The key thing is that it's a double-pole, double-throw switch. You can put any number of them in series, and the circuit essentially counts the parity of the switches, so flipping any of them toggles the state of the lights.

If it's only two switches, you only need SPDTs ('hot' & 'light' on each, throws connected, then toggling either switch takes either to the other).

That's what a 3-way or 4-way (should be called X-way) switch setup is - three or four or more switches that no matter what the state of the light is, flipping ANY switch will change the state.

https://www.electrical101.com/4way-switch-wiring-using-nm-ca...

Sorry then, I misunderstood what you meant by:

> a path is always found on one flip of the switch

I thought you were taking the comment you replied to to be seeking a simple parallel arrangement of SPST switches, so that turning any one on would turn the light on.

So I was explaining that No, point is you want 'off' on any switch (regardless of the state of the others) too. Seems you understood that though. :)

yes... A way to look at 2 way switches is to look at the connections of usual circuit setups to Boolean logic. Taking two on-off switches,

- parallel is or

- series is and

We can see that a 2 way switch is xor (or xnor depending on the setup).

So now, for 3 way it feels like you have to beyond 2 state (on-off) switches... i don't have a proof of impossibility that 2 state switches cannot do 3 way switching, but in practice they indeed involve more states (multi-throw etc.)

The closest you can get with '2 state' (by which you mean SPST - single pole single throw; SPDT is the standard way to achieve this but is usually still '2 state', it's just that 'off' (if you like) has a terminal too) switches is to add a 'don't care' state, and accept that first deliberate ('do care') switch wins.

To do this you have a live rail and a switched live rail, and two SPST each end: one switch SL hot or not; the other to break its connection to the light. Open the latter for 'off', close both for 'on', close break & open SL for 'don't care'. (Then in the 'don't care' state, SL is hot iff switched on the other end, and if so is also closed through to the light.)

Not legal, not (that I can imagine) desirable!

Proving it is actually simple if you continue your Boolean line of thought: you can't build XOR without an inverter; an SPST switch is only a sort of 'half inverter', a mux between '0' and its input. Since it can never give you '1' from '0' input, you need a schematic for your XOR where each constituent truth table in isolation has '0' out for its '0;0' in line. Which can't be done, because clearly then you could never get '0;0=>1' overall.