User talk:Addemf/sandbox/Technical Reasoning/Propositional Entailment

A few comments
Generally, I think this resource could be made shorter and more concise. Sentences like "Syntax is, to repeat myself a little, the study of a purely formal and almost "thoughtless" system of calculation. It is like a computer computing operations on 0s and 1s, with no sense of the interpretation or significance of the operations." should be removed. I know you want the course to be accessible and understandable, but really this sentence is extremely vague. Just define things as usual, e.g. ''Let A={ T,F,X1,X2,...,Y1,Y2,... }. This is the set of symbols we will use for logical formula. The set W of sentences or well-formed formula (which is a good phrase, as it's self-explanatory) is defined by the following rules: 1) Each member of A is a well-formed formula. If A and B are any pair of well-formed formulas, then 2) A v B is a well formed formula, 3) ~A is a well formed formula,... ''

Then one can explain that a formal theory (or whatever you want to call it) is defined by a language, a set of well-formed formula, axioms, and rules of inference. You could probably do the whole thing just as well if not better without ever using the word "syntax", which is really just a cryptic way to say "grammar". A student would understand perfectly if you explained that WFFs are just the set of grammatically correct sentences in propositional logic. Everyone knows what it means for an English sentence to be grammatically correct, and that grammatical correctness is defined by a set of rules. Why make it sound more complicated than it actually is? AP295 (discuss • contribs) 23:57, 9 May 2024 (UTC)

P.S. I almost forgot the point; once you explain it as above, you can just say that a proof (with respect to a theory) is a sequence of WFFs (w1,w2,...,wi,...) such that each element wi is either 1) an axiom or 2) Follows directly from element(s) of (w1,w2,....wi-1) using the theory's inference rules. That's pretty much how it's defined in Mendelson 5e and probably other books on mathematical logic. Dead simple. AP295 (discuss • contribs) 00:06, 10 May 2024 (UTC)

Actually I've wanted for some time now to do my own resource on propositional logic and FOL. I'm pretty sure I could cover propositional logic on a single page without leaving anything out and without leaving the reader confused. Many undergraduate textbooks e.g. for discrete math courses grossly overcomplicate this. Somehow they manage to be both excessively verbose and hopelessly vague at the same time, and I always wonder, what's the point of avoiding mathematical formality in a course on logic? It won't make it any simpler. Anyway, let me know if you have any questions on the subject. I wouldn't call myself an expert or anything, but logic was one of my very favorite subjects. AP295 (discuss • contribs) 00:26, 10 May 2024 (UTC)


 * Thanks for the feedback. Sounds like you might be more interested in making a reference text, which is very much what I would regard Mendelson and some others. There's a lot of value in that, and if you have a unique take in some way then there can absolutely be value in it.
 * I personally don't think I have a lot to add to a new reference text. Maybe I have some ideas which are different from other texts. But not as much as I would like to make in a more expository text, which is more what this aims to be. My aim is to be gradual and progressive, rather than delivering the endpoint of centuries of research.  I find that many students simply fail to understand and connect with that material, so I hope to present it in a way that grows a little more organically from common sense to formal mathematics. Addemf (discuss • contribs) 00:48, 10 May 2024 (UTC)


 * What I had in mind for my own resource was an introduction to logic for undergraduates. It would only cover a small chunk of what mendelson covers, and with more emphasis on applied reasoning/proofs. Looking at this page, I thought this would have frustrated me as an undergraduate. In the intro, the page states "In this lesson we will study the syntactic concept of proof, which is like the syntactic version of the same thing. ". Maybe I'm missing it, but I don't see that the page gives any concrete definition of what a proof is. I mean, a student would probably glean enough info from the page to get the gist of what a proof is, but why not make it easy for them? Do you really think the current page is easier to digest than the single-sentence definition I paraphrased from mendelson? Not a chance. I'm not saying the page should have only a definition (a few examples and a bit of explanation would be good), but it should certainly lead with one and go from there. The great thing about logic is that you need hardly any background knowledge to study it. It's the foundation, and students (really everyone) should understand these fundamentals in concrete and sufficiently abstract terms that they can then apply elsewhere. A colloquial or ad-hoc style is not the way to do this. If you feel there's some specific point that students would tend to get snagged on, say so and perhaps I'll understand what you mean. AP295 (discuss • contribs) 01:33, 10 May 2024 (UTC)
 * This particular page -- and really the entire course -- is far from a completed state. I think you may be judging it as if it were a finished product.
 * I intend to redesign much of what I've done in this page, although not to make it more economical. If this is not the kind of product that interests you, I understand.  It's not for everyone. Addemf (discuss • contribs) 01:39, 10 May 2024 (UTC)
 * I understand it's not finished, and I hope that my comments are helpful to that end. You asked for input on this course some time ago in the "colloquium", so I assume you don't mind my feedback. At the very least, I think the page needs to include (and preferably lead with) a concrete definition of what a proof is. Personally I'd also avoid using the word "syntax", for the reasons I've explained above. While it's not always possible or practical to avoid jargon, it should probably be omitted in favor of plain english when that is possible. I don't see any value in using "syntax" when the word "grammar" would do just as well if not better. You could probably avoid even the word "grammar" and be no worse off. Let's not conflate conflate jargon with precision and specificity. I imagine the former is what some people really have in mind when they bemoan "formality". Here I just mean that you should be more specific. When people go out of their way to avoid formal (or rather, precise/specific) style and definitions they tend to belabor the point - presumably because what they have does not seem to suffice - trying to explain it every which way and making bad analogies, using jargon, etc. Just give the reader an actual, concrete definition and get on with it. AP295 (discuss • contribs) 02:21, 10 May 2024 (UTC)
 * Indeed, the page needs to include a concrete definition of what a proof is. I was getting to that, hence why I point out that the page is not done yet.  As for syntax and semantics, I think it's a useful distinction common in logic and computer science, and so serves the student to keep track of which piece of formalism corresponds to which.
 * Let's be clear now, a proof is a sequence of WFFs such that each element follows directly from some inference rule applied to any of the previous elements. In other words, it's just the set of sequences of WFFs that satisfy a specific, inductively-defined condition. Proofs are not themselves WFFs. In grammatical terms, you could say WFFs are members of the object language, while proofs are written in the meta-language(see page 27 Mendelson 5e). The metalanguage does not really need to be formalized here, because it's not what we're studying. You really just need to explain (formally or informally) that proofs are just sequences of wffs that satisfy the aforementioned condition. I would not make a big fuss about the "syntax" of a proof. AP295 (discuss • contribs) 04:34, 10 May 2024 (UTC)
 * Thank you for your input. Addemf (discuss • contribs) 04:46, 10 May 2024 (UTC)
 * My pleasure. AP295 (discuss • contribs) 04:47, 10 May 2024 (UTC)
 * But mostly I think you want a text that is not the kind of text that I'm trying to write. So I thank you for the feedback, but if you want a text that just "gets on with it", then you don't want what I'm writing.  Which is fine, not everything is right for everybody. Addemf (discuss • contribs) 03:08, 10 May 2024 (UTC)
 * I'm already quite familiar with propositional logic and FOL. I don't need an undergraduate introductory text of any sort. I make these suggestions because I think they will make your resource more useful to an undergraduate studying discrete math.

If you want a second opinion you might ask, who claims to be a logician and at any rate does seem familiar with logic. AP295 (discuss • contribs) 04:57, 10 May 2024 (UTC)