I surely start with the simple understanding of multiplication in mathematics which would say -*-= + (negative*negative= positive). But again if you multiply one more negative, it becomes negative again and so on. So even in mathematics, negatives can give rise to positives. Leaving the stray thought aside, I will discuss some of the negative ideas, words to know their meaning.
We have a school of philosophy in Indian philosophy called advaita. There are systems like monism and dualism. But advaita is non-dualism, which is not monism (logically opposite of dualism is monism). A negative prefix ‘a’ was added to dvaita (non to dualism) to give it a richer meaning. This meaning is more than just being the opposite of dualism. Heidegger would use the prefix un-concealing which is not the opposite of concealing, but the combination of concealing and revealing. In many cases, negative prefix would say what it is not, but leaving many other possibilities open, given a richer meaning than what the positive term could mean. (I am not saying all negatives do give such a flavour).
Recently we had some presentations, and a friend was presenting on some aspects of the ten commandments (especially the prohibitions like don’t kill). Surely they are negative statements. But when it definitely cuts off some possibilities, it definitely leaves many possibilities open. It requires greater sensibility to see what it opens up rather than focusing on what it prohibits.
Heidegger would speak about the absence and presence. Absence is normally considered a negative term, and we give importance to the present and presence. Heidegger would say, absensing in the default mode of divine presencing. The absence is no longer an absence of divine, but a with-holding presence.
And probably my favourite is the Nietzsche’s proclamation of the ‘death of God’. From a believer’s (and specially christian, as he is attacking Christians) perspective, this is a negative statement. But when he proclaims the death of the false images of God, there is a hidden cry for the true picture of God. Some would say, Nietzsche has done great service to the church than many theologians by proclaiming death of God (which was a prophetic voice against the false images and understandings of God and human beings).
Thus looking at terms/concepts (especially logically negative ones) with an openness can help us to open up many new avenues of meanings for us. It would open up many more possibilities.