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The formula, in the old way of using it, can contain symbols in order and a mixture that does not meet the criteria of correctness (i.e. arbitrary symbols do not form a well-formed formula (WFF) and do not conform to the grammar).

Also, a sentence written in a natural language could be in such a form that it cannot be transformed to the WFF. WFF, on the other hand, always reduces to the boolean true/false values and two-valued truth tables in predicate logic.

It is possible that we can never make a parser good enough to do English sentence transformation to an informal one perfectly. Peter Norvig has the following parser made in Python to give an idea of the process: https://github.com/norvig/pytudes/blob/master/ipynb/PropositionalLogic.ipynb

So, at the end, when a sentence cannot be transformed, by automation or by human manual work, to the well-formed formula, we need to categorize it by some way not fitting to the classical propositional paradigm.

I want to examine this interface where we decide if a logical formula is correctly formed so that we can call it WFF or if a naturally written sentence is transformable to the correctly formed propositional and predicate logic formula. I assume that this logical system, or maybe it us anticipating a larger theoretical framework, would at least entail the use of one additional type? This is: to decide if the sentence is transformable to the boolean reducible format, thus if the sentence reduces to the boolean type or not.

So, my question has a few parts. On the other hand, I'd like to get a clarification what kind of theoretical framework is required to axiomate a system that has propositional and predicate logic functionality, but also can determine, if the formula or the sentence is a boolean type at all? Related to that, what is needed to determine theoretical wise if a given sentence is well-formed at all?

Does this have something to do with language parsers and syntax checking? From that perspective, how does this affect the minimum logical paradigm required?

I hope these questions intertwine in such a way that they can be handled in one answer.

Additional info

For example, the following paper is discussing similar topic and introduces the logic of presuppositions and the truth-relevance concepts: https://www.researchgate.net/deref/https%3A%2F%2Fphilpapers.org%2Farchive%2FNEWMPT.pdf

Logic systems that I'm talking about are introduced in: https://www.britannica.com/topic/logic#ref1049434

Possible candidate for such logical framework might be found from this document: http://homepages.inf.ed.ac.uk/gdp/publications/Framework_Def_Log.pdf where the S4 modal logic is mentioned to have both truth and validity relations.

Also this cstheory topic may relate to my question: https://cstheory.stackexchange.com/questions/30541/logical-framework-vs-type-theory

I have added these clarifications due to request by D.W. If the former forum is better for handling my question, I hope it can be forwarded there.

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    $\begingroup$ At least I don't understand what you're asking. It might help if you give us an example of what you're looking for. $\endgroup$ – Andrej Bauer Apr 20 at 14:07
  • $\begingroup$ I will formulate it better when on laptop. $\endgroup$ – MarkokraM Apr 20 at 18:09
  • $\begingroup$ I don't understand what it would mean for a "logical framework to implement a system". Also I'm not sure what "formal sentence is not boolean type" means. I suggest editing your question to provide additional context and explanation. Perhaps it might be helpful to provide a definition of what you mean by "framework", "system", "implement", "sentence", and "boolean type". Have you looked at en.wikipedia.org/wiki/Well-formed_formula? $\endgroup$ – D.W. Apr 20 at 22:37
  • $\begingroup$ I added much more stuff to the question. $\endgroup$ – MarkokraM Apr 21 at 5:35
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    $\begingroup$ I can't understand your language. Right from the first sentence, I don't know what "mixture that does not meet the criteria of correctness" means. When you say "this logical system", I don't know what "this" refers to; you haven't previously introduced any logical system. I don't know what "axiomate a system that has propositional and predicate logic functionality" means. $\endgroup$ – D.W. Apr 21 at 19:22
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There is no systematic algorithm that, given any English sentence, can always determine whether it can be translated to a well-formed formula in first-order logic. The English language allows expression of statements that are inherently subjective and imprecise; the meaning of some English sentences is unavoidably a matter of opinion.

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  • $\begingroup$ Should we agree with that and define the case when sentence cannot be transferred as "not known", "undefined", or "not boolean". What is this kind of logical system, what are the requirements and implications of such system? $\endgroup$ – MarkokraM Apr 21 at 19:52
  • $\begingroup$ @MarkokraM, Rather than responding here in the comments to my answer, it would probably be more useful to edit your question to make it clear what you are asking and clarify the points raised in my comments under your question. We are not a discussion forum, so we're not looking to support interactive back-and-forth. Instead, we want you to pose a clear, well-formulated, narrowly-focused technical question that can be answered without any discussion or back-and-forth interaction. (I notice you continue to use the phrase "logical system" without defining what you mean by it.) $\endgroup$ – D.W. Apr 21 at 20:00
  • $\begingroup$ If I'd knew better wordings from the beginning, I'd surely use them. Until that some feedback and forth might be needed to formulate question better. I'll take a look your points and try to edit the question later. $\endgroup$ – MarkokraM Apr 21 at 20:04
  • $\begingroup$ With logical system I mean this: britannica.com/topic/logic#ref1049434 $\endgroup$ – MarkokraM Apr 21 at 20:37
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    $\begingroup$ @MarkokraM, don't respond here in by leaving comments; edit your question. A logical system as defined there has nothing to do with translating an English sentence into a logical formula. $\endgroup$ – D.W. Apr 21 at 22:38

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