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 holds in every model then , and conversely if then 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.[1]

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— Mathematics – Stack Exchange

— Oct 1 ’11 at 2:52

— Carl Mummert

[1] should be replaced by ` \phi \models \psi`

in LaTeX code.

2013.09.18 Wednesday ACHK