Philosphical Logic 101 (aka Propositional Logic) and Digital Logic 101 are both based on Boolean Algebra.
In this example you have inverted both inputs and outputs around an AND gate which one of the basic reciprocal properties converting between AND and OR logic. In this case if either input is 1 the output is 1. or in other words, if both inputs are 0 the output is 0. (where 0 = low = false)
- Let one input be = The Sun is shining
- Let the other input = I have a powered flashlight
- Therefore (assuming one's eyesight is OK and your Logic has power amd there are no other inputs etc.)
- the output = I can see =1 if either A or B is true =1
- The de Morgan Rule is thus I can NOT see if both A AND B are off (=NOT on)
http://en.wikipedia.org/wiki/Boolean_algebra#Basic_operations This will explain more.
What we call De Morgan's Laws are the same as Boolean Logic but with different symbology with some simplification. These were later converted by Veitch to Tables with graphical simplification using an intuititve circle mapping method to reduce complex logic with a simpler realization. However the graphical presentation was not as obvious to the reader in 1952 and in 1953 Karnaugh published the mapping methods used by Veitch and got credit for it until a few decades later. So Karnaugh-Veitch Maps ought to be the proper designation.
All of these Laws of Logic above and Propositional Logic are a subset of the original laws of Logic defined by Aristotle and later amended by Plato and simplified for the purpose of easier understanding. They start with the simple truths of OR, AND and NOT logic. Aristotle documented them but the other's adapted it to Set theory, Electronics design, and the rapid solving of Sudoku puzzles among other things.
The design of Exclusive OR gates, counters, adders, multiplexors and PGA's are all based on these laws with the addition of memory states.
FWIW Philosphical vs Religious Logic ( not )
Now there are beliefs that logic can be extended beyond Laws of proof to experience based results and some religions have called these as truths based on experience and not proof. Although it is fair to call this a belief, it cannot be proven based on Aristotalean Logic. Just beware they exist and you cannot debate this with them as they have different laws of logic. They attempt to "raise the strength of analogy to that of a first order Aristotelian syllogism" This would be like adding to the truth table an analogy and given that the same validity as a proven law which can lead to illogical results. Like adding a hidden term in the Gate saying the Output could also be 1 if one saw the Illuminous One and there was no sun or flashlight. It is only true if you believe it is true but beliefs are not logic as logic is based on proveable assumptions and simple infallible logic of AND, OR and NOT. However this does lead into an interesting topic of Fuzzy Logic which are Rules based on Experience.