Cfrgtkky

Why is "material implication" called "material"? What does the word "material" imply or underline?

It seems that the term "material condition" even preferred over the term "implication", preferred also by Wikipedia.

Why? What is so special about "material" versus just "implication"?

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

Only historical origins... in fact, there is no "immaterial" implication.

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

• Two kinds of implication, the material and the formal.

See in Whitehead and Russell Principia Mathematica the "horseshoe" ($⊃$) notation.

In the "material" case it is used as a connective between propositions :

*1.2 $ ⊢ : p lor p . ⊃ . p$,

while in the "formal" usage it is a relation between propositional functions (the symbolic counterparts of classes) :

*10·02 $ φx ⊃_x ψx . = . (x). φx ⊃ ψx$.

While "implication" for "conditional" ?

Again, see :

• Alfred North Whitehead & Bertrand Russell, Principia Mathematica to *56 (2nd ed - 1927), page 7 :

"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 $q$."

Unfortunately, Russell is mixing here two concepts : the connective "if..., then..." and the relation of (logical) consequence (in this, following his "maestro" : Giuseppe Peano, that introduced the symbol $a ⊃ b$ reading it (1889) as "deducitur").

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).

See also Implication and Modal Logic.

Tarski on material implication:

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.

Source:

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