Air carrier Captain reported an aircraft refusal due to a worn tire.

Narrative

As we were accepting the aircraft in ZZZ; ground personnel brought to my attention Cord showing on #3 Tire. I called Maintenance on VHF and asked that they inspect it. They asked that I write it up in the logbook. I wrote it up as cord showing. There was also a write up from yesterday on the same tire. When Maintenance discussed the cord showing he said I have to put it on the maintenance discrepancy and it's good for two cycles. They will change it when the aircraft returns I explained that it was written up yesterday so that limitation would mean the tire should be changed already. I called dispatch and they agreed so I refused the aircraft unless the tire was changedHad the maintenance discrepancy been applied properly to begin with the delay would not have been 3 hours