# The meaning of Material in Material Implication

What does "material" mean when one talks of the "material implication"? Why call it "material" implication?

The term material comes from references to Alfred North Whitehead and Bertrand Russell's work (Russell, B. (1963). Principia Mathematica Volume 1. Cambridge, At the University Press.) He used terms such as molecular, elementary and atomic statements to describe structures in logic. The term which refers to statements that are at the bottom is atomic statements. They are what you would obtain if you kept expanding a statement until you could expand it no more. You have reached statements which have a value of either true or false. An example may help:

Let say we have a few statements A and B. If A and B are atomic and we choose to rename the statement A AND B to C, we call C a molecular statement. C is labeled such because it is made up of two different atomic statements.

If we have a different statement, say A|A , we may rename this to D. In this case D would be an elementary statement because only 1 type of atomic statement is used to make it up.

A material statement is used to describe anything made up of molecular or elementary statements.

Because atomic statements all have truth values; this means that a material statement has a truth value. They can be put onto truth tables. A material implication is an "if" statement that is made out of statements with truth values. This contrasts to a logical implication which is made out of other mathematical structures. These structures may not have truth values of their own.

• The reference to Principia is correct, but I think that R&W nowhere in it speak of "material statements". – Mauro ALLEGRANZA Jan 1 '17 at 9:51

Dictionary.com says that the origin of "material implication" is ca. 1900.

Thus a good answer to the question why call it material implication is quite likely somewhere in the works of

• Charles Sanders Peirce
• Bertrand Russell
• Ludwig Wittgenstein

(Particularly we should take a close look at early terminology of Peirce, Russell, Wittgenstein.)

A starting point to get to the answer might be arXiv:1108.2429 by Irving H. Anellis (2011)

The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883-84 in connection with the composition of Peirce’s "On the Algebra of Logic: A Contribution to the Philosophy of Notation" that appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truth table for the conditional.

• 1903 B. Russell Princ. Math. ii. 14 How far formal implication is definable in terms of implication simply, or material implication as it may be called, is a difficult question. — the online OED – bof Jan 1 '17 at 2:20
• The origin of truth tables does not explain the R&W use of "material implication". – Mauro ALLEGRANZA Jan 1 '17 at 9:52
• Source : Principles : Chapter III. Implication and Formal Implication. – Mauro ALLEGRANZA Jan 1 '17 at 9:53
• Source : Principia, page 7 : "When it is necessary explicitly to discriminate "implication" [i.e. "if $p$, then $q$" ] from "formal implication," it is called "material implication." – Mauro ALLEGRANZA Jan 1 '17 at 9:55
• Source : Principia, page 20 : "When an implication, say $\phi x . \supset . \psi x$, is said to hold always, i.e. when $(x) : \phi x . \supset . \psi x$, we shall say that $\phi x$ formally implies $\psi x$". – Mauro ALLEGRANZA Jan 1 '17 at 9:59

The adjective " material" in " material implication" refers to the content (i.e. "matter") , that is to the meaning, and more precisely to the denotation (reduced to truth value since Frege) of the propositions involved. It is distinguished from "logical implication" which, strictly speaking, is purely formal, that is, holds whatever the truth values of the atomic propositions involved might be.

Remark : in ancient philosophy , " content" was identified to "matter" and "matter" was opposed to " form"

Remark : according to Frege the "meaning" of a proposition has two aspects, (1) the " sense" ( conceptual aspect) and the (2) " denotation" ( or reference, which Frege identifies to truth-value). "1²+ sqrt(1)= sqrt(4)" has the same denotation as " 1+1=2" ( that is, the truth value True) , but the two propositions don't have the same sense ( they do not express the same thought).

Remark: the terms "material" and " formal" were used in medieval logic to distinguish two kinds of " consequences". But the two distinctions are not equivalent, since in medieval logic, the relation involved in a " consequance" was always a necessary one. In modern logic, a material implication is by itself CONTINGENT (depending on the contingent truth values of the propositions) while logical implication is a NECESSARY relation.

Reference: On this topic see: Lipschutz, Schaum's Outline Of Set theory and Related Topics, chapter 14. - Available at Archive.org.

Example. Suppose someone asks you " Does A imply (materially) (A&B)"? You wouldn't know what annswer to give, and would say " that depends on the truth values A and B are given in the case you are considering". Suppose now someone asks you: " Does (A&B) logically imply A?". Here, without hesitation, and whithout even knowing the truth values of A and of B, you would answer " Yes!".

Example : Lets's say that A is the proposition " America is a democracy in 2019" and that B is the proposition " D. Trump is the President of the USA in 2019".

Since both A and B are true in the actual world ( although one of them could logically be false, or both), it is actually the case that A MATERIALLY implies B. ( This can be seen on the truth table of material implication). The only case in which A would not imply ( materially) B is the case in which A would be true while B would be false. But notice that this is contingent: it could logically be the case that the first were true, and the second false.

Now, although A materially implies B, A does not LOGICALLY imply B. Logical implication is (1) material implication (2) WITH necessity. Lets's use the symbol ==> for logical implication, and the symbol --> for material implication.

             [ A ==> B ] if and only if [NECESSARILY  ( A-->B)]


that is

      [ A ==> B] if and only if [ there is no logically possible case,
in which A is True and B is False]


Saying that " USA is a democracy (in 2019)" logically implies that "D. trump is President of the USA in 2019" means: " Necessarily, if America is a democracy, then D. Trump is the President of the USA", which is absolutely false. There is no logical impossibility in " America is a democracy and D. trump is NOT the President of the USA".

In summary : A materially implies B , just if ( contingently) we are not in a case where A is true and B is false. Material implication depends on the truth values propositions have ( contingently) in the actual world.

A logically implies B if and only if there is no logically possible case ( no possible world as logicians/philosophers say , or, no possible "interpretation") in which A is true and B is false. A logical implication is an implication that has truth-value ' T ' on every row of it's truth table.

Remark : there is a weaker sense of "logical implication": necessary implication that, however, is not formal. for example I can correctly say that " John is a pianist" logically implies " John is a musician". The implication is necessary since there is no possible case in which a person is a pianist without being a musician. But this is not pure ( formal) logical implication, since the necessity depends on the meaning of the terms " pianist" and " musician". It is not true that any proposition of the form " b is a P" logically implies any proposition of the form " b is a M".