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

$$F \rightarrow F = T$$

$$F \rightarrow T = T$$

$$T \rightarrow F = F$$

$$T \rightarrow T = T$$

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?

there are a series of cognitive traps for students. First is that we have taught implication as a proposition that could be true or false, but we are now using it as an unfalsifiable predicate, so we would not even consider the entry in the truth table in which $A$ implies $B$ is false.

What I think happened

If we create a table of all of the possible truth results from two inputs (using 1 for true and 0 for false), we get:

all possible values

Using conjunction, exclusive and inclusive disjunction, not, and tautology/contradiction, we can label these thus:

enter image description here

This leaves us with the four slots in which three of the four answers are identical, and one of the two middle answers is the only different answer.

What I believe happened is that someone essentially looked at this table, realized that those four could be reduced to one asymmetric operation, and looked back to Aristotle or Philo and chose $\rightarrow$ and "implies" to fill in this relationship.

enter image description here

In essence, then, implication is a purely axiomatic relationship, and all of the "intuitive" (but not really) ways of thinking about it can be used as a good way to think about it, but not what it truly is.

However, George Boole did not address this in his treatises as far as I can see. So, where did we get the modern relationship that we now call "implies"?


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 $\supset$ (1890), and Peano's Ↄ. Later Peano, and after him Russell, adopted Schröder's $\supset$ (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.

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 $A\supset B=\lnot A \lor B$ 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.
  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:

"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."

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

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  • $\begingroup$ Interesting. Are you saying that Aristotle's syllogisms fully capture implication, or did Aristotle discuss the notions of intension and extension more generally. $\endgroup$ – Nick Sep 12 '19 at 0:55
  • $\begingroup$ @NickR Syllogisms do not capture the modern material conditional simply because of weak expressive power, they do not even capture propositional logic that Stoics already had. Interestingly, Stoic implication was closer to the relevance conditional than to the material one, which suggests that conceptual containment intuition does not match class containment beyond the syllogistic. But Aristotle was dogmatized during the middle ages, and later symbologists, like Leibniz and de Morgan, were drawn to the neat algebra of class containment that relevance conditionals lack. $\endgroup$ – Conifold Sep 12 '19 at 4:01
  • $\begingroup$ Another problem is that there is no neat correspondence between predicates and classes once we move beyond Aristotle's one-place predicates. De Morgan already introduced the logic of relations (multi-place predicates), and Peirce and Frege quickly realized that one has to switch to truth-functional (i.e. extensional) definition to keep the algebra going under the quantifiers. $\endgroup$ – Conifold Sep 12 '19 at 4:10
  • $\begingroup$ Thanks for your clear account. I was actually thinking that the relevantism so plain in Aristotle was a major hurdle to overcome, so it's encouraging that you make this point. $\endgroup$ – Nick Sep 12 '19 at 4:50
  • $\begingroup$ This is an amazing answer. What I really need to understand is packed up in your last two sentences. "Of them, conceptual containment in the syllogistic (a la Aristotle and Kant) is too narrow to cover our intuitive notion of inference, and the purely extensional entailment in modern mathematics (a la Tarski) simply does not match it. Hence the cognitive traps." Can you expand on this part a bit? $\endgroup$ – Ben I. Sep 12 '19 at 12:57

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.

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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 $P \rightarrow Q$ as $\lnot P \lor Q$ 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 "$\text {not-}p \text { or } q$" The symbol employed for "$p$ implies $q$" i.e. for "$\lnot p \lor q$" is "$p ⊃ q$." This symbol may also be read "if $p$, then $q$."

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Here we find a quote from Dorothy Edgington, The Stanford Encyclopedia of Philosophy, "Conditionals":

The truth-functional theory of the conditional was integral to Frege's new logic (1879). It was taken up enthusiastically by Russell (who called it "material implication"), Wittgenstein in the Tractatus, and the logical positivists, and it is now found in every logic text.

The date for Boole's Laws of Thought is 1854.

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