An imagined event, an event allowed by a physical law and an event reachable from a particular state are different kinds of possibility. Keeping them separate turns a broad philosophical question into an inquiry with explicit premises.
Possible relative to what?
In section IV of the Enquiry Concerning Human Understanding, Hume distinguishes relations of ideas from matters of fact. His example of tomorrow's sunrise challenges the inference from an expectation about the world to a logical demonstration that its contrary is contradictory.
The useful reading question is not whether regularities are worthless. It is what kind of support a claim possesses. A consequence of definitions, a consequence of a selected law and a well-supported empirical expectation do not acquire the same status merely because each feels secure.
When someone says that a state is impossible, ask what has been held fixed: logic, a mathematical model, measured parameters, initial conditions, available interventions or a finite observation period. Changing one of these can change the answer without producing a contradiction.
Eight states, three rules
The demonstration uses three binary entries, so its state labels run from 000 to 111. These are formal configurations, not particles or physical space. Choose a starting state and one of three stipulated transition rules.
The first rule flips exactly one of the three bits at each step. The second flips exactly two. The third advances the numerical label by one modulo eight: 000 goes to 001, and 111 returns to 000. The circular drawing is a layout for the graph; it is not a spatial geometry or an additional transition rule.
Compare states reachable after exactly n steps with those reachable in at most n steps. At zero steps, both sets contain only the starting state. A longer allowed duration can enlarge the second set, while the exact-step set can change without containing its predecessor.
Which possibilities does a rule allow?
Compare exact-step endpoints with states reachable within the selected number of steps. Counts describe allowed paths, without assigning probabilities.
Green: exact endpointsGold: reached earlier onlyPurple ring: starting state
Read endpoint counts
An invariant excludes possibilities
Define parity as whether the number of 1 entries is even or odd. Flipping exactly two bits preserves parity. Starting at 000, only 000,011,101,110 can ever be reached under that rule. The other four labels are perfectly valid configurations, but they are dynamically inaccessible from this start.
Flipping one bit reverses parity at every step. At exactly two steps from 000, the possible endpoints are 000,011,101,110. There are nine distinct two-step paths, but only four endpoints. In at most two steps, seven states are reachable; 111 first requires three flips.
The directed cycle has one successor per state. After n steps it has one exact endpoint, while the number visited within n steps is min(n+1,8). Thus a deterministic rule can eventually visit every state, and a branching rule can remain confined to a smaller sector.
Possibility is not probability
The table counts allowed paths ending at each state. It does not assign probabilities. Treating each outgoing option as equally likely would add a stochastic rule that is absent here. A physical system might favour one transition, depend on a control setting or require information missing from the three-bit state.
This distinction also protects the interpretation of a null result. Failing to observe 111 after two one-bit steps agrees with the model's exact exclusion. Failing to observe another permitted state in a small sample would have a different meaning, and its assessment would require a probability model.
From conditional necessity to physics
Parity conservation is necessary given the two-bit rule. Its proof does not establish that nature must use that rule. The mathematical conclusion and the physical selection of premises are separate achievements.
This is directly relevant to SMM/DTT. The book's account of its conditional results distinguishes explicit constructions from the further task of explaining why their assumptions hold together. A proposed source law becomes more informative when it excludes alternatives before their observed consequences are used to choose it.
Use tests with explicit alternatives to continue. Ask which histories each candidate law permits, what observation distinguishes them and which parameters were fixed independently. A compelling necessity claim states its domain instead of allowing the word necessary to carry an unstated assumption.
Sources and further reading
- David Hume, An Enquiry Concerning Human Understanding, section IV, paragraphs 20–22, Selby-Bigge edition.
- Mohamed Haj Yousef, What this edition establishes. The eight-state graph is an original finite teaching construction.