There is exactly one atomic sentence form of L (call it 'A'), which is always a tautology, and whose negation 'negate A', is always a formal contradiction. What is this special, general type of atomic sentence?

I know that it must be a two-place relation but beyond that I don't know what it could possibly be.

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    Can you specify a little further what exactly you'd like someone here to explain to you? What hypotheses have you formed, what has your research uncovered so far, etc.?
    – Joseph Weissman
    Commented Apr 12, 2014 at 23:58
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    Well I would like to know what atomic sentence the question is asking me to find. As far as research I have none, we have no book for the class , notes are very cryptic and taken from in class.
    – Achilles
    Commented Apr 13, 2014 at 0:14

1 Answer 1


Some logics have a nullary truth-functional connective (⊤) that evaluates to true under all valuations. You can call it 'A', you can call it 'atomic', if you so desire. Here are some facts about ⊤:

  • It's special in that its interpretation is fixed, i.e., ⊤ is a logical constant. That means that truth-assignments (like the rows of truth-tables) cannot alter its truth value.

  • It's general in that it can be used to represent the equivalence class of all tautologies in a given logic (the set of all φ, ψ s.t. φ ⇆ ψ). Because if φ and ψ are tautologies, then φ ≡ ψ ≡ ⊤.

I'm not sure if that's what you're looking for. I have to admit that the question wasn't very clear.

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