What does "material" mean when one talks of the "material implication"? Why call it "material" implication?
[Math] The meaning of Material in Material Implication
logicterminology
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There is one level at which they can be distinguished. The following definitions are relatively common.
Material implication is a binary connective that can be used to create new sentences; so $\phi \to \psi$ is a compound sentence using the material implication symbol $\to$. Alternatively, in some contexts, material implication is the truth function of this connective.
Logical implication is a relation between two sentences $\phi$ and $\psi$, which says that any model that makes $\phi$ true also makes $\psi$ true. This can be written as $\phi \models \psi$, or sometimes, confusingly, as $\phi \Rightarrow \psi$, although some people use $\Rightarrow$ for material implication.
In this distinction, material implication is a symbol at the object level, while logical implication is a relation at the meta level. In other words, material implication is a function of the truth value of two sentences in one fixed model, but logical implication is not directly about the truth values of sentences in a particular model, it is about the relation between the truth values of the sentences when all models are considered.
There is a close relationship between the two notions in first-order logic. It is somewhat immediate from the definitions that if $\phi \to \psi$ holds in every model then $\phi \models \psi$, and conversely if $\phi \models \psi$ then $\phi \to \psi$ is true in every model. This relationship becomes more fuzzy when we begin to look at other logics, and in particular it can be quite fuzzy when philosophers talk about material conditionals and logical implication independent of any formal system.
"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
Best Answer
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.