 Alternatively, the semantics can be reformulated so that the models feature no such privileged point and truth (in a model) is relativized to a pair of points, one of which plays the ‘real world’ role formerly played by the distinguished element, the other playing the role of the world of at which the formula is being locally evaluated, as in notes 4 and 16 of Davies and Humberstone (1980). General validity is a matter of truth or its preservation relative to arbitrary such pairs in all models, and diagonal validity (as real-world validity is often suggestively called in this setting) attends instead only to pairs whose first and second coordinates coincide. On either style of presentation, the formulation in terms of sequents gives a consequence relation in the obvious way, and for present purposes the following terminology will be convenient. Formulas will be said to be diagonally equivalent when each is a consequence of the other by the consequence relation capturing diagonal (or ‘real-world’) validity and to be unrestrictedly equivalent when this holds instead for the general consequence relation. (We avoid the phrase “generally equivalent”, since it is hard not to hear this as meaning “equivalent, for the most part”; this terminology is adapted from the reference to general consequence on p. 15 of Fusco 2015 as unrestricted consequence.) The current distinction between diagonal and unrestricted equivalence provides a useful formal modeling of Dummett’s well-known distinction between sameness of assertive (or assertoric) content, on the one hand, and sameness of ingredient sense on the  other.[50]  The idea of applying such a contrast to the case of specific pairs of connectives – as with the #, #′ pairs from the start of the preceding paragraph – appears, however, to be new with Fusco (2015), to which we turn presently. The consequence relation in play in the #, #′ discussion above is taken as the diagonal consequence relation (with the associated relation of weak equivalence), according to which these connectives are non-congruential, providing for mutual consequence without synonymy, rather than the finer-grained general consequence relation (associated with unrestricted equivalence). The latter consequence relation is congruential, but at the cost of not directly reflecting the felt a priori equivalence of Aφ with φ.