Commonsense and Non-Monotonic Reasoning

Classical first-order logic has a property called monotonicity: if a conclusion follows from a set of premises, it still follows from any larger set of premises that includes the original ones, since adding more true statements can only ever add more valid derivations, never remove one. This is exactly the right property for mathematical proof, but it is the wrong property for reasoning about an incompletely known, changing world, where the whole point of a default assumption — "birds fly," "cars have four wheels," "meetings start on time" — is that it should be retracted the instant a specific exception becomes known, as when the bird turns out to be a penguin. A reasoning system whose conclusion set can shrink as well as grow when new premises are added is called non-monotonic, and formalizing this behavior rigorously, without simply throwing away the guarantees that make logic reliable in the first place, became a central technical problem in knowledge representation from the late 1970s onward.

Ray Reiter's default logic gives one influential formal answer. A default rule has the schematic form "given prerequisite α, if justification β is consistent with what is currently believed, conclude consequent w," most commonly in the normal form where the justification and consequent coincide — capturing exactly the "birds fly unless something says otherwise" pattern. Applying a set of such default rules to a set of facts does not always yield a single set of conclusions; instead it yields one or more extensions, each a maximal, internally consistent set of beliefs that could result from applying the defaults in some consistent order, and different orders or different available exceptions can produce genuinely different extensions from the same starting theory. This is itself a meaningful modeling choice, not a flaw: when commonsense information genuinely supports more than one coherent way of filling in the gaps, a formalism that can represent multiple extensions is capturing a real ambiguity rather than papering over it.

John McCarthy's circumscription offers a different formal route to the same kind of default behavior. Rather than adding special default-rule syntax, circumscription works by minimizing the extension of chosen predicates — informally, assuming that the only objects known to have a property are the objects that actually have it, so nothing is assumed to be exceptional unless the theory is forced to say so. McCarthy developed circumscription specifically to address classic difficulties in representing commonsense knowledge about action and change, including the qualification problem, the difficulty of listing every possible precondition an action might require, and the closely related frame problem, the difficulty of stating everything that stays the same when an action occurs, since both problems become far more tractable if a reasoner can simply assume normality and abnormal exceptions are minimized by default rather than enumerated in full. Circumscription and default logic thus approach the same target behavior from different formal directions — one through a semantic minimization principle applied to predicates, the other through explicit syntactic default rules and the extensions they generate — and comparing where the two formalisms agree and where they diverge on a given example is itself a standard way researchers probe what any nonmonotonic formalism actually commits a reasoner to.

Default reasoning explains how a system can jump to a plausible conclusion; belief revision explains how it should update an entire body of prior beliefs when a new piece of information arrives that conflicts with something already believed. The foundational formal treatment, developed by Carlos Alchourrón, Peter Gärdenfors, and David Makinson and now generally called AGM theory after its authors' initials, frames belief change around three operations on a set of beliefs treated as a logically closed theory: expansion, simply adding a new belief without worrying about consistency; contraction, removing a belief (and whatever depends on it) so that it is no longer part of the theory; and revision, adding a new belief that may contradict existing ones, which is handled by first contracting whatever conflicts with the new information and only then expanding to include it. Framing belief change this way lets the same underlying question — which beliefs must give way when the world turns out to be different from what was assumed — be studied with the same logical precision classical logic brings to fixed, unchanging theories, rather than being left as an informal, ad hoc bookkeeping problem.

Because many different subsets of a belief set could be given up to restore consistency with new information, AGM theory does not settle for just any way of doing it — it specifies postulates that any rational contraction or revision operation should satisfy, including a minimal-change requirement often described informally as the principle of informational economy: give up as little of the prior belief set as necessary to accommodate the new information and restore consistency, rather than discarding beliefs that were never actually in conflict. The theory's technical device for capturing "as little as necessary" is the partial meet contraction function, which selects, among all the maximal subsets of the original beliefs that fail to imply the belief being removed, an intersection of the most epistemically preferable ones, and the original paper proves that functions built this way are exactly characterized by (are equivalent to satisfying) the rationality postulates the authors set out.

Belief revision and default reasoning address closely related but distinct problems: default reasoning is about which tentative conclusions to draw from incomplete information in the first place, while belief revision is about how to consistently update a standing set of beliefs once new, possibly conflicting information arrives — and a full account of commonsense reasoning over time needs both, since a system that draws sensible defaults but has no principled way to retract them when contradicted is only half-solved, and a system that can revise beliefs consistently but has no way to generate a default conclusion to revise later has nothing to start from. It is worth distinguishing revision from simple contraction alone: revision must both make room for the new belief by removing whatever conflicts with it and then actually incorporate that new belief, whereas contraction only removes, which is why AGM theory treats revision as a composite operation built out of the other two rather than defining it independently from scratch.

Commonsense reasoning routinely involves time — knowing not just that two events happened, but how they relate to one another (before, during, overlapping, immediately after) even when exact clock times are unavailable or irrelevant. James Allen's interval algebra formalized this by defining a small, complete set of thirteen possible qualitative relations that can hold between two time intervals, such as one interval occurring before another, one meeting another end-to-end, one overlapping another, or one occurring during another, and Allen showed how constraints stated in terms of these relations could be propagated across a whole network of events to detect inconsistencies or infer relations that were never stated directly. This qualitative approach lets a reasoning system represent and combine partial temporal knowledge — "the meeting was after lunch but before the flight" — without ever committing to exact timestamps that were never actually known, which is precisely the kind of representation commonsense temporal reasoning requires.

Reasoning under incomplete knowledge more broadly is what ties non-monotonicity, defaults, and belief revision together as a single coherent problem area rather than three unrelated topics: an agent operating in the real world essentially never has a complete description of its situation, must act on the best available default assumptions anyway, and must be prepared to revise those assumptions in a principled way as new information arrives. AIMA's treatment of default and nonmonotonic reasoning situates truth maintenance systems within this same picture: rather than only computing which extension or revision is correct after the fact, a truth maintenance system tracks the dependencies between beliefs as they are derived, so that when a supporting belief is later retracted, every conclusion that depended on it can be identified and withdrawn automatically, without having to re-derive the entire belief set from scratch. That dependency bookkeeping is what makes non-monotonic reasoning practical at any real scale: recomputing an entire extension or belief set from first principles every time one fact changes would be far too costly for a system managing thousands of interdependent default conclusions at once.

Non-monotonic and commonsense reasoning remain genuinely open research areas rather than settled engineering practice: circumscription, default logic, and AGM belief revision each solve their target problem in ways that do not fully agree with one another when applied to the same tricky examples (the literature contains many small scenarios, such as competing default rules yielding contradictory conclusions, where different formalisms diverge), computing extensions or performing minimal revision is computationally expensive in the general case, and connecting any of these symbolic formalisms cleanly to modern statistical and neural approaches to handling uncertainty is an active area of inquiry rather than a solved integration. A student encountering any single formalism in isolation should therefore resist treating it as the settled account of commonsense reasoning; the honest picture is a family of formal tools, each capturing part of how people handle incomplete, changing, and temporally structured knowledge, with the boundaries between them still an open subject of study.