Talk:Rule of inference

message: One glaring omission is not to mention derived rules of inference, that some rules of inference can be deduced giving others.  An example at least should be given, and mention made that there are some minimal sets of rules of inference that can be used to derive the rest.  This may form another article later on.
* I did this (a while ago) Brighterorange 22:03, 25 Apr 2005 (UTC)
replies: text: It is standard to say that theorems of a theory T are derivable from the empty set "with respect to (the axioms of) T". Specifically, the derivation relation is generally relativized to a system, so that if ⊢T denotes derivability in T, then if A is a theorem of T, it is written ⊢T P to mean that P is derivable in T from the empty set. If we take derivation in a more abstract form, then usually we write T ⊢ A to mean that A is derivable solely from the theory T, where T may be the axioms of classical logic, and ⊢ some abstract derivation relation. Nortexoid 17:36, 22 January 2007 (UTC)
thread_title:  glaring omission 

message: *Shouldn't this article point to Inference engine, somehow ? --Jerome Potts (talk) 14:46, 13 March 2009 (UTC)
thread_title:  missing links? 

message: I was thinking about this earlier, and I think nobody here has mentioned the ω rule as another example of a rule of inference that is not expressible as a formula.  The ω rule is an infinitary inference rule used to study first-order arithmetic, which says that if P(x) is a formula of Peano arithmetic and you can assert (individually) P(1), P(2), P(3), ..., so that you have separate assertions of P(n) for each standard natural number n, then you can deduce ∀x P(x) from this infinite collection of assertions. 
The ω rule is interesting here because it is a logical truth, but it isn't sound for classical first-order semantics. It isn't sound because its conclusion can fail in non-standard models even when all of its hypotheses hold, and so it does not preserve truth in arbitrary models of Peano arithmetic. 
But the ω rule is a logical truth in the sense that (disquotationally) the truth of ∀x P(x) follows logically from the (disquotational) truth of P(k) for every natural number k; this is just the T-schema for the universal quantifier.  This is the sense of "logical truth" that Ayer is talking about: not just "truth in every first-order model" but Logical Truth in the philosophical sense. The ω rule is that sort of Logical Truth, but it cannot possibly be represented as a formula in PA, because formulas of PA are finite. — Carl (CBM · talk) 17:48, 16 February 2010 (UTC)
replies: text: Carl, this doesn't speak to classical standard logic does it? Be well. Greg Bard (talk) 03:42, 17 February 2010 (UTC)
replies: text: Yes, ω logic is a classical logic, rather than an intuitionistic one. It is not the same as first-order logic, but that's a separate matter. The thing I'm pointing out is that the ω rule is a Logical Truth, in the philosophical sense, but it does not correspond to any logically valid formula in the language of arithmetic. — Carl (CBM · talk) 03:56, 17 February 2010 (UTC)
replies: text: There's a axiom schema in Lω1ω which corresponds to that rule.  I don't really want to side with the Bard, but he may have a point.  — Arthur Rubin  (talk) 09:07, 17 February 2010 (UTC)

text: Maybe we're going about this wrong, and what we need is to mention the deduction (meta)theorem? CRGreathouse (t | c) 15:38, 17 February 2010 (UTC)
thread_title:  ω rule 

message: The lead is supposed to be the most reader-friendly introduction to the topic (I often even add a picture that summarizes the lead). That having been said, I find the lead for this page far too dense. It needs a plain English summary. The currently detailed overview would fit nicely in an "Overview" section.-Tesseract2 (talk) 16:59, 15 October 2010 (UTC)
replies: text: I tried to help. ᛭ LokiClock (talk) 04:14, 3 September 2011 (UTC)
thread_title:  Dense lead 

message: I recall from one of Irving Copi's logic books that there is a pattern of deduction that goes:----
A
B
-------
A->B
Copi introduced this rule before quantifiers were discussed.  I don't see this rule in the Wikipedia's rules of inference.  Is it among them?  Should it be listed?
(I'm not making a claim that this is a valid abbrevation of a rule, but I think it suggests a common form of everyday argument, which is: assume A, show B follows, conclude "if A then B".  As I recall, Copi took pains to point out it that  this form of argument is not modus ponens.)
Tashiro (talk) 15:33, 1 October 2012 (UTC)
thread_title: The Rule:  A, B Hence A->B 

message: I have performed a copy edit of this article under the auspices of the Guild of Copy Editors. If I changed the meaning of a sentence in a way that makes it inaccurate, feel free to change it.
I have removed a number of "clarification needed" and similar tags. Given the text I started with, I have made the article as clear as I could. As with many articles, this one cannot be made entirely clear to a layperson with no background in logic or other related subjects. I think the goal should be to make the article as clear and accurate as possible to someone with a basic understanding of the terms used in it. – Jonesey95 (talk) 00:35, 30 January 2014 (UTC)
thread_title:  [[WP:GOCE]] copy edit 

message: In the section on "Axiom schemas and axioms" it mentions that certain statements are provable under the Peano axioms, but the Peano axioms page, when I follow the link, has nothing about provability.
It would be nice if this example was removed or better explained.
I'm fairly new to Wikipedia, so if this is not the best way to bring up things like this, please let me know.  — Preceding unsigned comment added by Jadiker (talk • contribs) 09:37, 19 November 2017 (UTC)
thread_title:  Peano Axioms and Provability