# What is the minimum type of logical system that recognizes if a formalized sentence is a well-formed formula thus reducible to the boolean value?

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.

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.

• 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. – Andrej Bauer Apr 20 at 14:07
• I will formulate it better when on laptop. – MarkokraM Apr 20 at 18:09
• 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? – D.W. Apr 20 at 22:37
• I added much more stuff to the question. – MarkokraM Apr 21 at 5:35
• 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. – D.W. Apr 21 at 19:22