 Suppose one wants to say more about the weather than that it rained on a particular Saturday. One wants to strengthen one’s commitments so that they include not only rain on Saturday but also rain on the following Sunday. With ∧ (assuming some background ∧-classical consequence relation) there is nothing easier. Abbreviating “It rained on the Saturday in question” to Sat and “It rained on Sunday” to Sun, one can convert one’s original statement into one making the stronger claim by passing from Sat to Sat ∧ Sun. Now suppose one wanted a similar device but working in the reverse direction. One starts with a claim whose content is given by Sat ∧ Sun and wants to modify this using a connective which removes commitments in much the way that ∧ adds them, and using this connective, often called logical subtraction (though a case could be made for ‘logical division’ in view of the usual association of conjunction with multiplication), one wants to take away the commitment to rain on Sunday, leaving only the original claim. That is, one would like a connective added to the language of one’s favoured consequence relation, in such a way that the added connective was governed by rules making the extension conservative, but for which one had the equivalence (writing ⊢) for the extended consequence relation: