I wonder why "Mathematical Logic" from Ebbinghaus et al. omits 0-ary relational symbols (which would normally be interpreted as propositions, similar to how 0-ary functional symbols are normally interpreted as constants). I have to admit that this omissions seems to have nearly no influence on the text, and I only noticed this omission in chapter 11, where propositional logic is introduced.
The book also omits the truth constants for "true" (1/T/⊤=top) and "false" (0/F/⊥=bottom). This seems to be a more common omission also in other books. At least the omission of a truth constant for "false" seems to have a noticeable impact on the text. This constant seems to be implicitly required for some theorems, and replacements for it like "¬x≡x" or "(φ ∧ ¬φ)" (which is an abbreviation for "¬(¬φ ∨ ¬¬φ)", but could be replaced by "¬(φ ∨ ¬φ)") keep to be invented on the fly (or the theorems are stated with more distinction of cases than necessary). It also reduces the expressive power of implication (→), which would otherwise suffice as the only logical connective (similar to the Sheffer stroke). Regarding implication, the "is implied by" connective (←) is also omitted.
These omissions are certainly intentional, but I don't understand why. Can you help me?