Full Idea
If it were possible to have A true, B false, and If A,B true, it would be unsafe to infer B from A and If A,B: modus ponens would thus be invalid. Hence 'If A,B' must entail ¬(A & ¬B).
Gist of Idea
'If A,B' must entail ¬(A & ¬B); otherwise we could have A true, B false, and If A,B true, invalidating modus ponens
Source
Dorothy Edgington (Conditionals [2001], 17.1)
Book Reference
'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.387
A Reaction
This is a firm defence of part of the truth-functional view of conditionals, and seems unassailable. The other parts of the truth table are open to question, though, if A is false, or they are both true.
Related Idea
Idea 13764 Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? [Edgington]