Fractional jet Captain reported there was no signage for Taxiway J on the west exit of Runway 22R at MDW.

Narrative

No taxiway sign for J as you taxi westbound across Runway 22R from F. Ground cleared us to cross Runway 22R; J and A to the ramp." As I consulted the taxi diagram to find J; we had reached the hold-short line for 22R. PM increased thrust for the runway crossing and I suspected he may not know where J was relative to Runway 22R. So as he said clear left; I said left turn on the runway for J; followed by cleared right. He said "heh?" so I pointed to his left and said that's Taxiway J; diagonally there to our left. He stepped on the brakes and said; "Are you sure?" I said yes; I've checked the diagram.Cause: No signage indicating "J" on the west exit of Runway 22R. ATC clearance that is delivered in the same way as the typical clearance where you cross the runway and then turn onto another taxiway.After the brief slowing and hesitation from the PF; we turned towards J on Runway 22R and exited as cleared by Ground.Suggestions: There needs to be at least one taxiway sign; preferably one on either side of Taxiway J visible from this crossing point on F; on the west exit Runway 22R. Suspect there is a sign further north on Runway 22R if you were landing on 22R to exit on J. This sign is not visible from where we were taxiing. I don't believe there is any; but with the lack of time involved I will say it was not identifiable by us.Also; the clearance we were given by Ground Control creates a confirmation bias. We expect to cross a runway; then turn onto the next taxiway. Our situation begs for something like "cross Runway 22R and EXIT ON J. Then A; cleared to the ramp" to be the clearance to emphasis the need to exit on a taxiway other than F; the one we entered the runway from."