The law of demand holds that ceteris paribis, a decrease of demand can be caused by an increase in price. However, it is also true that when demand decreases, the price will decrease, it will fall. How can one explain this seeming contradiction? I know it's because of the invisible hand of the market forces, but can I don't know how to specifically explain what is that makes this make sense, and ensures that it is not contradiction with the law of demand. Can someone please explain?
We have a problem of terms here. "Demand" is usually used to refer to the function where for a given price, a given quantity will be demanded. That quantity is referred to as the quantity demanded.
a decrease of demand can be caused by an increase in price.
This is a shift along the demand curve (function). It changes the quantity demanded, but the demand curve itself stays the same. I would say instead:
An increase in price will cause a decrease in the quantity demanded if all else is equal.
This could happen if the supply curve shifted.
However, it is also true that when demand decreases, the price will decrease, it will fall.
This is a shift of the curve itself. I.e. the curve or function changes as a result of a change in demand.
Even if the supply curve is fixed, the intersection could still change because the demand curve moved.
The point is that you are using demand in one place to refer to the curve and in another place you are using it to refer to the quantity demanded. If you use different terms for those two things, it's easier to see that your statements are not in conflict. They are talking about different things.
In an economic equilibrium, supply and demand, as functions of quantity "meet"
When an increase in price occur, as a result of a change in the supply function, for example increase in the cost of labour, it is because the new supply function will meet the same demand function, at a point where supply prices are higher and generally the demanded quantity in such prices is lower
The other case you suggested implies the same exercise on the demand function, for example: the effect of a health trend on fried food. In this case, consumers maxinum prices for a given price would be lower: they would need a bigger incentive to buy