(This article nicely explains the difference between proper and proper scoring rules)
According to the Wikipedia entry, and Merkle & Steyvers (2013), these are both strictly proper scoring rules. But in his article Damage Caused by Classification Accuracy and Other Discontinuous Improper Accuracy Scoring Rules Dr Harrel refers to these two as just proper scoring rules:
The two most commonly used proper scoring rules are the quadratic error measure, i.e., mean squared error or Brier score, and the logarithmic scoring rule...
Are these two strictly proper or just proper scoring rules?