The sorites paradox asks: at what point does a heap stop being a heap? The compact version: let’s take a large heap, remove one grain, then it surely is still a heap. Rinse and repeat. What is the grain count threshold after which a heap isn’t a heap any more? It’s one of those logic puzzles that you should try to solve for the journey, not the destination.
Athanassios Tzouvaras discusses in An observer-based approach to the sorites paradox and the logic derived from that” a formal framework for classifying a set of observations of a changing object (“Flux Object”) over time. The sorites paradox is solved by acknowledging that human perception is not a continuous video feed, but a series of snapshots. The “contradiction” of a vague transition never actually appears in our consciousness because the transition itself is “irreparably elusive”, it always happens during the gaps between our encounters with the world. This ties in with an ongoing interest of mine, namely where the discrete and continuos meet. Although the paper discusses human cognitive capabilities such as attention, it’s not a piece on psychology – it actually presents logical formalism which isn’t even too hard to follow. My takeaway is that there is a gray zone between states that eludes observation: the grass is green on this side of the fence, greener on the other side, but we can’t look under the fence.
Key concepts
The failure of induction: the paradox traditionally arises because if n is “small,” we assume n+1 must also be “small.” Mathematically, this implies a set without a least element, which contradicts the standard properties of natural numbers (N).
Fluxing-Object Semantics (FOS): truth depends on who and when they are looking. Fluxing objects are not mathematical (timeless) objects, but objects that can change between observations. FOS introduces three variables:
– Structure (A): a set of “object forms” (snapshots of an object)
– Agents (I): a set of observers (you, me, somebody else)
– Time (N): Represented as discrete moments (t, t+1, etc.)
Fluxing objects are partial functions: objects are represented as functions of time and observer. Crucially, these functions are partial, meaning there are gaps where they are undefined.
f: IxN -> A
The “observation gap” solution: transitions happen between intervals of time where the observer is not observing the object.
The paper makes use of three-valued logic for an observer’s observation at a point in time:
– true: the observer observes a property of the object as true
– false: the observer observes a property of the object as false
– undetermined: there is no observation
Imperceptibly changing objects
An object is “imperceptibly changing” if an observer who watches the object at two consecutive moments their observation of a property cannot change. The transition from “not bald” to “bald” cannot happen while you are looking. It can only occur during a watching gap (when you blink or turn away). This explains why we never “see” the exact moment a heap becomes a non-heap; our brain requires a gap in observation to reset the classification.
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