Let's assume for a moment that the transistor had never been invented and humankind were still using electromechanical relays, like in the early 1900's. We can still apply boolean logic to them, just as we can apply it to transistors, but what about the engineering side of things? Let's take a look:
Mechanical parts are larger, consume more electricity and wear more quickly than semiconductors.
Modern processors consist of billions of transistors. Let’s assume we had a 1 billion transistor chip and we wanted to build it with relays. If each transistor were to be replaced with one relay, then we would require 1 billion relays. Let’s further assume that 1 relay would require 1cm^3 (the size of a sugar cube) of space and that we need another 1cm^3 of space around each relay for wiring, cooling, etc. So we'd need 2cm^3 of space per relay. That would be 2 billion cm^3. for all our 1 billion relays. That’s 2,000,000,000 cm^3 = 2,000 m^3, a cube with a side length of 12.6m, equivalent to a 4 storey building. Ok, that's BIG.
What about powering these 1 billion relays?
If each relay required 50mA of current at 5V, then we’d need 50mA*1,000,000,000=50,000,000A.
50,000,000A*5V=250,000,000W, which is 250 Mega Watt. A smaller coal fired power plant produces 500 megawatt of electricity and burns 1.4 million tons of coal each year. We’d need half of this.
In summary: If we could build such a relays computer, it would be the size of a 4 storey house, require half a coal-fired power plant and consume 700,000 tons of coal each year. This would be a tad too big to carry around with us. Not to mention the heat that the 700,000 tons of coal generate.
I'm glad the transistor got invented :-)