# “实质蕴涵”术语溯源

https://hsm.stackexchange.com/questions/10978/where-does-the-material-implication-come-from-if-not-from-george-boole

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

𝐹→𝐹=𝑇

𝐹→𝑇=𝑇

𝑇→𝐹=𝐹

𝑇→𝑇=𝑇

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?

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.

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.

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.

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):

"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.

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:

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…

...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。

NWR :

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.

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 ».

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).

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

"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. »

Mauro ALLEGRANZA

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).

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.

https://math.stackexchange.com/questions/3098321/why-is-material-implication-called-material

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：

https://en.wikipedia.org/wiki/Material_conditional

Mauro ALLEGRANZA

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

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).

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.

https://m.sciencenet.cn/blog-2322490-1411471.html

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