What is the difference between the tags and ?

This question has been raised several times (here, here and here), and several users proposed in their comments to synonymize (or merge) them, but nobody concretely suggested them as synonymous on Mathematics.

Thus, I formally proposed to synonymize them here, you can upvote or downvote as you prefer.

I'm aware that, from a theoretical point of view, the tag can cover more than , since it can include also and , but concretely:

  1. and have already a considerable overlap (at the time of this post, 619 questions contain both tags out of 2267 questions in );
  2. almost all the questions tagged with without could coherently and meaningfully be tagged with ;
  3. the really few questions tagged with and that concern second or higher order logic could be tagged with or , plus possibly .
  • I was going to raise this, and I'm glad to see someone else taking interest in tagging. – Asaf Karagila Aug 9 at 18:11
  • My experience (both teaching and studying) is that predicate logic usually refers to the study of sentences and structures. First-order logic usually refers to the study of theories and models. While to the untrained eye this might seem the same, it's really not. – Asaf Karagila Aug 9 at 18:13
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    I guess my eye is untrained... – Andrés E. Caicedo Aug 9 at 18:49
  • @AsafKaragila - In the domain of proof theory, predicate logic is most of the time a synonymous of first-order logic. Even on Wikipedia, predicate logic redirects to first-order logic. – Taroccoesbrocco Aug 9 at 18:57
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    @Andrés: In Beersheba we talked about predicate logic. We talked about formalizations, about this structure satisfies that formula with that assignment, and this formula implies that other one if and only if the material implication is logically valid. We didn't talk about theories, about models of theories, about completeness, etc. For me, that is the key difference between the two tags. – Asaf Karagila Aug 9 at 19:11
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    (@Asaf I understand what you are saying. I do not think this difference in usage is universal.) – Andrés E. Caicedo Aug 9 at 19:12
  • @Andrés: I see. Well, then we agree. I guess it's just because this is how I was trained to think as a student and as a teacher, the distinction between the two tags is more natural to me. – Asaf Karagila Aug 9 at 19:13
  • @Taroccoesbrocco: I know that you've been pinged by all my comments to Andrés, those are also for you (and anyone else weighing in on this conversation). – Asaf Karagila Aug 9 at 19:13
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    To me, "predicate logic" is applicable to higher-order logics as well as first-order logic. But I am fine with merging them, because I think that the higher-order logic tag is useful in almost every case that predicate-logic would not mean first-order-logic. – Carl Mummert Aug 9 at 20:49

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