In classical logic if either A or B is false then "A and B" is false. But in natural language it's often the case to hear someone say "that's only partially true" or "that's partly false" as a retort to hearing a sequence of statements some of which are false (often followed by an explanation what part was false).
One way to go at this is if there are truth-value gaps in the logic, i.e. a many-valued logic. However, the existence of gaps alone doesn't entail that the conjunction of true and false is necessarily going to be a gap. In fact, many many-valued logics have it that "true and false = false", e.g. that's the case in (strong) Kleene, or LP, or Łukasiewicz's logic. Even in Belnap's four-valued logic, although it has notions of "neither true nor false" and "both true and false", the conjunction of something that's (just) true with something that's (just) false... is again (just) false. In a comment below, someone suggested fuzzy logics, which are continuously truth-valued (over [0, 1]), but from what I can tell the conjunction of 0 and 1 is 0 in this family: the "conjunctive truth function * [...] must be commutative [and have] (0*x)=0".
I see from a 2003 paper of Humberstone, which I've only looked at very briefly that one approach is to have some kind of operator other than the actual truth valuation defined. (This seems to be a kind of "alternative valuation" of sorts. The paper, which is quite long, doesn't seem incredibly cited, so either the problem I'm talking about about is obscure, or Humberstone's proposed solution to is so...)
So have any logics been proposed in which the conjunction of true and false is a gap, rather than false? Or if not, is there any other way to formalize this notion that can capture that a conjunction of true and false is "partly true", besides Humberstone's operator approach, which seems a bit unsatisfactory as it uses a different "scale of truth" for this sort of evaluation than the main truth-valuation used in the logic?
(N.B: I suspect this comes down to some "deeper" issues like what can be reasonably called a conjunction in a logic and what is "false", because in a [semi]lattice with false as the bottom element, if conjunction is taken as the greatest lower bound on the [semi]lattice, it's inescapable that "false and anything = false".)