Fair P/E Multiple for the S&P500 – What am I missing?

So this year (2022), earnings for the S&P are expected to be 240 according to Yardeni Research. I've seen other estimates saying 215-230 EPS for this year, assuming no recession. Given that interest rates are rising, I would argue that a fair P/E multiple for the S&P would be 15-18x, rather than the 20x+ we've seen the last 2 years. So here's the math using different EPS estimates:

Bull Case – S&P Earnings of 240:

15x = SPY 3600

16x = SPY 3840

17x = SPY 4080

18x = $4320

Base case – S&P Earnings of 220:

15x = SPY 3300

16x = SPY 3520

17x = SPY 3740

18x = SPY 3960

Bear case – S&P Earnings of 200:

15x = SPY 3000

16x = SPY 3200

17x = SPY 3400

18x = SPY 3600

None of these scenario's look great, and the math tells me we are in for potentially much lower lows.

Could someone tell me if any part of this analysis is wrong, and how I should be thinking about this if the above is incorrect?