This is a purely terminological (and tedious) question.

Given a language of first-order logic that includes the truth constant $\bot$ (falsehood), is this constant considered as an atomic formula or not?

I know that Prawitz's "Natural Deduction" (p. 14) defines $\bot$ as an atomic formula.

In contrast, Troelstra' and Schwichtenberg's "Basic Proof Theory" (p. 2) does not consider $\bot$ as an atomic formula.

Often an atomic formula in first-order logic is defined just as a $n$-ary predicate symbol applied to $n$ terms, but in a language where $\bot$ is not included. This is the case for example of Wikipedia's page defining well-formed formulas (but Wikipedia's page defining atomic formulas adopts Troelstra' and Schwichtenberg's terminology).

Are there other (well-known) handbooks in logic that consider $\bot$ as an atomic formula? Which is the most common terminological solution in the literature?

  • $\begingroup$ Related post: what is the correct reading of $\bot$ ?. $\endgroup$ – Mauro ALLEGRANZA Nov 3 '17 at 20:05
  • $\begingroup$ @MauroALLEGRANZA - Thank you for the link. I'm interested in the sentence you cited from Negri & von Plato (2001) "Often ⊥ is counted among the atomic formulas, but this will not work in proof theory". Do you have any clue of why they claim that? $\endgroup$ – Taroccoesbrocco Nov 3 '17 at 23:38
  • $\begingroup$ You have to read J Marcos' answer for a good argument supporting the view that $\bot$ is a (nullary) conncetive. $\endgroup$ – Mauro ALLEGRANZA Nov 4 '17 at 9:17

There are actually three ways in which you can treat the $\bot$ (and same for $\top$, if that is part of your language).

  1. As an atomic formula (different though as it is from the more 'typical' atomic formula that has a predicate term)

  2. As something that is not an atomic formula but still 'acts like' an atomic formula in its uses. For example, one could define the set of 'base formulas' as the set of 'atomic formulas' (the ones involving predicate symbols) together with $\bot$ (and $\top$), and those 'base formulas' can be combines with others using truth-functional connectives.

  3. As a syntactical shorthand for a generalized disjunction with $0$ disjuncts. That is, you can see the $\bot$ as neither an atomic formula nor acts like it, but rather as a complex formula (namely, a generalized disjunction (that disjuncts together any number of statements) ... with $0$ disjuncts!). This might be helpful when you have generalized disjunctions as part of your formal way of building up statements (as opposed to more 'traditional' approaches where disjunctions are typically understood as taking exactly two statements for its disjuncts), and when you do that, it could also be useful (e.g for your meta-theoretical proofs) to allow generalized disjunctions to have $0$ disjuncts ... and use $\bot$ as a way of expressing those.

Oh, and there is actually another use I sometimes see for the $\bot$, which is to use it as the truth-value $False$, or as the value $0$ in a boolean algebra. I really don't like that practice, as there is a big important difference between a truth-value and a statement (even if it is one, like the $\bot$ that always takes on a certain truth-value), so in my eyes it is not deserving of a 4. ... but like I said, you may see this in other books yet.

From my experience, I would say 3 is rather unusual, but both 1 and 2 are fairly common.

  • 1
    $\begingroup$ I regard $\top$ and $\bot$ as zero-place connectives. This is rather similar to your option 3, but it fits nicely into the usual framework of first-order logic without needing any notion of generalized conjunction and disjunction. (If that notion is available, then I'd go with your option 3 instead.) $\endgroup$ – Andreas Blass Nov 3 '17 at 22:45
  • $\begingroup$ @AndreasBlass Hey, interesting, thanks! $\endgroup$ – Bram28 Nov 3 '17 at 22:51
  • $\begingroup$ @Bram28 - Thank you for your answer. For each of the three options, do you have precise references of handbooks in logic or scientific papers that adopt it? I know that there are these three viewpoints about truth constants, I would like to know which one is the most common in the literature. I agree with you, the fourth option you mentioned is absolutely misleading, in my opinion it does not deserve any consideration. $\endgroup$ – Taroccoesbrocco Nov 3 '17 at 23:53

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