I want to know if proving the existence of something by using only logic is possible.
No, it general it can’t.
To show that something exists one has to either find it, like Mount Everest or to construct it, like an aeroplane.
Some people make an exception for mathematics arguing that anything that can be defined consistently neccesarily exists, this is the formalist programme. This misses the point that often things are found or constructed first without their being a consistent way of defining them. An important example of this is Feynmans Path Integral. Another important example is the calculus which is at the root of the modern effloresence of the sciences.
The other important concern missed by the formalist programme is of relevance. Far too much mathematics is shown to ‘exist’ whose relevance is dubious at best, and misleading at worst.
No, this is the lesson of Goedel's incompleteness theorem: no self-consistent system is sufficient. We also get this sense from Decartes: without some superfact such as "I exist", we can't even know anything.