Sec. 5.3 of the SEP article on constructive and intuitionistic set theories makes note of a property meant for theories that compromise on the LEM:
A theory T has the disjunction property (DP) if whenever T proves (ϕ∨ψ) for sentences ϕ and ψ of L(T), then T proves ϕ or T proves ψ.
What is the point of even having disjunction in such a system? If I am inclined to say, "It's raining or it's not," then I will be thinking that I don't know one way or the other; maybe I'm in a dungeon somewhere, for example, and have no chance of finding out (unless I somehow escape or am released on this very day, or a weather-reporter angel visits me, or whatever). I won't even have much of a "theory" (about what? the weather?) behind my claim; or, then, no theory such that I would believe I could know if it's raining by inference (deductive or not) from said theory.
For all that, I will still be thinking that if it's raining, then it's not not raining, and if it's not raining, then it's not raining. If I knew there was a way to find out (an "intuition" of raining/not-raining) at the same time as I was wondering, then I would have reason to just look to my intuition to tell me. If I didn't have such an intuition (because I'm in a dungeon, etc.), and I thought a disjunction indicated the availability of such an intuition, why would I make a disjunctive statement at all? Generally, I seem able to intend to refer in an exclusively disjunctive way, i.e. my will to refer can disjoin A from ~A in every case, so either I reject the display of my semantic will (in the syntax of disjunction) in the first place, or I just use the word "or" in the usual way.
Caveat: however, I can appreciate the sentiment:
- The absence of a term x is not the same as a term for the absence of x.
Then, modulo questions of trying to quantify over "all numbers" and the like, I understand where predicativists and intuitionists are coming from, even in the way of compromising on the LEM for infinity-talk (the absence of my knowledge of a number with a property P is not equivalent to knowledge that such a number is itself absent from the indeterminate ensemble of absolute infinity). Even so, I would think something like what I think about dialethic logic: even if there are (natural-language) versions of the concept/word "or" (or "not") that aren't exclusive, there is still the exclusive use of those terms regardless, and when I say, "For all assertions A, either A is true or A is not true," I mean by that the exclusion function (whether or not there is some broader notion of quasi-exclusion that intuitionists/dialethicists are working with on their side of things).