 impulses 总和才能兴奋现在被抑制的神经元。如今，一些抑制已被证明消耗 < 1 ms。这排除了中间神经元，并需要这样的突触：通过它们，impulses 抑制正在被其他突触的 impulses 刺激的那个神经元。目前实验尚未表明不应性是相对还是绝对。我们将假设后者，并证明这种差异对我们的论证无关紧要。任何一种不应性都可以用两种方式之一来解释。“抑制性突触”可能是一种产生提高神经元阈值的物质的突触，或者其位置使得其兴奋产生的局部扰动对抗了其他兴奋性突触诱发的改变。既然位置在电刺激的情况下已知有这样的效应，第一个假设应被排除，除非并直到被证实，因为第二个假设不涉及新假设。于是我们有了基于相同一般前提的两种抑制解释，仅在假设的神经网以及由此抑制所需的时间上有所不同。此后，我们将此类神经网称为广义等价。由于我们关心在等价下不变的网络性质，我们可以为演算选择最方便的物理假设。多年前，我们中的一人基于与此论证无关的考虑，认为任何神经元的反应实际上等价于一个提出其充分刺激的命题。因此他尝试用命题符号逻辑的记法记录复杂网络的行为。神经活动的“全或无”定律足以保证任何神经元的活动可被表示为一个命题。神经活动之间存在的生理关系对应于逻辑演算中的命题关系。

is impossible for single neurons and empirically depends upon structural properties of the net. Between the arrival of impulses upon a neuron and its own propagated impulse there is a synaptic delay of > 0.5 ms. During the first part of the nervous impulse the neuron is absolutely refractory to any stimulation. Thereafter its excitability returns rapidly, in some cases reaching a value above normal from which it sinks again to a subnormal value, whence it returns slowly to normal. Frequent activity augments this subnormality. Such specificity as is possessed by nervous impulses depends solely upon their time and place and not on any other specificity of nervous energies. Of late only inhibition has been seriously adduced to contravene this thesis. Inhibition is the termination or prevention of the activity of one group of neurons by concurrent or antecedent activity of a second group. Until recently this could be explained on the supposition that previous activity of neurons of the second group might so raise the thresholds of internuncial neurons that they could no longer be excited by neurons of the first group, whereas the impulses of the first group must sum with the impulses of these internuncials to excite the now inhibited neurons. Today, some inhibitions have been shown to consume < 1 ms. This excludes internuncials and requires synapses through which impulses inhibit that neuron which is being stimulated by impulses through other synapses. As yet experiment has not shown whether the refractoriness is relative or absolute. We will assume the latter and demonstrate that the difference is immaterial to our argument. Either variety of refractoriness can be accounted for in either of two ways. The “inhibitory synapse” may be of such a kind as to produce a substance which raises the threshold of the neuron, or it may be so placed that the local disturbance produced by its excitation opposes the alteration induced by the otherwise excitatory synapses. Inasmuch as position is already known to have such effects in the cases of electrical stimulation, the first hypothesis is to be excluded unless and until it be substantiated, for the second involves no new hypothesis. We have, then, two explanations of inhibition based on the same general premises, differing only in the assumed nervous nets and, consequently, in the time required for inhibition. Hereafter we shall refer to such nervous nets as equivalent in the extended sense. Since we are concerned with properties of nets which are invariant under equivalence, we may make the physical assumptions which are most convenient for the calculus. Many years ago one of us, by considerations impertinent to this argument, was led to conceive of the response of any neuron as factually equivalent to a proposition which proposed its adequate stimulus. He therefore attempted to record the behavior of complicated nets in the notation of the symbolic logic of propositions. The “all-or-none” law of nervous activity is sufficient to insure that the activity of any neuron may be represented as a proposition. Physiological relations existing among nervous activities correspond, of LOGICAL CALCULUS FOR NERVOUS ACTIVITY 101

当然，对应于命题之间的关系；这种表示的有效性取决于这些关系与命题逻辑关系的一致性。对任何神经元的每个反应，都有一个简单命题的断言与之对应。这又蕴涵着要么另一个简单命题，要么某些相似命题的合取或析取（带或不带否定），具体取决于所考虑神经元的突触配置和阈值。出现了两个困难。第一个涉及易化与消退，即先前活动暂时改变网络同一部位对后续刺激的反应性。第二个涉及学习，即先前同时发生的活动永久性地改变了网络，使得先前不充分的刺激现在变得充分。但对于经历这两种改变的网络，我们可以用由连接和阈值未改变的神经元组成的等价虚构网络来替代。但有一点必须澄清：我们两人都不认为形式等价是事实上的解释。恰恰相反！我们认为易化与消退依赖于与电化学变量（如后电位和离子浓度）相关的阈值连续变化；而学习是一种持久的改变，能经受睡眠、麻醉、抽搐和昏迷。形式等价的重要性在于：易化、消退和学习背后实际的变化丝毫不影响从神经活动形式处理中得出的结论，且相应命题的关系仍保持为命题逻辑的关系。神经系统包含许多环形路径，其活动如此再生任何参与神经元的兴奋，以至于对过去时间的参考变得不确定，尽管它仍意味着传入活动在时间上实现了某一类配置。通过递归函数精确指定这些蕴涵关系，并确定哪些可以在神经活动中实现，便完成了理论。

course, to relations among the propositions; and the utility of the representation depends upon the identity of these relations with those of the logic of propositions. To each reaction of any neuron there is a corresponding assertion of a simple proposition. This, in turn, implies either some other simple proposition or the disjunction of the conjunction, with or without negation, of similar propositions, according to the configuration of the synapses upon and the threshold of the neuron in question. Two difficulties appeared. The first concerns facilitation and extinction, in which antecedent activity temporarily alters responsiveness to subsequent stimulation of one and the same part of the net. The second concerns learning, in which activities concurrent at some previous time have altered the net permanently, so that a stimulus which would previously have been inadequate is now adequate. But for nets undergoing both alterations, we can substitute equivalent fictitious nets composed of neurons whose connections and thresholds are unaltered. But one point must be made clear: neither of us conceives the formal equivalence to be a factual explanation. Per contra!-we regard facilitation and extinction as dependent upon continuous changes in threshold related to electrical and chemical variables, such as after-potentials and ionic concentrations; and learning as an enduring change which can survive sleep, anaesthesia, convulsions and coma. The importance of the formal equivalence lies in this: that the alterations actually underlying facilitation, extinction and learning in no way affect the conclusions which follow from the formal treatment of the activity of nervous nets, and the relations of the corresponding propositions remain those of the logic of propositions. The nervous system contains many circular paths, whose activity so regenerates the excitation of any participant neuron that reference to time past becomes indefinite, although it still implies that afferent activity has realized one of a certain class of configurations over time. Precise specification of these implications by means of recursive functions, and determination of those that can be embodied in the activity of nervous nets, completes the theory.

2. 理论：无环网络。我们为演算做出以下物理假设。(1) 神经元的活动是一个“全或无”过程。(2) 必须在潜伏总和期内兴奋固定数量的突触才能在任何时刻兴奋一个神经元，且该数量与先前活动和神经元上的位置无关。(3) 神经系统内唯一显著延迟是突触延迟。(4) 任何抑制性突触的活动绝对阻止该时刻神经元的兴奋。(5) 网络结构不随时间变化。102 W. S. 麦卡洛克和 W. 皮茨

2. The Theory: Nets Without Circles. We shall make the following physical assumptions for our calculus. (1) The activity of the neuron is an “all-or-none” process. (2) A certain fixed number of synapses must be excited within the period of latent addition in order to excite a neuron at any time, and this number is independent of previous activity and position on the neuron. (3) The only significant delay within the nervous system is synaptic delay. (4) The activity of any inhibitory synapse absolutely prevents excitation of the neuron at that time. (5) The structure of the net does not change with time. 102 W. S. McCULLOCH AND W. PITTS

为了阐述理论，最合适的符号系统是 Carnap (1938) 的语言 II，并辅以 Russell 和 Whitehead (1927) 的各种记法，包括《数学原理》中的点约定。然而，由于排版需要，我们将用竖写‘E’代替倒写表示存在量词，用箭头（“-+”）代替马蹄铁表示蕴涵。我们还将使用 Carnap 的语法记法，但用粗体而非德文印刷；并引入一个函子 S，其对性质 P 的值是当 P 对某数成立时该数所具有的性质。

To present the theory, the most appropriate symbolism is that of Language II of Carnap (1938), augmented with various notations drawn from Russell and Whitehead (1927), including the Principia conventions for dots. Typographical necessity, however, will compel us to use the upright ‘E’ for the existential operator instead of the inverted, and an arrow (“-+“) for implication instead of the horseshoe. We shall also use the Carnap syntactical notations, but print them in boldface rather than German type; and we shall introduce a functor S, whose value for a property P is the property which holds of a number when P

它对其前驱成立；由“S(P)(t) . s . P(Kx) . t = x'”定义；其参数周围的括号常被省略，此时理解为右边最近的谓词表达式[Pr]。此外，我们将 S2Pr 写作 S(S(Pr))等。给定网络 N 的神经元可指定为“c1”、“c2”、……、“c_n”。据此，我们用下标数字 i 的“N”表示一个性质：神经元 c_i 在距时间原点若干突触延迟的时刻放电，因此 N_i(t)断言 c_i 在该时刻放电。

holds of its predecessor; it is defined by “S(P)(t) . s . P(Kx) . t = x'”; the brackets around its argument will often be omitted, in which case this is understood to be the nearest predicate-expression [Pr] on the right. Moreover, we shall write S2Pr for S(S(Pr)), etc. The neurons of a given net N may be assigned designations “c1”, “c2”, . . . , “c_n”. This done, we shall denote the property of a number, that a neuron c_i fires at a time which is that number of synaptic delays from the origin of time, by “N” with the numeral i as subscript, so that N_i(t) asserts that c_i fires at the time

称为 c_i 的作用。有时我们将“N”的下标数字视为属于对象语言，并处于函子参数的位置，从而可被数字变量[z]替换并量化；这使我们能用运算符缩写长而有限的析取和合取。我们将普遍地对 Pr 序列使用这种措辞；它可以通过明显的析取定义在形式上得到保证。谓词“N1”、“N2”、……构成句法类“N”。定义 N 的外周传入神经元为没有轴突突触于其上的神经元。令 N_1、……、N_n 表示这些神经元的作用，N_{p+1}、……、N_n 表示其余神经元的作用。那么 N 的一个解将是形如 S_i: N_{a_i}(z_1) .=. Pr_i(N_1, N_2, ..., N_n, z_1)的句子类，其中 Pr_i

is called the action of c_i. We shall sometimes regard the subscripted numeral of “N” as if it belonged to the object-language, and were in a place for a functoral argument, so that it might be replaced by a number-variable [z] and quantified; this enables us to abbreviate long but finite disjunctions and conjunctions by the use of an operator. We shall employ this locution quite generally for sequences of Pr; it may be secured formally by an obvious disjunctive definition. The predicates “N1”, “N2”, . . . , comprise the syntactical class “N”. Let us define the peripheral afferents of N as the neurons of N with no axons synapsing upon them. Let N_1, . . . , N_n denote the actions of such neurons and N_{p+1}, . . . , N_n those of the rest. Then a solution of N will be a class of sentences of the form S_i: N_{a_i}(z_1) .=. Pr_i(N_1, N_2, . . . , N_n, z_1), where Pr_i

除 z_1 外不含自由变量，除参数[Arg]中的 N 外不含描述符号，可能还包含一些常句子[sa]；并且每个 S_i 对 N 为真。反之，给定 Pr_i(φ_1, φ_2, ..., φ_j, z_1, s)，

contains no free variable save z_1 and no descriptive symbols save the N in the argument [Arg], and possibly some constant sentences [sa]; and such that each S_i is true of N. Conversely, given a Pr_i(φ_1, φ_2, . . . , φ_j, z_1, s),

除其 Arg 中的变量外不含自由变量，则称它在窄义上是可实现的，如果存在一个网络 N 及其中的一系列 N_i 使得

containing no free variable save those in its Arg, we shall say that it is realizable in the narrow sense if there exists a net N and a series of N_i in it such that

对它为真，其中 s_a 具有形式 N(0)。如果对于某个 n，S^n(Pr_i)(P_1, ..., P_p, z_1, s)在上述意义上是可实现的，则称它在广义上是可实现的，或简称为可实现的。φ_i 是这里的实现神经元。对于两个神经兴奋定律，如果在一个假定下任意意义上可实现的每个 S 在另一个假定下（可能通过不同网络）也是可实现的，则称它们在该意义上是等价假设。以下关于可实现性的定理均指广义。在某些情况下，可以得到关于窄义可实现性的更精确定理；但《神经活动的逻辑演算》103

is true of it, where s_a has the form N(0). We shall call it realizable in the extended sense, or simply realizable, if for some n S^n(Pr_i)(P_1, . . . , P_p, z_1, s) is realizable in the above sense. φ_i is here the realizing neuron. We shall say of two laws of nervous excitation which are such that every S which is realizable in either sense upon one supposition is also realizable, perhaps by a different net, upon the other, that they are equivalent assumptions, in that sense. The following theorems about realizability all refer to the extended sense. In some cases, sharper theorems about narrow realizability can be obtained; but LOGICAL CALCULUS FOR NERVOUS ACTIVITY 103

除了陈述上更为复杂之外，这几乎没有实际价值，因为目前我们的神经生理学知识只能将兴奋定律确定到扩展等价的程度，而更精确的定理则根据我们做出的不同可能假设而有所差异。然而，我们不太精确的定理在等价下是不变的，并且对于所有不需要精确知道脉冲通过整个网络的时间的目的而言仍然足够。我们的核心问题现在可以精确表述：第一，找到一种有效的方法来获得一组可计算的 S，构成任何给定网络的解；第二，以有效的方式刻画可实现的 S 的类别。具体来说，问题就是计算任何网络的行为，并在存在这样的网络时，找到一个具有指定行为的网络。如果一个网络包含一个回路，即存在一条链 ci，

In addition to greater complication in statement, this was of little practical value, since our present neurophysiological knowledge determines the law of excitation only to extended equivalence, and the more precise theorems differ according to which possible assumption we make. Our less precise theorems, however, are invariant under equivalence, and are still sufficient for all purposes in which the exact time for impulses to pass through the whole net is not crucial. Our central problems may now be stated exactly: first, to find an effective method of obtaining a set of computable S constituting a solution of any given net; and second, to characterize the class of realizable S in an effective fashion. Materially stated, the problems are to calculate the behavior of any net, and to find a net which will behave in a specified way, when such a net exists. A net will be called cyclic if it contains a circle, i.e. if there exists a chain ci,

链上的每个神经元与下一个形成突触，且起点和终点相同。如果一组神经元 ci, c2, ..., cp 使得从网络中移除它们后网络不再有回路，并且没有更小的神经元类具有此性质，则该集合称为循环集，其基数称为网络的

of neurons on it, each member of the chain synapsing upon the next, with the same beginning and end. If a set of its neurons ci, c2, ..., cp is such that its removal from the net leaves it without circles, and no smaller class of neurons has this property, the set is called a cyclic set, and its cardinality is the

阶。从重要意义上讲，正如我们将看到的，网络的阶是其行为复杂性的一个指标。特别地，零阶网络具有特别简单的性质；我们将首先讨论它们。让我们定义一个时间命题表达式（TPE），表示一个

order of the net. In an important sense, as we shall see, the order of a net is an index of the complexity of its behaviour. In particular, nets of zero order have especially simple properties; we shall discuss them first. Let us define a temporal propositional expression (a TPE), designating a

in addition to greater complication in statement this were of little practical value, since our present neurophysiological knowledge determines the law of excitation only to extended equivalence, and the more precise theorems differ according to which possible assumption we make. Our less precise theorems, however, are invariant under equivalence, and are still sufficient for all purposes in which the exact time for impulses to pass through the whole net is not crucial. Our central problems may now be stated exactly: first, to find an effective method of obtaining a set of computable Sconstituting a solution of any given net; and second, to characterize the class of realizable Sin an effective fashion. Materially stated, the problems are to calculate the behavior of any net, and to find a net which will behave in a specified way, when such a net exists. A net will be called cyclic if it contains a circle, i.e. if there exists a chain ci,

（1）（2） (1) (2)

(3) 一个 p(z) 是一个 TPE，其中 p1 是一个谓词变量。如果 S1 和 S2 是包含相同自由个体变量的 TPE，那么 S1S2、S1∨S2、S1·S2 和 ¬S1 也是 TPE。没有其他东西是 TPE。

(3) A p(z) is a TPE, where p1 is a predicate-variable. If S1 and S2 are TPE containing the same free individual variable, so are S1S2, S1∨S2, S1·S2, and ¬S1. Nothing else is a TPE.

每个 0 阶网络都可以用时间命题表达式来求解。

Every net of order 0 can be solved in terms of temporal propositional expressions.

设 ci 是 J1 的任意一个神经元，其阈值 \theta_i > 0\，并设 ci1, ci2, ..., cig 分别有 ni1, ni2, ..., nip 个兴奋性突触作用于它。

Let ci be any neuron of J1 with a threshold \theta_i > 0\, and let ci1, ci2, ..., cig have respectively ni1, ni2, ..., nip excitatory synapses upon it.

设 cj1, cj2, ..., cjq 有抑制性突触作用于它。令 rci 是 {ni1, ni2, ..., nip} 的子类的集合，使得这些子类中成员的和超过 \theta_i\。

Let cj1, cj2, ..., cjq have inhibitory synapses upon it. Let rci be the set of the subclasses of {ni1, ni2, ..., nip} such that the sum of their members exceeds \theta_i\.

然后我们将能够根据上述假设写出：

We shall then be able to write, in accordance with the assumptions mentioned above:

I9 (1) I9 (1)

其中‘v’和‘&’是表示析取和合取的语法符号，在每种情况下都是有限的。既然对于每个不是外周传入神经元的\(c_i\)都可以写出这种形式的表达式，那么通过代入……

where the 'v' and '&' are syntactical symbols for disjunctions and conjunctions which are finite in each case. Since an expression of this form can be written for each \(c_i\) which is not a peripheral afferent, we can, by substituting ...

将(1)中对应的表达式代入每个不是外周传入神经元的\(N_{jn}\)或\(N_i\)，并对结果重复这一过程，最终得到一个仅由外周传入神经元\(N\)表示的\(N_i\)表达式，因为网络无环。此外，这个表达式将是 TPE，因为显然(1)是 TPE；并且直接从定义可知，将……代入一个 TPE 的结果也是 TPE。

the corresponding expression in (1) for each \(N_{jn}\) or \(N_i\), whose neuron is not a peripheral afferent, and repeating the process on the result, ultimately come to an expression for \(N_i\) in terms solely of peripherally afferent \(N\), since the net is without circles. Moreover, this expression will be a TPE, since obviously (1) is; and it follows immediately from the definition that the result of substituting a

TPE 中的组成部分\(p(z)\)也是一个 TPE。

for a constituent \(p(z)\) in a TPE is also one.

TPE 可由零阶网络实现。

TPE is realizable by a net of order zero.

函子 S 显然与析取、合取和否定可交换。显然，将任意窄意义（i.n.s.）可实现的\(S_i\)代入可实现表达式 S 中的\(p(z)\)，所得结果本身在窄意义下可实现；通过将 S 网络中的外周传入神经元替换为\(S_i\)网络中的实现神经元，即可构造出实现网络。单神经元网络在窄意义下实现\(p_i(z_i)\)，图 1a 展示了一个网络，它实现\(S'(z)\)从而在窄意义下实现\(S S_i\)（如果 S 可在窄意义下实现）。现在，如果\(S_1\)和\(S_2\)是可实现的，那么对于适当的 m 和 n，\(S^m S_2\)和\(S^n S_1\)在窄意义下可实现。因此\(S^m S_2\)和\(S^{m+n} S_3\)也是如此。图 1b-d 的网络分别实现\(S(p_1(z_1) \lor p_2(z_2))\)、\(S(p_1(z_1) \cdot p_2(z_2))\)和\(~S(p_1(z_1) \cdot ~p_2(z_2))\)（窄意义下）。因此\(S^{m+n+1} (S_1 \lor S_2)\)、\(S^{m+n+1} (S_1 \cdot S_2)\)和\(S^{m+n+1} (S_1 \cdot ~S_2)\)在窄意义下可实现。故如果\(S_1\)和\(S_2\)是可实现的，则\(S_1 \lor S_2\)、\(S_1 \cdot S_2\)、\(S_1 \cdot ~S_2\)是可实现的。通过完全归纳，所有……

The functor S obviously commutes with disjunction, conjunction, and negation. It is obvious that the result of substituting any \(S_i\), realizable in the narrow sense (i.n.s.), for the \(p(z)\) in a realizable expression S, is itself realizable i.n.s.; one constructs the realizing net by replacing the peripheral afferents in the net for S, by the realizing neurons in the nets for the \(S_i\). The one neuron net realizes \(p_i(z_i)\) i.n.s., and Fig. 1a shows a net that realizes \(S'(z)\) and hence \(S S_i\) i.n.s., if S can be realized i.n.s. Now if \(S_1\) and \(S_2\) are realizable then \(S^m S_2\) and \(S^n S_1\) are realizable i.n.s., for suitable m and n. Hence so are \(S^m S_2\) and \(S^{m+n} S_3\). Now the nets of Figs. 1b-d respectively realize \(S(p_1(z_1) \lor p_2(z_2))\), \(S(p_1(z_1) \cdot p_2(z_2))\), and \(~S(p_1(z_1) \cdot ~p_2(z_2))\) i.n.s. Hence \(S^{m+n+1} (S_1 \lor S_2)\), \(S^{m+n+1} (S_1 \cdot S_2)\), and \(S^{m+n+1} (S_1 \cdot ~S_2)\) are realizable i.n.s. Therefore \(S_1 \lor S_2\), \(S_1 \cdot S_2\), \(S_1 \cdot ~S_2\) are realizable if \(S_1\) and \(S_2\) are. By complete induction, all

是可实现的。这样，所有网络都可视为由图 1a—d 的基本元素构建，正如时间命题表达式由前件、析取、合取和合取否定运算生成。特别地，对应于任何状态描述（即网络所有神经元动作的真值分布，除了使它们全为假的那个），可以构造一个单一的神经元，其发放是该描述有效性的充分必要条件。此外，总是存在无限多个拓扑不同的网络来实现任何 TPE。

are realizable. In this way all nets may be regarded as built out of the fundamental elements of Figs 1a-d, precisely as the temporal propositional expressions are generated out of the operations of precession, disjunction, conjunction, and conjoined negation. In particular, corresponding to any description of state, or distribution of the values true and false for the actions of all the neurons of a net save that which makes them all false, a single neuron is constructible whose firing is a necessary and sufficient condition for the validity of that description. Moreover, there is always an indefinite number of topologically different nets realizing any TPE.

时间命题表达式（TPE）当且仅当其组成式\p(z_1 - z_2)\为假时才为假。

TPE if and only if it is false when its constituent \p(z_1 - z_2)\

后三个条件显然是等价的（Hilbert and Ackermann, 1938）。通过归纳我们看到第一个条件是必要的，因为当\p(z_1 - z_2)\被假语句替换时它变为假，并且\S_1 \lor S_2\，

These latter three conditions are of course equivalent (Hilbert and Ackermann, 1938). We see by induction that the first of them is necessary, since \p(z_1 - z_2)\ becomes false when it is replaced by a false sentence, and \S_1 \lor S_2\,

\S' \cdot S\和\S \cdot \overline{S'}\如果其两个组成式都为假则全为假。我们看到最后一个条件是充分的，注意到当析取的组成式都是 TPE 时它也是 TPE，并且任何项：神经活动的逻辑演算 105

\S' \cdot S\ and \S \cdot \overline{S'}\ are all false if both their constituents are. We see that the last condition is sufficient by remarking that a disjunction is a TPE when its constituents are, and that any term: LOGICAL CALCULUS FOR NERVOUS ACTIVITY 105

图 1. 神经元\c_i\始终在其胞体上用数字 i 标记

Figure 1. The neuron \c_i\ is always marked with the numeral i upon the body of the

细胞，相应的动作用带下标的 N 表示，如文中所述：(a) \(N^*(t) = N_1(t-1)\);

cell, and the corresponding action is denoted by N with its subscript, as in the text: (a) \(N^*(t) = N_1(t-1)\);

(e) \(N_1(t) := N_1(t-1) \lor (N_1(t-3) \land \lnot N_1(t-2))\); \(N_1(t) = N_2(t-2) \land N_2(t-1)\); (f) \(N_4(t) := \lnot N_1(t-1) \land N_1(t-1) \lor N_1(t-1) \lor N_1(t-1) \land N_1(t-1)\); \(N_J(t) := \lnot N_1(t-2) \land N_1(t-2) \lor N_1(t-2) \lor N_1(t-2)\);

(e) \(N_1(t) := N_1(t-1) \lor (N_1(t-3) \land \lnot N_1(t-2))\); \(N_1(t) = N_2(t-2) \land N_2(t-1)\); (f) \(N_4(t) := \lnot N_1(t-1) \land N_1(t-1) \lor N_1(t-1) \lor N_1(t-1) \land N_1(t-1)\); \(N_J(t) := \lnot N_1(t-2) \land N_1(t-2) \lor N_1(t-2) \lor N_1(t-2)\);

\(N_1(t-2) \land N_1(t-2)\); (g) \(N_1(t) = \lnot N_1(t-2) \land \lnot N_1(t-3)\);

\(N_1(t-2) \land N_1(t-2)\); (g) \(N_1(t) = \lnot N_1(t-2) \land \lnot N_1(t-3)\);

(h) \(N_1(t) = N_1(t-1) \land N_1(t-2)\); (i) \(N_1(t) := N_2(t-1) \lor N_1(t-1) \lor (\exists x) t-1 N_1(x) \land N_1(x)\). 106 W. S. MCCULLOCH AND W. PITTS

(h) \(N_1(t) = N_1(t-1) \land N_1(t-2)\); (i) \(N_1(t) := N_2(t-1) \lor N_1(t-1) \lor (\exists x) t-1 N_1(x) \land N_1(x)\). 106 W. S. MCCULLOCH AND W. PITTS

神经网络的构造方法示例 s, .s, . . . Sm.4m+l .-. . .- s,,

上述定理的方法实际上提供了一种非常方便且可行的步骤，用于按需构建神经网络，适用于条件说明中不涉及无限遥远过去事件的情况。举例来说，我们可以考虑瞬态冷却产生的热感情形：若将一个冷物体短暂接触皮肤后移开，会感到热；若接触时间较长，则仅感到冷，而没有任何短暂的暖意。已知皮肤中有一种感受器对热敏感，另一种对冷敏感。设 \(N_1\) 和 \(N_2\) 分别为这两种感受器的活动，\(N_3\) 和 \(N_4\) 为神经元的活动，其活动分别代表热感和冷感，则我们的要求可写为：\(N_3(t) \coloneqq N_1(t-1) \lor \bigl(N_1(t-3) \land \lnot N_2(t-2)\bigr)\)，\(N_4(t) \equiv N_2(t-2) \land N_3(t-1)\)，为简化起见，假设冷感所需的持续时间为两个突触延迟，而热感为一个突触延迟。这些条件显然属于定理 3 的情形。因此，可通过定理 2 的方法构造一个网络来实现它们。我们首先以一种能展示其由基本运算（如图 1a–d 所示）组合而成的方式写出它们，即形式如下：

The method of the last theorems does in fact provide a very convenient and workable procedure for constructing nervous nets to order, for those cases where there is no reference to events indefinitely far in the past in the specification of the conditions. By way of example, we may consider the case of heat produced by a transient cooling. If a cold object is held to the skin for a moment and then removed, a sensation of heat will be felt; if it is applied for a longer time, the sensation will be only of cold, with no preliminary warmth, however transient. It is known that one cutaneous receptor is affected by heat, and another by cold. If we let \(N_1\) and \(N_2\) be the actions of the respective receptors and \(N_3\) and \(N_4\) of neurons whose activity implies a sensation of heat and cold, our requirements may be written as: \(N_3(t) \coloneqq N_1(t-1) \lor \bigl(N_1(t-3) \land \lnot N_2(t-2)\bigr)\), \(N_4(t) \equiv N_2(t-2) \land N_3(t-1)\), where we suppose for simplicity that the required persistence in the sensation of cold is say two synaptic delays, compared with one for that of heat. These conditions clearly fall under Theorem 3. A net may consequently be constructed to realize them, by the method of Theorem 2. We begin by writing them in a fashion which exhibits them as built out of their constituents by the operations realized in Figs 1a–d, i.e. in the form:

网络构建与相对抑制 N&) -= . SW, WV XW,(t)). - N,(t)l)

首先，我们为包含最多层括号的函数构建一个网络，并向外扩展；在此情况下，我们运行一个如图 1a 所示的网络，从 c2 到一个神经元（比如 c1），使得：\(N_1(t) = S[N_2(t)]\)。

First we construct a net for the function enclosed in the greatest number of brackets and proceed outward; in this case we run a net of the form shown in Fig. 1a from c2 to some neuron c1, say, so that: \(N_1(t) = S[N_2(t)]\).

接下来引入两个网络，形式分别为 1c 和 1d，都从 c1 和 c2 出发，分别终止于 c1 和某个 c_b。于是：

Next introduce two nets of the forms 1c and 1d, both running from c1 and c2, and ending respectively at c1 and say c_b. Then:

\(N_3(t) = S[N_1(t) \lor N_2(t)]\), \(N_4(t) = S[N_1(t) \lor S[N_2(t)]]\), 神经活动逻辑演算 107

\(N_3(t) = S[N_1(t) \lor N_2(t)]\), \(N_4(t) = S[N_1(t) \lor S[N_2(t)]]\), LOGICAL CALCULUS FOR NERVOUS ACTIVITY 107

最后，运行一个形式为 1b 的网络，从 c1 和 cb 到 c3，得到：\(N_3(t) = S[N_1(t) \lor N_2(t)] = S[N_1(t) \lor S[N_2(t)]] = N_4(t)\)。

Finally, run a net of the form 1b from c1 and cb to c3, and derive: \(N_3(t) = S[N_1(t) \lor N_2(t)] = S[N_1(t) \lor S[N_2(t)]] = N_4(t)\).

First we construct a net for the function enclosed in the greatest number of brackets and proceed outward; in this case we run a net of the form shown in Fig. la from c2 to some neuron c,, say, so that: N,(t). = . SN,(t).

相对抑制与绝对抑制在扩展意义上是等价的。

Relative and absolute inhibition are equivalent in the extended sense.

我们可以仿照公式(I)的形式写出神经兴奋定律，但采用相对抑制的假设；检查表明该表达式是一个 TPE。图 1f 给出了用绝对抑制替代相对抑制的一个例子。反向替代甚至更容易；我们给传入 ci 的抑制性轴突分别分配任意足够数量的抑制性突触。其次，考虑消逝的情况。我们可以将其写成阈值⟀(⟀theta_i⟀)的变化形式；在神经元 ci 放电后，取最接近的整数——并且仅在此近似下，阈值变化在自然形式的兴奋中才显著——这可以写成一个序列⟀(⟀theta_i + b_j⟀)，对应于放电后 j 个突触延迟，其中对于足够大的 j（比如 j ≥ M），有⟀(b_j = 0⟀)。于是我们可以陈述定理 5。

We may write out a law of nervous excitation after the fashion of (I), but employing the assumption of relative inhibition instead; inspection then shows that this expression is a TPE. An example of the replacement of relative inhibition by absolute is given by Fig. 1f. The reverse replacement is even easier; we give the inhibitory axons afferent to ci any sufficiently large number of inhibitory synapses apiece. Second, we consider the case of extinction. We may write this in the form of a variation in the threshold ⟀(⟀theta_i⟀); after the neuron ⟀(c_i⟀) has fired; to the nearest integer—and only to this approximation is the variation in threshold significant in natural forms of excitation—this may be written as a sequence ⟀(⟀theta_i + b_j⟀) for j synaptic delays after firing, where ⟀(b_j = 0⟀) for j large enough, say j = M or greater. We may then state Theorem 5.

消逝等价于绝对抑制。

Extinction is equivalent to absolute inhibition.

因为，假设当前采用相对抑制，我们只需运行 M 个回路⟀(Y_1, Y_2, ⟀ldots, Y_M⟀)，它们分别包含⟀(1, 2, ⟀ldots, M⟀)个神经元，使得任一回路中每个环节的放电足以触发下一个环节，从神经元 ci 回到它自身，其中回路⟀(Y_j⟀)的末端恰好有⟀(b_j⟀)个抑制性突触作用于 ci。显然，这将产生所需结果。108 W.S. McCulloch 和 W. Pitts 的反向替代可通过图 1g 的示意图完成。由替代的传递性，我们推断出该定理。属于这组定理的还有以下众所周知的定理。

For, assuming relative inhibition to hold for the moment, we need merely run M circuits ⟀(Y_1, Y_2, ⟀ldots, Y_M⟀) containing respectively ⟀(1, 2, ⟀ldots, M⟀) neurons, such that the firing of each link in any circuit is sufficient to fire the next, from the neuron ⟀(c_i⟀) back to itself, where the end of the circuit ⟀(Y_j⟀) has just ⟀(b_j⟀) inhibitory synapses upon ⟀(c_i⟀). It is evident that this will produce the desired results. The 108 W.S. McCulloch and W. Pitts reverse substitution may be accomplished by the diagram of Fig. 1g. From the transitivity of replacement, we infer the theorem. To this group of theorems also belongs the following well-known theorem.

Relative and absolute inhibition are equivalent in the extended sense.

这是显而易见的：只需在兴奋细胞与需要时间总和维持的神经元之间引入适当顺序的延迟链，且突触数量递增，那么空间总和的假设就会给出所需结果（例如图 1h）。这一方法曾用于证明，在粗网络中观察到的时间总和并不暗示单个神经元相互作用中存在这种机制。学习现象具有在神经活动的大多数生理变化中持续存在的特征，似乎要求网络结构具有永久性改变的可能性。最简单的此类改变是新突触的形成或等效的局部阈值降低。我们假设一些轴突末梢最初不能兴奋后续神经元；但如果神经元在任何时候发放且轴突末梢同时兴奋，它们就会变成普通突触，从此能够兴奋该神经元。抑制性突触的丢失会产生完全等效的结果。接下来我们将有

This is obvious: one need merely introduce a suitable sequence of delaying chains, of increasing numbers of synapses, between the exciting cell and the neuron whereon temporal summation is desired to hold. The assumption of spatial summation will then give the required results (see e.g. Fig. 1h). This procedure had application in showing that the observed temporal summation in gross nets does not imply such a mechanism in the interaction of individual neurons. The phenomena of learning, which are of a character persisting over most physiological changes in nervous activity, seem to require the possibility of permanent alterations in the structure of nets. The simplest such alteration is the formation of new synapses or equivalent local depressions of threshold. We suppose that some axonal terminations cannot at first excite the succeeding neuron; but if at any time the neuron fires, and the axonal terminations are simultaneously excited, they become synapses of the ordinary kind, henceforth capable of exciting the neuron. The loss of an inhibitory synapse gives an entirely equivalent result. We shall then have

这是通过图 1i 的方法实现的。另外值得注意的是，一个变得并保持自发活动的神经元同样可以被一个环取代，该环在活动开始时由外周传入激活，在活动停止时由某个传入抑制。3. 理论：带圆环的网络。处理不满足先前无环假设的网络比那种情况困难得多。这主要是因为活动可能在回路中建立并无限期地持续回荡，因此可实现的活动可能涉及对无限遥远过去事件的参照。考虑这样一个网络 Jlr（设为 p 阶），令 c1, c2, ..., cp 为 Jlr 的一个神经元循环集。从定义首先清楚：每个 N3（即网络中的每个神经元）都可以表示为 N1, N2, ..., N 以及绝对传入的时间乘积表达式（TPE）；那么 X 的解只需确定循环集的表达式。完成这一步后，我们将推导出一组表达式[A]：

This is accomplished by the method of Fig. 1i. It is also to be remarked that a neuron which becomes and remains spontaneously active can likewise be replaced by a circle, which is set into activity by a peripheral afferent when the activity commences, and inhibited by one when it ceases. 3. The Theory: Nets with Circles. The treatment of nets which do not satisfy our previous assumption of freedom from circles is very much more difficult than that case. This is largely a consequence of the possibility that activity may be set up in a circuit and continue reverberating around it for an indefinite period of time, so that the realizable Pr may involve reference to past events of an indefinite degree of remoteness. Consider such a net Jlr, say of order p, and let c1, c2, ..., cp be a cyclic set of neurons of Jlr. It is first of all clear from the definition that every N3 of JV can be expressed as a TPE, of N1, N2, ..., N, and the absolute afferents; the solution of X involves then only the determination of expressions for the cyclic set. This done, we shall derive a set of expressions [A]:

\(N_i(z) = \mathrm{Pr}_i[ S_{i1}N_1(z_1), S_{i2}N_2(z_1), \ldots , L W N_p(q) ]\),

\(N_i(z) = \mathrm{Pr}_i[ S_{i1}N_1(z_1), S_{i2}N_2(z_1), \ldots , L W N_p(q) ]\),

其中也涉及外周传入。现在如果 n 是 y_{ij}的最小公倍数

where Pri also involves peripheral afferents. Now if n is the least common

我们将根据（2）把（3）中的 N_j 替换为其等价形式，并重复此过程足够多次，从而得到一组形式如下的表达式：

multiple of the y_{ij}, we shall, by substituting their equivalents according to (2) in (3) for the N_j, and repeating this process often enough on the result, obtain a set of the form:

(3) 这些表达式可以写成希尔伯特析取范式：\exists z\. C S, \cap\ S' N_j(z_1) \cup\ N S' N_j(z_1)，对于合适的 K, B, j, w, k, B\in\K

(3) These expressions may be written in the Hilbert disjunctive normal form as: \exists z\. C S, \cap\ S' N_j(z_1) \cup\ N S' N_j(z_1), for suitable K, B, j, w, k, B\in\K

(4) 其中 S 是 N 的绝对传入的 TPE。存在一些 2p 个不同的句子，通过对其中一些集合的合取与其余否定集合的合取而形成。将它们命名为 X_1(z_1), X_2(z_1), ..., X_{2p}(z_1)，我们可以利用表达式(4)得到一组等价的方程形式：

(4) where S is a TPE of the absolute afferents of N. There exist some 2p different sentences formed out of the variables by conjoining to the conjunction of some set of them the conjunction of the negations of the rest. Denumerating these by X_1(z_1), X_2(z_1), ..., X_{2p}(z_1), we may, by use of the expressions (4), arrive at an equipollent set of equations of the form:

(5) 现在我们将带下标的数字 i, j 引入对象语言，即定义 Pr_1 和 Pr_2，使得 Pr_1(z_1, z_2) \equiv\ \&\(z_1)和 Pr_2(z_1, z_2, ...)在 z_1 和 z_2 分别表示 i 和 j 时可证明。然后我们可以将(5)重写为：

(5) Now we import the subscripted numerals i, j into the object-language, i.e. define Pr_1 and Pr_2 such that Pr_1(z_1, z_2) \equiv\ \&\(z_1) and Pr_2(z_1, z_2, ...) are provable whenever z_1 and z_2 denote i and j respectively. Then we may rewrite (5) as:

(3) These expressions may be written in the Hilbert disjunctive normal form as: Ni(zl)*z. C S, fl S”Nj(Zl) JJ N S”Nj(z1), for suitable K, BEK jw .k& B&K

(z1)z1: Pr1(z1, z2, z3 - z1). Pr1(z1, z3 - z1), (6) (zl)zzp:Prltzl~z,) .- .tEz,)zz, -Pr,(zl , z2, z3 - zz,). Pr, (z, , z3 - zz,), (6)

其中 z1 表示 y1，z2 表示 2p。经过反复代入，我们得到表达式：

where z1 denotes y1 and z2 denotes 2p. By repeated substitution we arrive at an expression:

\(E z1) z1. Pr2(z1, z2, z1(z1 - 1)) * Pr1(z1, z3) z3(z3 - 1) * ... (7)\

(E z1) z1. Pr2(z1, z2, z1(z1 - 1)) * Pr1(z1, z3) z3(z3 - 1) * ... (7)

Pr1(z1, z2, 0)，对于任何表示 s 的数字 z2。通过归纳容易证明这与下式等价：\(z1)z1: *Pr1(z1, z2, z1): = : (Ef) (z2)z3 - 1 (Z, Z3) S z1 . f(z1, z2) = z1 . Pr2(~(z1(z1 + 1)),\

Pr1(z1, z2, 0), for any numeral z2 which denotes s. This is easily shown by induction to be equipollent to: (z1)z1: *Pr1(z1, z2, z1): = : (Ef) (z2)z3 - 1 (Z, Z3) S z1 . f(z1, z2) = z1 . Pr2(~(z1(z1 + 1)),

f(zz,zd) . Pr, (f(o), Oh (8) 110 W. S. 麦卡洛克和 W. 皮茨 f(zz,zd) . Pr, (f(o), Oh (8) 110 W. S. McCULLOCH AND W. PI-I-E

由于这对所有 z_2 都成立，因此也有：

and since this is the case for all z_2, it is also true that:

循环集的表达式 w tz,&J: WZl, z,) -= (Ef) (z,) (z.$--1) .f(z,) ~zz,.f(24)=Zlf(Z4)=Zl JWY(z~+ l),f(Zz), z,. 4 Cf(res(z,, zz,)), res(z,, zz,), (9)

其中 \(t_z\) 表示 \(n\)，\(res(r,s)\) 是 \(r\) 模 \(s\) 的余数，\(z_z\) 表示 \(2p\)。这可以不太精确地写为：\(N_t(t)\)、\( (E_f)(x) t-1 \) 和 \(2' I \sim(t)=i\)。

where \(t_z\) denotes \(n\), \(res(r,s)\) is the residue of \(r\) mod \(s\) and \(z_z\) denotes \(2p\). This may be written in a less exact way as: \(N_t(t)\). \( (E_f)(x) t-1 \). and \(2' I \sim(t)=i\).

其中假设 \(x\) 和 \(t\) 也能被 \(n\) 整除，且 \(P_r\) 表示 \(P\)。根据前面的论述，我们将得到定理 8。

where \(x\) and \(t\) are also assumed divisible by \(n\), and \(P_r\) denotes \(P\). From the preceding remarks we shall have Theorem 8.

8. 表达式 (9) 对于网络 \(J_1'\) 的循环集上的神经元，连同用它们表示其他神经元动作的某些 TPE，构成了 \(J_1\) 的一个解。

8. The expression (9) for neurons of the cyclic set of a net \(J_1'\) together with certain TPE expressing the actions of other neurons in terms of them, constitute a solution of \(J_1\).

现在考虑一组 \(S_i\) 的可实现性问题。一个可以通过简单归纳证明的首要必要条件是：该条件应成立，对 \(S_i\) 中其他自由变量也有类似陈述，即没有神经系统可以考虑未来的外周传入。任何满足此要求的 \(S_i\) 都可以被替换为一个等价的 \(S\)，形式如下：\( (E_f)(\tilde{z}_1)\tilde{z}_1(\tilde{z}_2)\tilde{z}_2: \dots: f(z_1, z_2, z_3)=1 = p_{z_a}(z_2)\, (11)

Consider now the question of the realizability of a set of \(S_i\). A first necessary condition, demonstrable by an easy induction, is that: the condition should be true, with similar statements for the other free variables in \(S_i\), i.e. no nervous net can take account of future peripheral afferents. Any \(S_i\) satisfying this requirement can be replaced by an equipollent \(S\) of the form: \( (E_f)(\tilde{z}_1)\tilde{z}_1(\tilde{z}_2)\tilde{z}_2: \dots: f(z_1, z_2, z_3)=1 = p_{z_a}(z_2)\, (11)

其中 \(z_z\) 表示 \(p\)，通过定义：\(P_{r,i} = (E_f)(z_1)z_1(z_3)z_2: f(z_1, z_2, \tilde{z})=0 . Y Z (z_2, z_3) = I : (z_2, z_3)=1 . p_{,}(z_3 - KS] .\

where \(z_z\) denotes \(p\), by defining: \(P_{r,i} = (E_f)(z_1)z_1(z_3)z_2: f(z_1, z_2, \tilde{z})=0 . Y Z (z_2, z_3) = I : (z_2, z_3)=1 . p_{,}(z_3 - KS] .\

现在考虑这些类 \(a_j\) 的序列，满足：

Consider now these series of classes \(a_j\), for which:

对某个网络成立。这些类将被称为 prehensible 类。让我们定义

holds for some net. These will be called prehensible classes. Let us define the

由类 \(K\) 生成的布尔环为这些类的集合 LOGICAL CALCULUS FOR NERVOUS ACTIVITY 111

Boolean ring generated by a class of classes \(K\) as the aggregate of the classes LOGICAL CALCULUS FOR NERVOUS ACTIVITY 111

这些类可以通过重复应用逻辑运算从 \(K\) 的成员形成，即我们设：\(\mathfrak{B}(K) = \&(a, p): a \subset K\)

which can be formed from members of \(K\) by repeated application of the logical operations, i.e. we put: \(\mathfrak{B}(K) = \&(a, p): a \subset K\)

\(\mathfrak{B}(K) = \hat{p} \times [(\alpha, \beta): \alpha \subset K \to \alpha \subset \ldots] \quad \mathfrak{B}_e(K) = \mathfrak{B}(K) - t \& f - \mathfrak{K}\)

\(\mathfrak{B}(K) = \hat{p} \times [(\alpha, \beta): \alpha \subset K \to \alpha \subset \ldots] \quad \mathfrak{B}_e(K) = \mathfrak{B}(K) - t \& f - \mathfrak{K}\)

类 W(K)的构造方式与 L(X)类似，但通过重复应用不仅包括逻辑运算，还包括将性质类 P(a)替换为 S(P)~S(a)的运算。于是我们得到以下引理。

The class W(K) is formed from K in analogy with L(X), but by repeated application not only of the logical operations but also of that which replaces a class of properties P(a) by S(P)~S(a). We shall then have the following lemma.

Pr(p1, p2, ..., pm, z1)是一个 TPE（真命题表达式）当且仅当(z1)(p1, ..., pm)(\exists\ pm+1): pm+1 \in\ ... e((p1, p2, ..., pm)) (13) P_{m+1}(z1) ... (p1, p2, ..., pm, z1)，

Pr(p1, p2, ..., pm, z1) is a TPE if and only if (z1)(p1, ..., pm)(\exists\ pm+1): pm+1 \in\ ... e((p1, p2, ..., pm)) (13) P_{m+1}(z1) ... (p1, p2, ..., pm, z1),

该式为真；并且它是一个不涉及“S”的 TPE 当且仅当在将“S”替换为...时该式成立，于是我们得到定理 9。

is true; and it is a TPE not involving “S” if and only if this holds when “S” is replaced by ... , and we then obtain Theorem 9.

类序列 a1, a2, ..., aY 是一个 prehensible 类序列当且仅当：(\exists\ m)(\exists\ n)(p)n(i? (t, &): . (x)m. \phi\(x)=0 \lor\ \psi\(x)=1: +: (Ef1) (Ey)m. \phi\(y)=0 . & S?[~((Ei).y=ai)). v. (x)m. 1(\phi\(x)=0 . \phi\ is B[\phi\((Ei). y=ai)]: (t) (4): @ai.

A series of classes a1, a2, ..., aY is a series of prehensible classes if and only if: (\exists\ m)(\exists\ n)(p)n(i? (t, &): . (x)m. \phi\(x)=0 \lor\ \psi\(x)=1: +: (Ef1) (Ey)m. \phi\(y)=0 . & S?[~((Ei).y=ai)). v. (x)m. 1(\phi\(x)=0 . \phi\ is B[\phi\((Ei). y=ai)]: (t) (4): @ai.

The class W,(K) is formed from k: in analogy with L%(X), but by repeated application not only of the logical operations but also of that which replaces a class of properties P&a by S(P)~S”a. We shall then have the following lemma.

\(n(t+1)+p, nx+p, w) = f(nt+p, nx+p, w)\。

(n(t+1)+p, nx+p, w) = f(nt+p, nx+p, w).

(14) 证明：证明直接由引理得出。条件是必要的，因为每个可以写出形如(4)的表达式的网络显然满足该条件，其中 I/S 是 S 的特征函数，而每个$对应的 b 是指称形式为 niea Pri nj&fla Prj 的类。

(14) Proof: The proof here follows directly from the lemma. The condition is necessary, since every net for which an expression of the form (4) can be written obviously verifies it, the I/S being the characteristic functions of the S, and the b for each $ being the class whose designation has the form niea Pri nj&fla Prj,

其中 Pv 表示所有 k 的 ak。反之，对于满足可理解类且满足(14)的网络 N，我们可以写出形如(4)的表达式，通过将 112 W. S. McCULLOCH 和 W. PITTS

where Pv, denotes ak for all k. Conversely, we may write an expression of the form (4) for a net N fulfilling prehensible classes satisfying (14) by putting for 112 W. S. McCULLOCH AND W. PITTS

这些 Pr 表示ψ，而一个用类析取范式的类比写出的 Pr，表示与那个ψ合取的对应 c。由于每个形如(4)的 S 显然是可实现的，故定理得证。考虑我们能在多大程度上通过当前知识确定各种特殊网络的整个过去是有趣的，即何时我们可以构造一个网络，其循环神经元的激发要求外周传入具有由给定函数φ_i 指定的过去值集。在这种情况下，上一个定理的类α_i 简化为单元类；且条件可转化为：

the Pr, Pr denoting the ψ's, and a Pr, written in the analogue for classes of the disjunctive normal form, and denoting the c corresponding to that ψ, conjoined to it. Since every S of the form (4) is clearly realizable, we have the theorem. It is of some interest to consider the extent to which we can by knowledge of the present determine the whole past of various special nets, i.e. when we may construct a net the firing of the cyclic of whose neurons requires the peripheral afferents to have had a set of past values specified by given functions φ_i. In this case the classes α_i of the last theorem reduced to unit classes; and the condition may be transformed into:

\(∃m, n) (p)n(i, ψ) (Iψ): . (x)m: ψ(x) = 0 ∨ ψ(x) = 1: ¬i & α(ψ, nt+p):→: (W)m X t-1 . ¬i(n(t+1)+p, nx+p, W) = φ_j(nt+p, nx+p, w) ⇒ (∃u, v) (w)m α_i(n(u+1)+p, nv+p, w) = α_i(n(v+1)+p, nv+p, w)\

(Em, n) (p)n(i, $) (I$): . (x)m: $(x) = 0.v.$(x) = 1: ~i&a(l, nt+p):~: (W)m(X)t-1 .~i(n(t+1)+p, nX+p, W)=j(nt+P, nx+p, w)” (u9 v, (w)ma$i(n(u+ ')+p, n”+pY w, =~i(n(v+l)+p, n”+p, w)’

由于篇幅所限，以上论证非常简略；我们拟在后续出版物中对其进行扩展并讨论其某些含义。最后一个定理的条件在原则上相当简单，但细节上并非如此；然而，将其应用于实际案例需要探索大约 22 个函数类，即某个集合的成员。由于这些每个都是定理 9 的可能 p，因此该结果无法进一步强化。但我们可以得到一个关于 S 可实现性的充分条件，该条件易于应用且可能覆盖大多数实际目的。这由定理 10 给出。

On account of limitations of space, we have presented the above argument very sketchily; we propose to expand it and certain of its implications in a further publication. The condition of the last theorem is fairly simple in principle, though not in detail; its application to practical cases would, however, require the exploration of some 22 classes of functions, namely the members of a certain set. Since each of these is a possible p of Theorem 9, this result cannot be sharpened. But we may obtain a sufficient condition for the realizability of an S which is very easily applicable and probably covers most practical purposes. This is given by Theorem 10.

让我们通过以下递归定义一个\F_3\的集合 K：(1) 任何 TPE

Let us define a set K of \F_3\ by the following recursion: (1) any TPE

以及任何自变量已被 K 的成员替换的 TPE 都属于 K；(2) 如果\Pr_1(z_1)\是 K 的一个成员，则\(\forall z_1) . Pr_1(z_1)\、\(\exists z_1) . Pr_1(z_1)\以及它们都属于 K，其中\C_m\表示与 m 同余的性质。

and any TPE whose arguments have been replaced by members of K belong to K; (2) if \Pr_1(z_1)\ is a member of K, then \(\forall z_1) . Pr_1(z_1)\, \(\exists z_1) . Pr_1(z_1)\, and these belong to it, where \C_m\ denotes the property of being congruent to m.

On account of limitations of space, we have presented the above argument very sketchily; we propose to expand it and certain of its implications in a further publication. The condition of the last theorem is fairly simply in principle, though not in detail; its application to practical cases would, however, require the exploration of some 22” classes of functions, namely the members of =Q%&,., a,>). Since each of these is a possible p of Theorem 9, this result cannot be sharpened. But we may obtain a sufficient condition for the realizability of an S which is very easily applicable and probably covers most practical purposes. This is given by Theorem 10.

Let us dejne a set K off3 by thefollowing recursion: (1) any TPE

模 n, m < n; (3) 集合 K 没有其他成员。

modulo n, m < n; (3) The set K has no further members.

那么 K 的每个成员都是可实现的。因为，如果 Pr, (z,)是可实现的，那么神经网……

Then every member of K is realizable. For, if Pr, (z,) is realizable, nervous nets for which:

分别是方程(4)的表达式，实现(z_i)z_i . Pr_i(z_2)和(∃z_i)z_i . Pr_i(z_i)逻辑演算神经活动 113；并且一个由 n 个链接组成的简单回路 c_1, c_2, ..., c_n，每个链接都足以兴奋下一个，为最后的形式给出了一个表达式。通过归纳我们推导出定理。最后还要提到一点。很容易证明：首先，每个神经网，如果配备一条磁带、连接到传入神经的扫描器和执行必要运动操作的适当传出神经，只能计算图灵机可以计算的数；其次，每个这样的数都可以由这样的神经网计算；并且带有回路的神经网可以在没有扫描器和磁带的情况下计算机器能算的一些数，但其他数不行，而且也不是所有数都能算。这很有趣，因为它为图灵可计算性定义及其等价定义——丘奇的λ可定义性和克林的原初递归性——提供了心理学上的正当性：如果一个生物体可以计算某个数，那么它根据这些定义就是可计算的，反之亦然。4. 结论。因果性，需要描述状态以及将它们联系起来的必然联系规律，已在几种科学中以几种形式出现，但除了统计学外，从未像在本理论中这样不可逆。在任何时刻对传入刺激和所有组成神经元的活动（每个都是“全或无”事件）的说明决定了状态。神经网的说明提供了必然联系规律，据此人们可以从任何状态的描述计算出后续状态，但析取关系的包含阻止了前一个状态的完全确定。此外，组成回路的再生活动使得对过去时间的指称变得不确定。因此，我们对外部世界（包括我们自己）的知识在空间上是不完全的，在时间上是不确定的。这种蕴含在所有大脑中的无知，正是使我们的知识有用的抽象的对立面。大脑在决定我们的理论与其观察之间以及这些观察与事实之间的认识论关系中所起的作用再清楚不过了，因为很明显，每个想法和每个感觉都是由该网内的活动实现的，并且没有这样的活动能完全决定实际的传入。如果网被改变，我们持有的任何理论和进行的任何观察都无法保留其原有对事实的缺陷指称。耳鸣、感觉异常、幻觉、妄想、混淆和定向障碍随之出现。因此，经验证实，如果我们的网未定义，我们的事实也未定义，我们无法赋予“实在”哪怕一个性质或“形式”。随着网的确定，不可知的知识对象——“物自体”——不再是不可知的。无论心理学如何定义，神经网的说明将对该领域所有可达到的成就做出贡献——即使分析被推向终极心理单位或“心理元”，因为一个心理元至少是一个单个神经元的活动。由于该活动本质上是命题性的，所有心理事件都具有意向性或“符号学”特征。这些活动的“全或无”定律，以及它们的关系与命题逻辑关系的一致性，确保了心理元之间的关系是二值命题逻辑的关系。因此，在心理学中，无论是内省心理学、行为主义心理学还是生理心理学，基本关系都是二值逻辑的关系。由此产生了关于感觉意识的分化连续统以及知觉和执行的规范性、完成性和解决性的整体问题的构造性解决方案。从因果性的不可逆性可知，即使网是已知的，虽然我们可以从现在活动预测未来，但我们既不能从中央活动推导出传入活动，也不能从传出活动推导出中央活动，更不能从现在活动推导出过去活动——这些结论得到了目击者矛盾证言、区分器质性病变、癔症和诈病的诊断困难，以及将自己的记忆或回忆与同期记录进行比较的支持。此外，系统对传入再生网的传入活动与网内特定活动之间的差异作出响应以减少差异，从而表现出目的性行为；生物体拥有许多这样的系统，服务于稳态、欲求和注意。因此，我们习惯称之为精神的那种活动的形式方面和终极方面都可以从当前的神经生理学中严格推导出来。精神科医生可以从关于因果性的明显结论中得到安慰——即对于预后，历史从来不是必需的。但他从同样有效的结论中获益甚少，即他的可观察量只能以神经活动来解释，而直到最近，这些神经活动还超越了他的认知范围。这种无知的症结在于，从任何外显行为样本对神经网的推断都不是唯一的，而在可想象的网中，实际上只存在一个，并且可能随时表现出不可预测的活动。当然，对精神科医生来说更重要的是，在这样的系统中，“精神”不再“比鬼魂更飘渺”。相反，病态心理可以在神经生理学的科学术语中得到理解，而不会丧失范围和严谨性。对于神经学，该理论区分了对于给定活动是必需的还是仅仅足够的神经网，从而澄清了结构紊乱与功能紊乱之间的关系。在其自身领域内，等价网和狭义等价网之间的差异指示了神经活动时间研究的适当用途和重要性；而对于数学生物物理学，该理论贡献了一个对已知网进行严格符号处理的工具，以及构建所需性质的假设网的简便方法。逻辑演算神经活动 115

are the expressions of equation (4), realize (z_i)z_i . Pr_i(z_2) and (∃z_i)z_i . Pr_i(z_i) LOGICAL CALCULUS FOR NERVOUS ACTIVITY 113 respectively; and a simple circuit, c_1, c_2, ..., c_n of n links, each sufficient to excite the next, gives an expression: for the last form. By induction we derive the theorem. One more thing is to be remarked in conclusion. It is easily shown: first, that every net, if furnished with a tape, scanners connected to afferents, and suitable efferents to perform the necessary motor-operations, can compute only such numbers as can a Turing machine; second, that each of the latter numbers can be computed by such a net; and that nets with circles can compute, without scanners and a tape, some of the numbers the machine can, but no others, and not all of them. This is of interest as affording a psychological justification of the Turing definition of computability and its equivalents, Church's λ-definability and Kleene's primitive recursiveness: if any number can be computed by an organism, it is computable by these definitions, and conversely. 4. Consequences. Causality, which requires description of states and a law of necessary connection relating them, has appeared in several forms in several sciences, but never, except in statistics, has it been as irreciprocal as in this theory. Specification for any one time of afferent stimulation and of the activity of all constituent neurons, each an "all-or-none" affair, determines the state. Specification of the nervous net provides the law of necessary connection whereby one can compute from the description of any state that of the succeeding state, but the inclusion of disjunctive relations prevents complete determination of the one before. Moreover, the regenerative activity of constituent circles renders reference indefinite as to time past. Thus our knowledge of the world, including ourselves, is incomplete as to space and indefinite as to time. This ignorance, implicit in all our brains, is the counterpart of the abstraction which renders our knowledge useful. The role of brains in determining the epistemic relations of our theories to our observations and of these to the facts is all too clear, for it is apparent that every idea and every sensation is realized by activity within that net, and by no such activity are the actual afferents fully determined. There is no theory we may hold and no observation we can make that will retain so much as its old defective reference to the facts if the net be altered. Tinnitus, paresthesias, hallucinations, delusions, confusions and disorientation intervene. Thus empiry confirms that if our nets are undefined, our facts are undefined, and to the "real" we can attribute not so much as one quality or "form." With determination of the net, the unknowable object of knowledge, the "thing in itself," ceases to be unknowable. To psychology, however defined, specification of the net would contribute all 114 W. S. MCCULLOCH AND W. PITTS

即使分析被推向终极心理单位或“心理元”，因为一个心理元至少是一个单个神经元的活动。由于该活动本质上是命题性的，所有心理事件都具有意向性或“符号学”特征。这些活动的“全或无”定律，以及它们的关系与命题逻辑关系的一致性，确保了心理元之间的关系是二值命题逻辑的关系。因此，在心理学中，无论是内省心理学、行为主义心理学还是生理心理学，基本关系都是二值逻辑的关系。由此产生了关于感觉意识的分化连续统以及知觉和执行的规范性、完成性和解决性的整体问题的构造性解决方案。从因果性的不可逆性可知，即使网是已知的，虽然我们可以从现在活动预测未来，但我们既不能从中央活动推导出传入活动，也不能从传出活动推导出中央活动，更不能从现在活动推导出过去活动——这些结论得到了目击者矛盾证言、区分器质性病变、癔症和诈病的诊断困难，以及将自己的记忆或回忆与同期记录进行比较的支持。此外，系统对传入再生网的传入活动与网内特定活动之间的差异作出响应以减少差异，从而表现出目的性行为；生物体拥有许多这样的系统，服务于稳态、欲求和注意。因此，我们习惯称之为精神的那种活动的形式方面和终极方面都可以从当前的神经生理学中严格推导出来。精神科医生可以从关于因果性的明显结论中得到安慰——即对于预后，历史从来不是必需的。但他从同样有效的结论中获益甚少，即他的可观察量只能以神经活动来解释，而直到最近，这些神经活动还超越了他的认知范围。这种无知的症结在于，从任何外显行为样本对神经网的推断都不是唯一的，而在可想象的网中，实际上只存在一个，并且可能随时表现出不可预测的活动。当然，对精神科医生来说更重要的是，在这样的系统中，“精神”不再“比鬼魂更飘渺”。相反，病态心理可以在神经生理学的科学术语中得到理解，而不会丧失范围和严谨性。对于神经学，该理论区分了对于给定活动是必需的还是仅仅足够的神经网，从而澄清了结构紊乱与功能紊乱之间的关系。在其自身领域内，等价网和狭义等价网之间的差异指示了神经活动时间研究的适当用途和重要性；而对于数学生物物理学，该理论贡献了一个对已知网进行严格符号处理的工具，以及构建所需性质的假设网的简便方法。逻辑演算神经活动 115

that could be achieved in that field—even if the analysis were pushed to ultimate psychic units or "psychons," for a psychon can be no less than the activity of a single neuron. Since that activity is inherently propositional, all psychic events have an intentional, or "semiotic," character. The "all-or-none" law of these activities, and the conformity of their relations to those of the logic of propositions, insure that the relations of psychons are those of the two-valued logic of propositions. Thus in psychology, introspective, behavioristic or physiological, the fundamental relations are those of two-valued logic. Hence arise constructional solutions of holistic problems involving the differentiated continuum of sense awareness and the normative, perfective and resolvent properties of perception and execution. From the irreciprocity of causality it follows that even if the net be known, though we may predict future from present activities, we can deduce neither afferent from central, nor central from efferent, nor past from present activities—conclusions which are reinforced by the contradictory testimony of eye-witnesses, by the difficulty of diagnosing differentially the organically diseased, the hysteric and the malingerer, and by comparing one's own memories or recollections with his contemporaneous records. Moreover, systems which so respond to the difference between afferents to a regenerative net and certain activity within that net, as to reduce the difference, exhibit purposive behavior; and organisms are known to possess many such systems, subserving homeostasis, appetition and attention. Thus both the formal and the final aspects of that activity which we are wont to call mental are rigorously deducible from present neurophysiology. The psychiatrist may take comfort from the obvious conclusion concerning causality—that, for prognosis, history is never necessary. He can take little from the equally valid conclusion that his observables are explicable only in terms of nervous activities which, until recently, have been beyond his ken. The crux of this ignorance is that inference from any sample of overt behavior to nervous nets is not unique, whereas, of imaginable nets, only one in fact exists, and may, at any moment, exhibit some unpredictable activity. Certainly for the psychiatrist it is more to the point that in such systems "Mind" no longer "goes more ghostly than a ghost." Instead, diseased mentality can be understood without loss of scope or rigor, in the scientific terms of neurophysiology. For neurology, the theory sharpens the distinction between nets necessary or merely sufficient for given activities, and so clarifies the relations of disturbed structure to disturbed function. In its own domain the difference between equivalent nets and nets equivalent in the narrow sense indicates the appropriate use and importance of temporal studies of nervous activity: and to mathematical biophysics the theory contributes a tool for rigorous symbolic treatment of known nets and an easy method of constructing hypothetical nets of required properties. LOGICAL CALCULUS FOR NERVOUS ACTIVITY 115

modulo n, m < n; (3) The set K has no further members.

Carnap, R. 1938. 《语言的逻辑句法》。纽约：Harcourt-Brace. Hilbert, D. and W. Ackermann. 1927. 《理论逻辑基础》。柏林：Springer. Russell, B. and A. N. Whitehead. 1925. 《数学原理》。剑桥大学出版社。

Carnap, R. 1938. The Logical Syntax of Language. New York: Harcourt-Brace. Hilbert, D. and W. Ackermann. 1927. Grundzüge der theoretischen Logik. Berlin: Springer. Russell, B. and A. N. Whitehead. 1925. Principia Mathematica. Cambridge University Press.