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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"?

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3 Answers 3

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

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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:

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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 :

"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.

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  • $\begingroup$ Why do you say Russell is mixing up if/then and logical consequence? The statement you quote doesn't seem to say anything about logical consequence. I always interpreted material implication to be indicating what examples would falsify a statement like "all A are B" in a given domain of discourse; for example, if you find an entity x with property B but not property A, that example is consistent with (doesn't falsify) "all A are B", so it makes sense that A(x) -> B(x) is defined to be true if A(x) is false and B(x) is true. But a statement like "all A are B" needn't mean A logically implies B. $\endgroup$
    – Hypnosifl
    Dec 8, 2019 at 22:39
  • $\begingroup$ @Hypnosifl - see PM: Primitive proposition 1.1 [page 94]: "Anything implied by a true elementary proposition is true." This sounds more as the def of logical consequence [except for the "elementary"]. "The above principle is used whenever we have to deduce a proposition from a proposition." Again : deduce is not "if..., then...". $\endgroup$ Dec 9, 2019 at 8:02
  • $\begingroup$ Why do you think "elementary" excludes the possibility he was just talking about formal implication, not material implication? Assuming elementary proposition means the same thing as what is now more commonly called an atomic sentence, it is possible to derive new propositions from an atomic sentence p, for example p logically implies ~~p, and it logically implies p OR q where q can be any other proposition. $\endgroup$
    – Hypnosifl
    Dec 9, 2019 at 14:53
  • $\begingroup$ @Hypnosifl - correct; the "elementary" stands because for W&R elementary does not mean atomic. See page xvii : "elementary propositions are atomic propositions together with all that can be generated from them by means of the stroke applied any finite number of times." $\endgroup$ Dec 9, 2019 at 15:31
  • $\begingroup$ "the stroke" refers to the NOT symbol "~", correct? What about my example of logically deducing "p OR q" from the atomic sentence p? If "p OR q" is not itself considered an elementary proposition, doesn't this show it's possible that when he said "Anything implied by a true elementary proposition is true" he was talking about logical implication rather than material implication? Also I wonder if that statement could have been intended to cover the possibility of propositions that are logically implied by several true elementary propositions, not necessarily an implication from just one. $\endgroup$
    – Hypnosifl
    Dec 9, 2019 at 16:37

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