All-or-nothing-law: law or general principle?
I would argue the all-or-none principle is a rule of thumb. It is generally true if voltage hits a certain threshold then there will be a action potential with the same amplitude regardless the strength of the stimuli. But as you point out with the logic gate example, the threshold is notoriously hard to define mathematically. Furthermore Action Potentials don't always have the same amplitude, but as I will argue that doesn't matter all that much to the function of the neuron.

As most rules in biology we can find exceptions. I'll list some phenomena that confound the all or none principle. 


Accommodation is the prime example that not all action potentials have the same voltage amplitude. Accommodation is the when the amplitude of the voltage of successive action potentials drops, and may even stop firing (Depolarization Block). This is usually the result of a too strong of a stimulating current, and occurs under pathological conditions.   
Resonating Neurons These neurons fire when a certain frequency of input is achieved and can even fire when hyperpolarizing (negative) currents are injected. The reason I bring this particular neuron  type up is it shows that thresholds are not necessarily more depolarized (positive) than rest. They are complex manifolds (or shapes) relating to the gating variables of Ion channels in the membrane.





Another thing to note, is even though amplitudes of Action potential can change, the difference of a few millivolts in amplitude will not make a difference to the synapse and in that sense the all-or-none principle holds. Recall that voltage gated Calcium entry is required for a synapse to release it's neurotransmitter. Much like the logic gate, it will only enter at sufficiently depolarized (positive values) but the more positive the voltage, does not correspond to more Calcium entry. In other words it saturates at high voltages. In this way, to cause a stronger response at the synapse the neuron increases the firing frequency to stay "on average" depolarized longer. Thus more Calcium entry and thus more neurotransmitter release.