# Talk:List of rules of inference

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Four things, first it is not clear on what the organization of this page is, whether the three categories of the rules of inference make sense, second is the list complete and what rules are missing or incorrent, third many rules can be derived from others and this should be mentioned somewhere perhaps in another article or a link give, fourth it would be very nice if there was a short symbolic description of each rule (like for modus ponens). - 208.38.7.220

I personally think that this page confuses axioms and rules badly. (And used the notation for axioms, and calls them rules.) I think it needs an overhaul, and a conversion to more standard notation. Note that as an example, the modal logic system S10 has one basis that has the RULE "if ${\displaystyle \vdash \Box p}$ then infer ${\displaystyle \vdash p}$". But if you take exactly the same base, and replace the rule with the AXIOM "${\displaystyle \Box p\rightarrow p}$" you get a DIFFERENT system (specificly S1) that can prove things not in S10. That distinction doesn't seem to be practical to make with the notation used here. Another example is the "Conjunction introduction (or Conjunction)" [Back when I took logic that was called the rule of Adjunction.] "If you have ${\displaystyle \vdash p}$ and you have ${\displaystyle \vdash q}$, then infer ${\displaystyle \vdash p\wedge q}$" The use of redundant prens in the article is very non-standard, and (I think) confusing. Anybody object if I replace it all in total? Nahaj 03:01, 19 November 2005 (UTC)

I was the one who layed it out like that, a long time ago when I also added a few more rules. As you may've guessed, I really don't know the difference between rules and axioms, and I don't know of a 'standard' use for parens. So, it'd probably be good if you replaced it with more standard conventions.-Shadro 04:11, 10 February 2006 (UTC)

## Rules of inference vs. rules of replacement

Rules of inference and rules of replacement are lumped together in this article, so my question is this: are all rules of replacement also rules of inference? In my logic class, the two were treated seperately, since when doing natural deduction proofs, inference rules could only apply to whole lines where replacement rule could apply to parts of lines.—jiy (talk) 14:13, 9 December 2005 (UTC)

In my opinion, the rules of replacement shouldn't be there at all. But, as you can see above, I'm not that happy with the article anyway. Nahaj 21:30, 9 December 2005 (UTC)

## Some fixes

Nahaj did not get any objections in reply to his question "Anybody object if I replace it all in total?" So I was bold and fixed some of the worst problems on this page. It still needs work though. Some of the worst problems remaining are: (1) the "hypothetical rules" still need to be reformated. (2) Universal Instantiation needs its restrictions added (variable of instantiation not free in preceeding assumptions). (3) Existential Elimination needs to be fixed so that it is a "hypothetical" rule (and appropriate restrictions need to be added). Less important but useful, some of the "non-hypothetical" rules at the end probably should be pared away. Nahaj should not take my fixes to constitute an objection to replacing the whole thing. --JMRyan 00:48, 7 March 2006 (UTC)

I've now fixed the formatting for hypothetical rules (which I've renamed to "discharge rules") and removed some of the cruftier rules. I've also corrected and fixed the formatting for the predicate calculus rules. --JMRyan 11:08, 7 March 2006 (UTC)
Thank you. Nahaj 03:38, 10 March 2006 (UTC)

## Negation Elimination

I removed the subtitle "Negation Elimination" from (the nonconstructive version of) Reductio ad Absurdum. This subtitle was incorrect, since no negation is in fact eliminated; the only negations occur in hypothetical assumptions (which if anything is the opposite of eliminating them, albeit not the same sort of opposit as introducing them would be). I will add in an actual Negation Elimination rule, which seems to be missing. (Perhaps I should also add subtitles explaining the link to Aristotle's famous principles of noncontradiction and the excluded middle, although I'm not sure what the proper formatting of those should be.) -- Toby Bartels 11:40, 29 June 2006 (UTC)

## NEW REMARKS 25 august 2006

can you split the inference rules in categories

1. that hold in minimal logic (and intuitionistic and classical)
2. that hold in intuitionistic logic (and classical) but not minimal logic
3. that hold in classical logic but not intuitionistic logic (nor minimal)

As far as i know (but i am not so familiar with the subject yet, and i don't say anything about the first order/ predicate calculus)

• reductio ad ad absurdum (negative conclusion) category 1
• reductio ad ad absurdum (positive conclusion) category 3
• non contradicion category 2
• Double negation elimination category 3
• Double negation introduction category 1 or 2
• modus tollens positive conclusion category 3

The rest belongs (in my not so knowledgable opinion) to category 1

## Wikipedia as a learning aid

I am aware that Wikipedia is often criticized as an unreliable source due to the difficulty of performing adequate academic policing, but articles like the current "List of rules of inference" are useless wastes of server space. Due to the foregoing concern about scholarship, Wikipedia is used by academics and laymen alike almost solely as a tertiary source (an overview/introductory source). But articles like this one are so dense and theoretically confused, that only people with a great deal of expertise can even read them--it is highly unlikely that such an expert will actually learn anything from them though. Hence, while pages like this one exclude the main reader-base of Wikipedia, they also bore-to-death (and annoy) scholars. If that is true, what use does the page serve? I strongly urge that any logician with teaching experience and a couple hours of free time rewrite this page.

Key Points

• Inference rules should appear both extremely simple and intuitive to the unfamiliar reader.
• If such pages as this are not both accurate and accessible, they are useless.
• A great motto in Wikipedia editing: "where possible, keep it simple."

I agree with you to some extends : as I also think that keeping explanations as simple as possible is a really good way to write articles, I think it shouldn't involve "cutting information" from a complex topic. Furthermore, the way information was expressed in this page helped me quickly remember the rules in a convenient way -- I'm working on computer science and this formalism widespread. But you're right saying the actual form is not acceptable : to be really interesting, this page should contain more details for each rule, explain the difference between sentence and predicate calculus, and provide references. To sum up : I don't want this page to be deleted, but I hope it will be completed by a capable one. MrHobjo (talk) 10:22, 26 July 2008 (UTC)

I added some simple summary for the rules of inference and 2 examples from a well known discrete mathematics book, maybe this helps to make things clear. Infinity ive (talk) 11:18, 7 November 2009 (UTC)

## Existential Introduction

The rule for EI indicates that we must replace every occurrence of the free variable ${\displaystyle \alpha }$ with the newly bound variable ${\displaystyle \beta }$, but this is wrong on two counts:

• ${\displaystyle \alpha }$ may be any term in this rule, not just a free variable.
• we may choose to replace only some occurrences of ${\displaystyle \alpha }$.

In this respect, EI is not like the other quantifier rules. This is important, as well. As it stands, I don't see how one can prove ${\displaystyle (\forall x)(\exists y)s(x)=y}$ from the rules given (and using ${\displaystyle (\forall x)x=x}$). Phiwum (talk) 15:15, 14 February 2011 (UTC)

I think you have misread the article. It says that β can be a term, not just a variable. So if we begin with the formula ${\displaystyle \phi (y)\equiv S(x)=y}$, then ${\displaystyle \phi (S(x)/y)\equiv S(x)=S(x)}$. So EI gives ${\displaystyle (\exists y)\phi (y)}$, which is ${\displaystyle (\exists y)(S(x)=y)}$. — Carl (CBM · talk) 17:26, 14 February 2011 (UTC)
Quite right. Just a brain fart on my part. Somehow I was picturing the replacement in the consequence, not the premise. Thanks. Phiwum (talk) 19:19, 14 February 2011 (UTC)

This fragment seems wrong to me:

Noncontradiction (or Negation Elimination)
${\displaystyle \varphi \,\!}$
${\displaystyle {\underline {\lnot \varphi }}\,\!}$
${\displaystyle \psi \,\!}$

Where did that psi come from? -AlanUS (talk) 02:48, 28 February 2011 (UTC)

This rule is somewhat unique: it says that from φ and ~φ you can prove any other statement ψ. This rule is also called ex falso quodlibet. — Carl (CBM · talk) 02:55, 28 February 2011 (UTC)

## Greek To Me

As a novice to symbolic logic, I wonder what, other than obfuscation-based job security, can be the advantage of using Greek letters as terms?

Wmicawber (talk) 18:06, 25 February 2012 (UTC)

## Universal Introduction

"In the following rules, ${\displaystyle \varphi (\beta /\alpha )\,\!}$ is exactly like ${\displaystyle \varphi \,\!}$ except for having the term ${\displaystyle \beta \,\!}$ everywhere ${\displaystyle \varphi \,\!}$ has the free variable ${\displaystyle \alpha \,\!}$.
${\displaystyle {\underline {\varphi {(\beta /\alpha )}}}\,\!}$
${\displaystyle \forall \alpha \,\varphi \,\!}$
Restriction 1: ${\displaystyle \beta }$ does not occur in ${\displaystyle \varphi }$.
Restriction 2: ${\displaystyle \beta }$ is not mentioned in any hypothesis or undischarged assumptions."

— Really? Is ${\displaystyle \beta }$ just a term, not a variable? Boris Tsirelson (talk) 12:27, 1 October 2013 (UTC)