There is a 'no mans land' region between the lower and upper thresholds in a Schmitt device. The threshold is adjusted according to the present logic output state so you need to go below the lower threshold to make the output high and it will stay that way until you go above the upper threshold. The same applies the other way around, you have to go beyond the other threshold for the output to change.
A non-Schmitt inverter would have an abrupt state change at a single threshold, above it produces low output and below it produces a high output, the threshold satays constant though.
This is exactly why Schmitt inputs are used, for example, if the threshold was 2V and the input was 1.99999V the output would theoretically be high and at 2.00001V it would be low but in real life the electrical noise would randomly make it change state. By having two thresholds, the noise immunity is greatly improved.
The 'floating input' voltage on a logic gate means nothing, it is just leakage through the internal circuits. Because it is an input, you have to actively provide it with a voltage or current source from outside.
Brian.