Formal implication is usually represented by symbolization such as ‘(x) φx ⊃ Ψx,’ which may be read, “for all values of ‘x’, φx (materially) implies Ψx.” If the values of the variable ‘x’, in ‘φx’ and ‘Ψx’ be ‘x1’ ‘x2’ ‘x3’, etc., then … ‘φx’ formally implies ‘Ψx’ if and only if, whatever values of ‘x’, ‘xn’, be chosen, ‘φxn’ materially implies ‘Ψxn’ …
However, this still leaves it doubtful which of two possible interpretations of expressions having the form ‘(x) φx ⊃ Ψx’ is to be taken as correct. … It means one thing to say, “Every existent having the property φ … has also the property Ψ,” and it means quite a different thing to say, “Every thinkable thing which should have the property φ must also have the property Ψ.” The second of these holds only when having the properly φ logically entails having the property Ψ; when ‘Ψx’ is deductible from ‘φx’. … The first of them, however, holds not only in such cases … but also in every case where among existent things, one property is universally accompanied by another. (C. I. Lewis, An Analysis of Knowledge and Evaluation, pp. 217–8. I am responsible for the italicizing of the sentence, otherwise the italics follow Lewis.)