Mr B. H. Slater (‘A Grammatical Point About Disjunction’, Philosophy, April 1976) distinguishes between various pairs of questions:
1(a) Did they agree either that they would go to the pictures or that they would go out for a meal?
1(b) Did they agree that either they would go to the pictures or they would go out for a meal?
2(a) Is it known either that Jones was guilty or that Smith was guilty?
2(b) Is it known that either Jones was guilty or Smith was guilty?
3(a) Did he say either that he would get the puzzle solved or that he would eat his hat?
3(b) Did he say that either he would get the puzzle solved or he would eat his hat?
According to Slater, if we can only say ‘Yes’ to 1(b) (or, correspondingly, to 2(b) or 3(b)), then we immediately know what was agreed, namely, that either they would go to the pictures or they would go out for a meal; but if we can just say ‘Yes’ to 1(a) (or, correspondingly, to 2(a) or 3(a)), then we cannot immediately say what was agreed; all we know is that either they agreed to go to the pictures or they agreed to go out for a meal. Or, generalizing somewhat, a ‘Yes’ answer to the question ‘Did they agree that either p or q?’ tells us that they agreed that either p or q; but, with the question ‘Did they agree either that p or that q?’, while ‘Yes, they agreed that p’, and ‘Yes they agreed that q’ are possible answers (and Slater thinks that ‘Yes they agreed either that p or that q’ is also a possible answer), there is no answer, ‘Yes they agreed that r’, which is invariably correct, when ‘Yes’ is correct. In particular, the answer certainly cannot be ‘Yes, they agreed that p or q’ (that is, where ‘r’ = ‘p or q’).
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