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I ask because Kleene mentions the possibility in Mathematical Logic, but I am not sure if he meant this as something he's aware of, or as a hypothetical possibility. I also enjoy the first part of Gensler's Introduction to Logic where he develops a simple formal calculus based on Aristotle's term logic where the WFF's must be in one of the forms: all A is B, some A is B, no A is B, some A is not B, a is B, a is not B, a is b, and a is not b. Capital letters represent groups (sets) and small letters represent individuals (members). More here on his system.

To me, this represents a pretty good example of a restricted English formal system. I also enjoy how many arguments, both philosophical and informal, can be analyzed in this simple notation. But now I wonder about extending his notation so that more of first order logic can be translated into this restricted English formal system. For instance, I'm wondering about the possibilities of extending the syntax the include extensions of the Aristotlean form where the predicate term is another categorical proposition, such as "all A is some B", or including relations like "all A is R to some B". I doubt the entire first-order logic can be translated in this way without creating a syntax more difficult, or convoluted, than the terse symbolic notation that is now in common use.

It could be asked why I would want something like this, why don't I just use the existing notation. I think the main reason is that my primary interest isn't in mathematics, and I've just found it difficult to translate arguments I read in works of philosophy, or even with informal reasoning, into this terse symbolic notation. I've been studying logic on my own for years, but never had much success actually applying the discipline until I started reading Gensler's book. I think it might be better for non-mathematical purposes to develop a notation for logic that is closer to the idioms of English to ease transcription both ways, but can still be just as easily transcribed into the full first-order notation when the correctness of the logic is in dispute.

So basically, does something like this already exist that goes further than Gensler's simple system?

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    I wonder what exactly you are looking for. I would say that it is fairly straightforward to develop a restricted English to express concepts out of set-theory: not that you're looking for an English version of set-theory, but that anything expressible in mathematics can be expressed in a so-called "natural" language if you use the words carefully (though almost any ideas you wish to convey will likely involve a great many non-logical axioms: merely to express the concept of time, for instance).
    – Niel de Beaudrap
    Commented Dec 1, 2013 at 18:06
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    In some sense, a number of research programmes in logic (such as modal logic and deontic logic specifically) are motivated by trying to formalize certain naive expectations of "should do" and "could have been"; these would lend themselves fairly naturally to transliteration back into English. The same could be said of any programme in logic which attempts to extend or alter the mainstream logical framework by accomodating features which are motivated by the causal daily use of language.
    – Niel de Beaudrap
    Commented Dec 1, 2013 at 18:11
  • My interest is not in mathematics, but in philosophical or informal reasoning. I'm looking to see if someone has already developed a syntax for first order, or some fragment of first order, logic that is more idiomatic while straightforwardly translatable into standard first order logic (for establishing correctness, etc). I just wonder if anyone knows if this has been done, in a comprehensive and rigorous way, or if this is perhaps not something of interest to logicians. Basically, it is something that I think I would like to work on, if it hasn't been done.
    – Kevin Holmes
    Commented Dec 2, 2013 at 19:53
  • Gensler's system above is sort of what I have in mind, but I would like to avoid the use of variables as much as possible (when you convert (x)(cat(x) -> mammal(x)) to All cats are mammals, the variables disappear), and I would like to go beyond the Aristotlean forms. I would also like to include modal logic, and dealing with logic in intensional contexts, but I do worry about making the logic too computationally complex, as I don't have the knowledge or skills personally to determine if the system remains decidable.
    – Kevin Holmes
    Commented Dec 2, 2013 at 19:58
  • Basically, I want to see a set of logical constants that aren't terse mathematical symbols, but are idiomatic words like "some", "all", "implies", and "not. Parenthesis will probably be required. Instead of single letters to represent non-logical constants, you should be able to insert entire English expressions that will represent non-analyzed expressions that might be broken down further. Basically, the point is to adapt logical symbolism for analysis of reasoning than for use in mathematics or for the investigation into logic as such.
    – Kevin Holmes
    Commented Dec 2, 2013 at 20:07

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If I understand what it is you are looking for, then it might be "Controlled English". One well-known specific system is Attempto Controlled English (I would start with the wikipedia article), but I think that there are others as well. I'm pretty sure Attempto goes beyond syllogism, but I don't know whether it actually captures all of first order logic.

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    I like the inherent humbleness in how they named it...
    – Chris Sunami
    Commented Oct 12, 2017 at 15:52
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Professors Fred Sommers and George Englebretsen have developed a logical system comparable in power to the first-order predicate calculus, but which is "Aristotelian" in spirit. That is to say, it's more like natural language and thus easier to translate natural language expressions into logical notation.

See Something To Reckon With : The Logic of Terms (Englebretsen)

and An Invitation To Formal Reasoning : The Logic of Terms (Sommers & Englebretsen)

Both available from Amazon.

See also Hanoch Ben-Yami's Logic & Natural Language, which you can download from here.

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