A logic circuit with a NOT condition
The first thing to do is draw a truth table. This will clarify your thinking and is much easier to understand than a list of bullet points.

Table 1. The required functionality. 'X' is don't care.

Timer   Above low Above high  Humidity
On      temp SP   temp SP     high       Out
-------+---------+----------+----------++-----
 1     | 1       | 1        | X        || 1
 X     | X       | 0        | 1        || 1


This has neatly reduced the problem to two lines. The two lines will require an OR gate to combine the logic.

Note that you don't need to check the LOW TEMP OK signal on line 2 as if if it is over the high temperature setpoint then it must be over the low temperature setpoint.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. A possible solution.

This uses four different types of logic gates. You can use DeMorgan's Theorem and Laws to modify the logic to use fewer gate types.