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I have been somewhat confused about the definition of atoms, or atomic formulae.

Some sources say that verum (⊤) and falsum (⊥) are atoms, some not. Is there any consensus within the community or is just a matter of vernacular?

Note: I am using resources particular to Oxford.

Thanks for your help

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The verum (⊤) and the falsum (⊥) are 0-ary connectives. Thus, we can "read" them also as formulae.

If we are working in first-order logic, ⊤ and ⊥ are atomic formulae, because they are "indecomposable" into sub-fomulae.

See e.g.:

But see J Marcos'answer: they are not propositional variables.

See also the post: what-is-the-correct-reading-of-bot for a more detailed discussion.

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This is not just a matter of vernacular. Nullary connectives such as ⊤ and ⊥ do not usually behave the same way as atomic variables: for instance, while the latter allows for consequence-preserving (uniform) substitution (check here the notion of a formal logical system over a substitution-invariant consequence relation) the former does not.

Note: if you do not like the terminology "atomic variables", above, you may substitute it by "propositional variables". The whole point of my anwer is the need of distinguishing this class (whatever you call it) of syntactic objects that allow for consequence-preserving substitution from the class of somewhat similar objects called "nullary" / "0-ary" / "0-place" connectives, that do not in general allow for such forms of substitution. In many aspects the difference is analogous to the difference between a variable and a constant.

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  • I agree with you that they are not propositional variables; but I've specified: "in f-o logic". In prop logic, if we want to speak of atomic formulae (that seems not useful to me) we have to define them as prop variables.
    – Mauro ALLEGRANZA
    Commented Mar 15, 2016 at 13:40
  • @MauroALLEGRANZA My answer does not apply only to classical logic. Note anyway the conceptual distinction between variables and atoms (you might prefer to turn these into adjectives applied to "propositions" or "sentences"): one may have the latter include the former, but not the other way around. In the original question it is not clear to me whether the author wishes to distinguish between atomic formulae and propositional variables.
    – J Marcos
    Commented Mar 15, 2016 at 17:18
  • @MauroALLEGRANZA If that was the case (namely, if the question was about atomic formulae as opposed to propositional variables), then you may just forget what I wrote: the question was after all just a matter of terminological convention, and conventions are what they are.
    – J Marcos
    Commented Mar 15, 2016 at 17:19

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