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The right tack, ⊢, is typically read as “entails”, which is ambiguous due to the difference between syntactic entailment and semantic entailment. It is sometime read as “yields”, but although I agree with this word in the way it suggests something is produced, I feel it too much suggests something imperative, I mean “will produce” where I would expect “may produce”. So I was wondering if it could be read and written in natural language as “allows”. I mean to read or write A ⊢ b as, “given what is written in A, one can write b” (ex. like when ⊢ is used to express a typing judgement).

Then, there is the semantic entailment, ⊨. To help distinguish it from the syntactic entailment which allows (and not requires) to produce something, I was thinking about a word with no production connotation, like “means” or “says”. Ex. A ⊨ B would be read as “A means B” or “A also says B” (I feel to prefer the former).

I feel to not have that much doubt ⊨ can be read as “means” (I still may be wrong), I am more unsure about reading ⊢ as “allows”. What is your feeling about it?

--- Edit ---

After some comments, I believe I have to clarify. I was unclear in the way I had two point of view in mind with this question. To a target language, there may be two different point of view: the one from someone whose write in the language, to which syntactic derivations, allows sentences to be written, and the one from someone who proves or is interested in properties of the language, to which syntactic derivations yields cases to be covered.

The question is not in the contexte of teaching, I’m not a teacher (nor a student), it’s in the context of how to explain some things and provide material for people to be confident of some things about it.

Hibou57
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    No ambiguities: $\vdash$ is syntactical entailment while $\vDash$ is semantical entailment. See the post Meaning of symbols $⊢$ and $⊨$ – Mauro ALLEGRANZA Apr 29 '20 at 12:55
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    We can simply say: "$A$ proves $B$". IMO, the dichotomy writing/saying does not add clarity. – Mauro ALLEGRANZA Apr 29 '20 at 13:01
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    But it is imperative, so to speak. If $T\vdash \varphi$, there is no option for $\varphi$ to maybe not be a consequence of $T$; it is, with zero doubt or flexibility, necessarily a consequence of $T$. – Malice Vidrine Apr 29 '20 at 13:46
  • @MauroALLEGRANZA, the word “entailment” is used for both, that’s why it produces some confusion. Yes, may be I should have mentioned confusion instead of ambiguity, since this is not formally an ambiguity, rather it is often ambiguously understood. The use of “proves” is not clear enough when the reader is precisely not used to prove. – Hibou57 Apr 29 '20 at 13:49
  • @MaliceVidrine. What I mean it that given x: int, f: int → int one may write f(x) but this is not required to write it. At the syntactic level we are at the level of what we are writing and we may or may not write it. By the way, there is a Unicode symbol which seems to allow to stress the difference: the character ⊩ which is named “forces”. – Hibou57 Apr 29 '20 at 13:55
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    @Hibou57 - (A) that's not what "forces" means, mathematically. (B) The relation $\vdash$ isn't about what one writes down. It's about what it's possible to obtain as the consequence of a proof. And that is 100% fixed by the theory and the deductive system no matter whether you ever write down a jot of logic. – Malice Vidrine Apr 29 '20 at 14:08
  • @MaliceVidrine, so “allows” would be fine? If so, feel free to post it as an answer, or feel free to propose any wording you like too. That said, there are cases where one is forced to take all possible cases into account: ex. when one is to prove the soundness of a type system. Please, what is the meaning of “forces” as you know it? – Hibou57 Apr 29 '20 at 14:30
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    No, I'm arguing that "allows" is misleading and unhelpful, and one should read $T\vdash\varphi$ as "$T$ proves $\varphi$" or "$\varphi$ is deducible from $T$" where there's no ambiguity. And I am not going to write down the definition of the forcing relation because I would have to describe the entire aparatus of Kripke semantics, which you can now Google yourself. – Malice Vidrine Apr 29 '20 at 14:34
  • Well, don’t mind, I feel there is a misunderstanding, will just tell which one without arguing anymore: what I am seeking for, is a wording in natural language, which helps to distinguish things. Using the word “prove” for conclusions in the syntactic level, does not match my requirement for the same reason the same word “entailment” for both language levels is an issue for my requirements. – Hibou57 Apr 29 '20 at 15:05
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    I understand your requirements, but I don't think your requirements can actually be fulfilled in any meaningful way. If your audience doesn't understand syntactic proof versus semantic entailment, no natural language gloss is going to separate these two concepts in any informative way. – Malice Vidrine Apr 29 '20 at 15:14

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I think the pedagogical ambiguity here is best resolved exactly by not introducing a new term: instead use the totally unambiguous phrases "syntactic entailment" and "semantic entailment" (until fluency is achieved of course).

This is especially true since logic already suffers from an abundance of terminology (signature/vocabulary/type/alphabet; countable/denumerable/enumerable; recursively enumerable/computably enumerable/recognizable/semidecidable/; etc.).

That said, suppose one is absolutely dead-set on introducing a new notation; what's a least-bad choice?

"Allows" is in my opinion a poor choice here: there's no sense in which "$\vdash$" is less compulsory than "$\models$," so I don't see what difference is being emphasized. It's also highly misleading in that it would lead to the conclusion that more restrictive axiom sets allow more things. Adding axioms in the hope of removing inconsistencies is a common mistake students make ("Let's get rid of Russell's paradox by forbidding self-containing sets"), and this wouldn't help.

"Deduces" has the advantage of being fairly unambiguous and connecting with existing terminology ("natural deduction"). However, the grammar is horrible: "$\Gamma$ deduces $\varphi$" isn't right at all. What we should say is "From $\Gamma$ we can deduce $\varphi$" or similar, but that's a mouthful. I personally do think grammar matters in this case: using a strange grammar makes the terminology feel more alien.

The best one I can think of is "justifies." The idea here is that we think of statements involving $\vdash$ as taking place in some dialogical process, with our goal being to construct an argument.

But again, I really think that no new terminology should be introduced; rather, the existing terms "syntactic entailment" and "semantic entailment" should be used until comfort is achieved. Besides the reasons mentioned above, this terminology has one very useful advantage: it emphasizes the similarity between $\vdash$ and $\models$, which is really the surprising feature (ultimately justified by the completeness theorem).

  • Of course, in more advanced topics in logic we'll sometimes want to go the other way and emphasize the difference between the two notions, or work in a context where one or the other doesn't even exist, but those situations won't arise until well after we've achieved a basic level of competence - at which point this won't be an issue anymore.
Noah Schweber
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  • Why can't we just say "proves"? – user21820 Apr 29 '20 at 15:58
  • I like a lot the “justifies”, indeed. I was hopefully not to introduce a new terminology (the tag is not from me, it was added, but I don’t mind), rather wording. At least, the symbols remains the same. Wording is more to explain or as an alias. About the difference being emphasized, I believe now I was unclear: I implicitly saw two different point of view on one language: the point of view of one who write in the language and the point of view of one who proves or is interested in properties of the language. For the former, justifications allows, for the latter, justifications yields. – Hibou57 Apr 29 '20 at 16:25
  • I also agree with the need to emphase the relation between both. – Hibou57 Apr 29 '20 at 16:28
  • @user21820 This is due to the OP's comment "The use of “proves” is not clear enough when the reader is precisely not used to prove." I somewhat agree with this actually: the notion of "proof" that students come in with is often at least partly semantic, and I've seen this cause a lot of confusion. By contrast - in my limited experience - students don't seem to attach much meaning at all to "entails" ab initio, so the admittedly-clunkier language avoids this issue. – Noah Schweber Apr 29 '20 at 16:55
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    @Hibou57 "I implicitly saw two different point of view on one language [...] For the former, justifications allows, for the latter, justifications yields." I think bringing the former stance into the language that we use will only confuse the students since the second perspective is already new to them. Using language which emphasizes the first perspective won't clarify the second perspective, it will obscure it, and the second perspective is really the crucial one for studying proofs as mathematical objects. – Noah Schweber Apr 29 '20 at 16:58
  • I agree (I was thinking about the same a few minutes ago). Just that I'm not a teacher :-p (nor a student, at least not strictly). – Hibou57 Apr 29 '20 at 17:25
  • @NoahSchweber: I see. Hmm, I guess I agree with MaliceVidrine then that if the learner does not understand what a "proof" is, then that is precisely what they are going to have to understand first, and using different words wouldn't really make any difference. As long as we don't give "⊢" silly names like "pokes" or "pins" it should be fine. =P – user21820 Apr 29 '20 at 18:18