Here 'logic' means "a thing is not what it is not." Imagine if logic did not exist. That fact would be an example of logic. So, logic exists.

Imagine if nothing at all existed. Nothing is a concept: "not something". So, concepts exist. But "not something" implies two concepts: "something" and "not". So, two concepts exist.

"Two" is another concept: so we have three concepts. Continue this reasoning and we have "four", "five", etc. So, numbers exist. Apply that same reasoning to numbers as a group, and we have numbers of numbers (dimensions), which implies scales, relationships, etc. Continue this reasoning and we have all of mathematics.

Everything is mathematics
Pythagoras (the first great philosopher) was right. (See Aristotle, Metaphysics, 985-6)

Every point in dimensional space (e.g. "x=3, y=4") is connected to every other point by different equations: e.g. (3,4) + (1,1) = (4,5). Some equations require more steps than others: they are more difficult. We could call the dimensions "space-time" and the total number of steps in an equation could be called "difficulty" or "energy".

To move between adjacent points requires little energy, unless we take a very indirect route. We could even calculate the steps back to the zero point ("Big Bang" or before) then the energy needed to create a very different universe. So everything is connected.

Energy (difficulty, or inertia) is mass: see Einstein
for details. So, a physical universe exists.
The rest is physics, etc.
and is well known.