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一,论坛StackExchangeHistory of Science and Mathematics栏目中关于实质蕴涵术语的讨论


Ben I

I looked through both of George Boole's treatises (1 and 2), but there is nothing like implication as I have seen it, with






So, if George Boole didn't create this construct, where did it come from?


The reason I ask is that there are a few cognitive traps from implication, particularly when we try to view it "intuitively", and I suspect that this is because the intuitive explanations are all somewhat nonsensical.

Just as a quick for instance of what I mean: if I ask a the classic question



When we say A implies B, what can we say about whether 𝐴 or B are necessary or sufficient for one another?

当我们说 A 蕴含 B 时,对于𝐴B彼此是必要的还是充分的,我们可以说什么?

Conifold : 

It may be surprising, but the material implication does not come from truth tables, the truth table definition is a late development. Neither de Morgan, nor Peirce, nor Frege, nor even Russell came up with it or justified it by matching Boolean operations to something in Plato and Aristotle. A detailed story can be found in Cajori's History of mathematical Notations, vol. II.

这可能令人惊讶,但实质蕴涵并非来自真值表,真值表定义是后来发展的。 ·摩根、皮尔士、弗雷格,甚至罗素都没有提出这个概念,也没有通过将布尔运算与柏拉图和亚里士多德的某些东西相匹配来证明它的合理性。详细的故事可以在Cajori's History of mathematical Notations, vol. II. 中找到。

It came from a very common idea of classical logicians of identifying propositions with classes (intensions), and classes with sets (extensions). Accordingly, the early definitions of implication interpret "X implies Y" as "X is contained in Y". Originally, it was only applied to syllogisms, where it matches the intuition.

它来自古典逻辑学家的一个非常普遍的想法,即用类(内涵)来识别命题,用集合(外延)来识别类。 因此,早期的蕴含定义将“X蕴涵Y”解释为“X包含在Y 最初,它只应用于三段论,与直觉相匹配。

In the 19th century, its scope was expanded with the algebraization of logic by Boole and de Morgan. The chain of transmission went from de Morgan's  (1847, same year as Boole's first treatise), to Peirce's claw ―< (1867), to Schröder’s (1890), and Peano's . Later Peano, and after him Russell, adopted Schröder’s (note that the meaning is reversed compared to the modern set inclusion). Peirce called the material implication "the copula of inclusion" (also "illation"), and Frege (whose notation for it was clumsy and never reproduced later) even criticized Schröder for "confusing" it with class inclusion.

19 世纪,随着布尔和德·摩根的逻辑代数化,其范围得到了扩展。 传播链从德·摩根的 1847 年,与布尔的第一篇论文同年),到皮尔士的“claw ―<”1867 年),到施罗德的 1890 年),再到皮亚诺的 后来皮亚诺和他之后的罗素采用了施罗德的(注意,与现代集合包含相比,其含义是相反的)。 皮尔士将实质蕴涵称为包含的系词(也称为“illation”),而弗雷格(其对此的表示法很笨拙,后来从未被复制)甚至批评施罗德将其与类包含混淆

The identification itself predates Boole and even Leibniz, it can be traced back to Aristotle, and was implicit in scholastic logic (for syllogisms). Russell's Principia still has a trace of it, the class/set identification only goes away after Hausdorff's Grundzuge der Mengenlehre (1914), see Kanamori's The empty set, the singleton, and the ordered pair.

这种识别本身早于布尔甚至莱布尼茨,它可以追溯到亚里士多德,并且隐含在经院逻辑(对于三段论)中。 罗素的《原理》中仍然有它的痕迹,类/集合的识别只有在豪斯多夫的《Grundzuge der Mengenlehre》(1914)之后才消失,参见金森的《空集、单例和有序对》。

Of course, X is not contained in Y if and only if there is something in X which is not in Y. Frege and Peirce understood this truth functional consequence of the proposition/class identification, and made it definitional when transitioning to a logic with quantifiers. For example, Peirce wrote in 1883 (quoted from Dipert, Peirce's Propositional Logic):

当然,当且仅当 X 中存在 Y 中不存在的东西时,X 不包含在 Y 中。弗雷格和皮尔士理解命题/类识别的这一真值函数结果,并在过渡到带有量词的逻辑时将其定义。 例如,Peirce 1883 年写道(引自 DipertPeirce 的命题逻辑):

"To say that an inference is correct is to say that if the premisses are true the conclusion is also true; or that every possible state of things would be included among the possible state of things in which the conclusion would be true. We are thus led to the copula of inclusion ».


Frege's work remained buried until Russell brought it back from obscurity in Principia. The rest (including Peano and, through him, Russell) adopted notation and conventions of Schröder's Algebra of Logic, which followed Peirce, where an equivalent of 𝐴⊃𝐵𝐴∨𝐵

already appears, see Dipert Peirce, Frege, the logic of relations, and Church's theorem. But Peirce used truth tables only sporadically and in unpublished manuscripts (1893 and 1902), so they did not become common until Russell and Wittgenstein reinvented them in 1912.

弗雷格的著作一直被埋没,直到罗素在《原理》中将其从默默无闻中带了回来。 其余的人(包括皮亚诺,以及通过他,拉塞尔)采用了施罗德的逻辑代数的符号和约定,该逻辑代数遵循皮尔士,其中 𝐴⊃𝐵𝐴∨𝐵 的等价物已经出现,参见迪珀特·皮尔士、弗雷格、关系逻辑和丘奇定理。 但皮尔士只是偶尔在未发表的手稿中使用真值表(1893 年和 1902 年),因此直到罗素和维特根斯坦于 1912 年重新发明它们后,它们才变得普遍。

So the material conditional gradually emerged from a cluster of intuitions about propositions, classes and sets. But there are only two cases where it fully applies in its modern form:

因此,实质条件逐渐从一系列关于命题、类和集合的直觉中出现。 但只有两种情况可以充分应用它的现代形式:

  1. Conceptual containment in syllogism (a la Aristotle and Kant). This form is too narrow to cover our intuitive notion of inference.三段论中的概念包含(亚里士多德和康德)。 这种形式太狭窄,无法涵盖我们直观的推理概念

  2. The model-theoretic definition of extensional entailment in modern mathematics, a.k.a. semantic consequence, a la Tarski. This model does not entirely match the intuitive indicative conditional. Hence the cognitive traps:现代数学中外延蕴涵的模型理论定义,又名语义推论(Tarski)。 该模型并不完全符合直观的指示条件。 因此出现了认知陷阱:

"The material conditional allows implications to be true even when the antecedent is irrelevant to the consequent. For example, it's commonly accepted that the sun is made of plasma, on one hand, and that 3 is a prime number, on the other. The standard definition of implication allows us to conclude that, if the sun is made of plasma, then 3 is a prime number. This is arguably synonymous to the following: the sun's being made of plasma renders 3 a prime number. Many people intuitively think that this is false, because the sun and the number three simply have nothing to do with one another…

即使前件与后件无关,实质条件也允许蕴含成立。例如,人们普遍认为,一方面太阳是由等离子体组成的,另一方面 3 是一个素数。 蕴涵的标准定义让我们得出这样的结论:如果太阳是由等离子体构成的,那么 3 就是一个素数。这可以说与以下内容同义:太阳是由等离子体构成的,因此 3 是一个素数。许多人直觉这是错误的,因为太阳和数字 3 根本没有任何关系......

...Another issue is that the material conditional is not designed to deal with counterfactuals and other cases that people often find in if-then reasoning... A further problem is that the material conditional is such that (P ¬P) Q, regardless of what Q is taken to mean. That is, a contradiction implies that absolutely everything is true. »

...另一个问题是实质条件不是为了处理反事实和人们在 if-then 推理中经常发现的其他情况...另一个问题是实质性条件是这样的 (P ØP) Q,无论 Q 的含义是什么。 也就是说,矛盾意味着一切绝对都是真的。

An interesting reconstruction of how truth functional connectives became implicit in the vernacular of mathematical proofs is in Azzouni's paper, pp. 37-38.

Azzouni 的论文第 37-38 页对真值函数连接词在数学证明的白话中如何成为蕴含进行了有趣的重构。


Charles Sanders Peirce is credited with the introduction of truth tables in an unpublished manuscript dated 1893. This includes a truth table for what we now call material implication. A detailed account is provided in the paper Peirce’s Truth-Functional Analysis and the Origin of Truth Tables by I. Anellis.

查尔斯-桑德斯-皮尔斯(Charles Sanders Peirce)在 1893 年的一份未发表手稿中引入了真值表。其中包括我们现在所说的实质蕴涵的真值表。I. Anellis 撰写的论文《皮尔斯的真值函数分析和真值表的起源》中有详细介绍。

Peirce used the term illiation to denote material implication. In his 1880 paper The Algebra of Logic, Peirce explicitly defines illiation as "P implies Q ».

皮尔斯用illiation一词表示实质蕴涵。皮尔斯在其 1880 年的论文《逻辑代数》中明确将illiation 定义为 "P 蕴涵 Q"

A typed manuscript of one of Russell's 1912 lectures features a handwritten truth table for material implication on the verso (in the hand of Wittgenstein) along with a truth table for negation (in Russell's hand).

罗素 1912 年演讲的一份打字手稿的背面有一张手写的实质蕴涵真值表(维特根斯坦手写)和一张否定真值表(罗素手写)。

The definition of material implication 𝑃→𝑄 as ¬𝑃∨𝑄 is found in Russell and Whitehead's Principia Mathematica.

实质蕴涵 𝑃→𝑄 作为 ¬𝑃∨𝑄 的定义见于罗素和怀特海的《数学原理》。

"implies" as used here expresses nothing else than the connection between 𝑝 and 𝑞 also expressed by the disjunction "not-𝑝 or 𝑞" The symbol employed for "𝑝 implies 𝑞 " i.e. for "¬𝑝∨𝑞" is "𝑝⊃𝑞." This symbol may also be read "if 𝑝 then 𝑞. »

这里使用的 "蕴含 "表达的不过是𝑝𝑞之间的联系,也是由 "not-𝑝 or 𝑞"这个析取表达的。用于表示 "𝑝蕴含 𝑞",即𝑝∨ 𝑞"的符号是 "𝑝⊃ 𝑞"。该符号也可理解为 "if 𝑝 then 𝑞. "


The "discoverer" of what we call today material conditional, i.e. the truth-functional definition of "if ... then", is Philo the Dialectician (ca.300 BCE).

我们今天所说的实质条件,即 "如果......那么 « 的真值函数定义的发现者是辩证法家斐洛(约公元前 300 年)。

See Ancient Logic :

A conditional was considered a non-simple proposition composed of two propositions and the connecting particle ‘if’. Philo, who may be credited with introducing truth-functionality into logic, provided the following criterion for their truth: A conditional is false when and only when its antecedent is true and its consequent is false, and it is true in the three remaining truth-value combinations.



See also Benson Mates, Stoic Logic (California UP, 2nd ed.1961), Ch.4 Propositional connectives, page 43.



问:Why is "material implication" called « material »?

为什么 "实质蕴含 "被称为 « 实质 »

Alberto Takase

"Material" highlights that the relationship between P and Q in the notation 𝑃→𝑄 is not causal. For more insight, see https://en.wikipedia.org/wiki/Material_conditional

"实质 "强调符号𝑃→𝑄 P Q 之间的关系不是因果关系。欲知更多详情,请参阅:



The term material implication originated with Bertrand Russell, The Principles of Mathematics (1903); see Part I : Chapter III. Implication and Formal Implication 

实质蕴涵一词起源于伯特兰-罗素(Bertrand Russell)的《数学原理》(1903 年);见第一部分:第三章。蕴涵与形式蕴涵 

It is worth noting that G.Frege, in his groundbraking Begriffsschrift (1879) called the connective symbolizing "if...,then..." : Bedingtheit (tranlated into in English with Conditionality).

值得注意的是,弗雷格(G.Frege)在他的 "groundbraking Begriffsschrift"1879 年)中把 "如果......,那么...... « 称为:Bedingtheit(英文译为条件性)。

Rodrigo de Azevedo

Tarski on material implication:


Alfred Tarski, Introduction to Logic and to the Methodology of Deductive Sciences, Dover, 2013.

The logicians, with due regard for the needs of scientific languages, adopted the same procedure with respect to the phrase "if..., then..." as they had done in the case of the word "or". They decided to simplify and clarify the meaning of this phrase, and to free it from psychological factors. For this purpose they extended the usage of this phrase, considering an implication as a meaningful sentence even if no connection whatsoever exists between its two members, and they made the truth or falsity of an implication dependent exclusively upon the truth or falsity of the antecedent and consequent. To characterize this situation briefly, we say that contemporary logic uses IMPLICATIONS IN MATERIAL MEANING, or simply, MATERIAL IMPLICATIONS; this is opposed to the usage of IMPLICATION IN FORMAL MEANING or FORMAL IMPLICATION, in which case the presence of a certain formal connection between antecedent and consequent is an indispensable condition of the meaningfulness and truth of the implication. The concept of formal implication is not, perhaps, quite clear, but, at any rate, it is narrower than that of material implication; every meaningful and true formal implication is at the same time a meaningful and true material implication, but not vice versa.

逻辑学家适当考虑到科学语言的需要,对短语如果……,那么……”采取了与处理一词相同的程序。 他们决定简化和澄清这句话的含义,并将其从心理因素中解放出来。 为此目的,他们扩展了这个短语的用法,将一个蕴涵视为一个有意义的句子,即使它的两个成员之间不存在任何联系,并且他们使蕴涵的真假完全取决于前件和后件的真假。 为了简单地描述这种情况,我们说当代逻辑使用实质蕴涵意义的蕴涵,或者简称为实质蕴涵; 这与使用形式含义中的蕴涵形式蕴含相反,在这种情况下,前件和后件之间存在某种形式联系是蕴涵的意义和真实性的不可或缺的条件。 形式蕴涵的概念也许不是很清楚,但无论如何,它比实质蕴涵的概念要窄。 每一个有意义和真实的形式含义同时也是有意义和真实的实质蕴涵,但反之则不然。


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