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?

  • 1
    I have a guess now, why the authors did these omissions. They intentionally moved propositional logic to chapter 11, in order to focus on the "formal language" aspect of logic. By omitting propositions and the truth constants, they further help the reader to get less distracted by his preconceptions from propositional logic. Commented Mar 14, 2013 at 23:04
  • Sorry, are you asserting that "¬x≡x" and "(φ ∧ ¬φ)" are theorems? Commented Oct 21, 2013 at 2:35
  • @ChristopherE It's the opposite, "x≡x" and "φ ∨ ¬φ" can be proved to be true, so there negation can be used as a replacement for false. Commented Oct 21, 2013 at 7:35

1 Answer 1


Often authors omit these kind of details because they want to keep proofs simpler. Many proofs are by induction over the structure of formulas, the more rules you have the more cases you have to consider. That's probably the reason why Ebbinghaus et al. omit these details. On the other hand, when you 'use' the logic rather than proving meta-theorems, it can become cumbersome having to expand definitions all the time, so in that case you might prefer to introduce more expressions into the language directly rather than defining them. It depends on the purpose and in textbooks also on your pedagogical goals.

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