Talk:Hilbert system

message: This article reads like a cheatsheet for 3, which, I might add, is written in Hungarian. The article doesn't define any of its syntax nor explain much of anything else. I appreciate the translation work that Physis has done here, but it's so much more technical than similar articles that I think it needs a complete rewrite. Vagary 02:02, 4 February 2007 (UTC)
thread_title: Rewrite

message: I tried to reply these concerns with writing a new approach to introduce the topic. I kept this on a subpage. I shall follow Carl's proposal and will not develop this subpage any more, but contribute to the article indirectly. If the subpage contains anything that can be usable in any way, please feel free to use it. Physis 16:18, 15 November 2007 (UTC)
thread_title:  Availability of a forked, rewritten version 

message: Till now, I have found the following three approaches:
Relying on Extension by definition I
:R1. \alpha \to \left( \beta \to \alpha \right) 
:R2. \left ( \alpha \to \beta \rightarrow \xi \right) \to \left( \left( \alpha \to \beta \right) \to  \left( \alpha \to \xi \right) \right)
:R3. \left ( \lnot \alpha \to \lnot \beta \right) \to \left( \beta \to \alpha \right)  (reductio ad absurdum)
:R4.  \forall x \left( \alpha \right) \to \alpha[x:=t] where t may be substituted for x in \alpha
:R5. \forall x \left( \alpha \to \beta \right) \to \left( \forall x \left( \alpha \right) \to \forall x \left( \beta \right) \right)
:R6.  \alpha \to \forall x \left( \alpha \right)  where x is not a free variable of \alpha.
:R7. \mathfrak x = \mathfrak x for variable \mathfrak x where the notation \mathfrak x is not meant as a metavariable, but it is directly part of object language: any fixed variable selected from the object language. Thus, R7 need not be an axiom scheme: it can be a standalone axion instance of its own. Of course, we could use a x metavariable ranging over variables of the object language, but it can be proven that doing so would be superfluous.
:R8. \left( x = y \right) \to \left( \alpha[z:=x] \to \alpha[z:=y] \right)
Relying on extension by definition II
:D1. \alpha \to \left( \beta \to \alpha \right) 
:D2. \left ( \alpha \to \beta \rightarrow \xi \right) \to \left( \left( \alpha \to \beta \right) \to  \left( \alpha \to \xi \right) \right)
:D3. (\lnot\alpha \to \beta) \to (\lnot\alpha \to \lnot\beta) \to \alpha, mentioning that instead of this (contraposition), also reductio ad absurdum can be used, as at the above approach (R3).
:D4. \forall x(\alpha) \to \alpha[x:=t]
:D5. \forall x (\alpha \to \beta) \to \forall x (\alpha) \to \forall x (\beta)
:D6. \alpha_0 \to \forall x (\alpha_0) where x is not a free variable in \alpha_0
:D7. t = t where t is a metavariable ranging over terms
:D8. (s_1 = t_1) \to \dots \to (s_n = t_n) \to f(s_1, \dots, s_n) = f(t_1, \dots, t_n) where f is a metavariable  ranging over function symbols, and s_1, \dots, s_n, t_1, \dots, t_n are metavariables ranging over terms.
:D9. (s_1 = t_1) \to \dots \to (s_n = t_n) \to P(s_1, \dots, s_n) \to P(t_1, \dots, t_n) where P is a metavariable ranging over predicate symbols, and s_1, \dots, s_n, t_1, \dots, t_n are as above
Conservative extension
The approach I found is not a mere superset of any of the former approaches, because the most interesting axiom for universal quantifier differs: instead of
:\forall x (\alpha \to \beta) \to \forall x (\alpha) \to \forall x (\beta)
it postulates
:\forall x (\alpha_0 \to \beta) \to \alpha_0 \to \forall x (\beta) where x is not a free variable in \alpha_0
In details:
:C01. \alpha \to \left( \beta \to \alpha \right) 
:C02. \left ( \alpha \to \beta \rightarrow \xi \right) \to \left( \left( \alpha \to \beta \right) \to  \left( \alpha \to \xi \right) \right)
:C03. (\lnot\alpha \to \beta) \to (\lnot\alpha \to \lnot\beta) \to \alpha
:C04. \lnot\lnot\alpha \to \alpha. (In both "extension by definition" former approach, this was lacking.)
Conjunction and disjunction
:C05. \alpha \to \beta \to (\alpha \land \beta)
:C06. \alpha \land \beta \to \alpha
:C07. \alpha \land \beta \to \beta
:C08. (\alpha_1 \to \beta) \to (\alpha_2 \to \beta) \to (\alpha_1 \lor \alpha_2) \to \beta
:C09. \alpha \to \alpha \lor \beta
:C10. \beta \to \alpha \lor \beta
Universal
:C11. \forall x (\alpha_0 \to \beta) \to \alpha_0 \to \forall x (\beta) where x is not a free variable in \alpha_0
:C12. \forall x (\alpha) \to \alpha[x:=t] where x is a metavariable ranging over variables of the object language, t is also a metavariable ranging over terms of the object language
:C13 \alpha_0 \to \forall x (\alpha_0) where x is not a free variable in \alpha_0
Existential
:C14. \forall x (\alpha \to \beta_0) \to \exists x (\alpha) \to \beta_0 where x is not a free variable in \beta_0
:C15. \alpha[x:=t] \to \exists x (\alpha)
:C16. \exists x (\alpha_0) \to \alpha_0 where x is not a free variable in \alpha_0. The reference does not use it, I put it here only becauase I think it fits here, forming a kind of dual pair with C13. 
Identity
:C17. t = t where t is a metavarable ranging over terms
:C18. (s_1 = t_1) \to \dots \to (s_n = t_n) \to f(s_1, \dots, s_n) = f(t_1, \dots, t_n) where f is a metavariable ranging over function symbols, and s_1, \dots, s_n, t_1, \dots, t_n are metavariables ranging over terms.
:C19. (s_1 = t_1) \to \dots \to (s_n = t_n) \to P(s_1, \dots, s_n) \to P(t_1, \dots, t_n) where P is a metavariable ranging over predicate symbols, and s_1, \dots, s_n, t_1, \dots, t_n are as above
Questions
Unfortunatelly, I lack the knowledge and overview. which could be needed to write a good summary article. The most interesting thing for me in the differences and common parts in the above approaches:
* In the "conservative extension" approach, can reduction ad absurdum be postulated in a from, so that it resembes more to \lnot-introduction? (while double negation axiom would playing the role of \lnot-elimination.) Thus, instead of
::(\lnot\alpha \to \beta) \to (\lnot\alpha \to \lnot\beta) \to \alpha
:could we postulate 
::(\alpha \to \beta) \to (\alpha \to \lnot\beta) \to \lnot\alpha
:because I think, so  it would form a nice \lnot-introduction / \lnot-elimination pair with 
::\lnot\lnot \alpha \to \alpha
* Why is the most interesting axiom for universal quantifier in "conservative extension" treatment:
::\forall x (\alpha_0 \to \beta) \to \alpha_0 \to \forall x (\beta) where x is not a free variable in \alpha_0
:different from the corresponding axiom in "extension by definition" treatment:
::\forall x (\alpha \to \beta) \to \forall x (\alpha) \to \forall x (\beta)
Maybe these questions are not really important here, thus, if they are not really relevant for the article and the topic itself, please do not waste time for them.
Physis 13:56, 15 November 2007 (UTC)
thread_title:  Three approaches I found till now 

message: In a book by Stolyar,
:Abram Aronovich Stolyar 1970 Introduction to Elementary Mathematical Logic, Dover Publications Inc, NY , 
I found a reference to Alonzo Church 1956 'Introduction to Mathematical Logic, I [?? sic], Princeton University Press, 1956, and also David Hilbert and William Ackermann, Principles of Mathematical Logic, Chelsea, New York, 1950 (translation of 1928 German edition.)
On p. 102 Stolyar writes that "In one of Hilbert's systems, one takes the symbols for disjunction and implication" to form an axiom system for the propositional calculus. On page 167 Stolyar introduces the four formulas of the predicate calculus that appear in Kleene 1952:82, i.e. the same four that appear in the section above, in "Group A2". Bill Wvbailey (talk) 19:01, 18 November 2007 (UTC)
thread_title:  Possible references 

message: Maybe I have made an error as I wrote in the lead that an essential feature Hilbert-style deduction system (distinguishing it from other ones) is the specific trade-off between logical axioms versus rules of inference. I should have grasped the essence better. What is the main motivation? What is the most essential feature that sort of "defines" what can be regarded as a [variant of the many] Hilbert style deduction systems?
Two things could be explained in lead:
# The main motivation. I lack the knowledge to undertake it.
# Motivations, trade-offs are things that we can move along a continuous scale. But there is also a very clear, sharp, not-blurrable border that separates the various deduction systems, manifesting itself even in use of metasigns. This clear distinguishing (maybe defining?) characteristic feature is the way they define (and exploit) the auxiliary concept of “judgment”.
In Sequent calculus article, I read an approach that grasps the differences among deduction systems in the way they define the auxiliary concept of "judgment". Hilbert-style (HS), natural deduction (ND) and sequent calculus (SC) seem fro me increasingly exploiting this concept:
* from coinciding with a standalone formula (HS),
* through asserting deduction explicitly from a given context (ND),
* to reaching a sophisticated form (SC).
A standalone article could be written for Judgment (mathematical logic), detailing the roles of "judgment" in each of all the three deduction systems, in a similar way as it is worked out in Sequent calculus article. I lack the necessary sincerity of knowledge, thus I only have only written a stub (and extended also Judgment (disambiguation) slightly).
As for the other characteristics of Hilbert-style deduction system (trade-off between logical axioms and rules of inference, metatheorems that are not explicitly declared as building blocks, but can be demonstrated): are they only secondary phenomena following from the way judgments are defined? Theoretically, the number of rules of inference (or logical axioms) can be augmented (with declaring both-side rules for conjunction, disjunction, both quantifiers), but the resulting system will still be not the same as natural deduction, because judgments will still coincide with standalone formula themselves, lacking the explicit mentioning of context. Thus, it seems to be for me an independent feature, a casual coincidence, not giving the defining essence of Hilbert-style deduction systems (albeit tradition may suggest so).
In summary: how do these many things join to each other, and which are primary factors, which are of secondary nature, which are independent?
* motivation
* metatheorems
* trade-off between logical axioms and rules of inference
* exploiting auxiliary concept of "judgment"
Physis (talk) 23:45, 2 January 2008 (UTC)
replies: text: Could you give a reference for this? I don't think I've seen it in, for example, the Handbook of Proof Theory, but I might have missed it. The "party line" is that Hilbert-style systems trade fewer inference rules for more axiom schemes. — Carl (CBM · talk) 02:42, 3 January 2008 (UTC)
replies: text: Thank You very much for the quick feedback. I admit I cannot provide a reference it yet (except for article Sequent calculus, of couse I know Wikipedia cannot not be referenced by Wikipedia). But I remember vaguely a book that discusses the topic in a similar way, unfortunately I had to return it to the library, I try to take it again tomorrow. I hope I can provide exact references and more details then. Best wishes, Physis (talk) 03:42, 3 January 2008 (UTC)

text: Everything has also an article about "judgment" in formal theories], I shall check its references tomorrow (e.g. Pfenning, Frank and Rowan Davies. A Judgmental Reconstruction of Modal Logic. In Mathematical Structures in Computer Science, 11(4):511--540, August 2001). I shall check tomorrow if I can avail this article or someting refered by it. Physis (talk) 04:00, 3 January 2008 (UTC)

text: Thank you very much. I'll need some time to look through those before I can comment sensibly here. My main concern is first to understand what is being referred to as a judgement. A secondary concern is judging the correct amount of weight our articles should give to the concept. I'll be away for a few days, as well. — Carl (CBM · talk) 12:59, 4 January 2008 (UTC)
thread_title:  Concept of “judgment” as a more characteristic difference among various deduction systems 

message: The article uses ϕ in LaTeX formulas and φ in the text itself, as if they are the same symbol. It should be consistent and use only one of them. --salty-horse (talk) 22:44, 3 January 2009 (UTC)
thread_title:  ϕ vs φ 

message: Why is the frequently used Hilbert system a redirect to the infrequently used Hilbert-style deduction system?  I propose to reverse this redirect. — Charles Stewart (talk) 10:01, 26 February 2009 (UTC)
replies: text: Move applied. — Charles Stewart (talk) 07:33, 15 April 2009 (UTC)
thread_title:  Article name 

message: from A. S. Troelstra in “basic proof theory”:
"2.5.3. Hilbert systems. Kleene [1952a] uses the term “Hilbert-type system”;
this was apparently suggested by Gentzen [1935], who speaks of “einem dem Hilbertschen Formalismus angeglichenen Kalkiil”. Papers and books such as Hilbert [1926,1928], Hilbert and Ackermann [1928], Hilbert and Bernays [1934] have made such formalisms widely known, but they date from long before Hilbert; already Frege [1879] introduced a formalism of this kind (if one disregards the enormous notational differences). We have simplified the term “Hilbert-type system” to “Hilbert system”. There is one aspect in which our system differs from the systems used by Hilbert and Frege: they stated the axioms, not as schemas, but with proposition variables and/or relation variables for A, B, C, and added a substitution rule. As far as we know, von Neumann [1927] was the first to use axiom schemas."
Even in Hilbert and Ackermann [1928], they mentioned that this was first done by Frege. I think we should change the subject to something like "Hilbert-style system" or "Hilbert-type system" but I should take a look at A. Church’ Great Book for more information on history.  — Preceding unsigned comment added by Mrmusavi (talk • contribs) 12:08, 8 October 2012 (UTC)
replies: text: I support this idea. I would support a change to the article's title. I've been bothered from day one by the confusion between Hilbert's system (see Hilbert's papers in van Heijenoort, where it's clearly spelled out) and the various "systems" that followed. Kleene 1952:441ff indeed uses the words "Hilbert-type system" (see the index p. 543) in contrast to a formal system derived from Gentzen, as follows: "The formal system of propostiional and predicated calculus studied previously (Chapters IV ff.) we call now a Hilbert-type system, and denote by H. Precisely, H denotes any one or a particular one of several systems, according to wheter we are considering propositional calculus or predicate caluculus, in the classical or intuitionsitic version . . .."(p. 441) He then introduces the "Gentzen-type system G1". Previously in setting up his formal system (cf pages 69ff) Kleene makes use of "Hilbert and Ackermann 1928, Hibert and Bernays 1934, 1939, Gentzen 1934-5, Bernays 1936, and less immediate sources" (p. 69). Bill Wvbailey (talk) 14:40, 8 October 2012 (UTC)

text: I'd go even farther and rename it "Hilbert-style deductive system". Then it is very clear from the title what the article is about. Right now the title is completely opaque to me. — Carl (CBM · talk) 14:48, 8 October 2012 (UTC)
replies: text: A 1989 paper I'm reading now says, I quote: "natural deduction as opposed to Hilbert-style deduction" 67.198.37.16 (talk) 02:38, 15 December 2018 (UTC)
thread_title:  is it really a work by Hilbert? 

message: I believe that the axioms listed for the Hilbert system have a small error. The logical axioms each need to be preceded by a list of arbitrary universal quantifiers. E.g., instead of
\phi \rightarrow (\psi \rightarrow \phi),
it should read
\forall^\star (\phi \rightarrow (\psi \rightarrow \phi)),
where \forall^\star denotes a sequence of arbitrarily many universal quantifiers over arbitrary variables. And so forth for all the other axioms. Otherwise you can't infer, e.g., \forall x (R(x) \rightarrow R(x)) (supposing that R is a unary relation). (The universal introduction rule requires that the quantified variable not be free in the formula of interest.) See, e.g., https://cs.uwaterloo.ca/~david/cs245/cs-firstorder.pdf.
Nick Thomas <nwthomas@asu.edu>  — Preceding unsigned comment added by 68.230.67.135 (talk) 10:06, 22 November 2012 (UTC)
replies: text: The axioms do not need to be explicitly preceded by universal quantifiers, as non-closed propositions are implicitly assumed to be universally quantified in the semantics.  In fact new rules need to be added to the calculus if the axioms are so changed, in order to eliminate occurences of the quantifiers when axioms are invoked, making the calculus both more complicated to formalise, more complicated to use and not the same calculus as that proposed by Hilbert and Ackermann. — Charles Stewart (talk) 08:54, 5 April 2016 (UTC)
thread_title:  Possible error in axiom list 

message: The page says: "Universal quantification is often given an alternative axiomatisation using an extra rule of generalisation (see the section on Metatheorems), in which case the rules Q6 and Q7 are redundant."
I can't see how this can be the case. It is easy to prove Q6 and Q7 using universal generalisation and the deduction theorem, but proving the deduction theorem as a metatheorem with universal generalisation as a rule (as well as modus ponens) requires Q6 and Q7. 
Conversely, I don't see how you could do anything without universal generalisation as a rule. How would you prove something as simple as \forall x (x=x)? You can't use substitution of equivalents because x=x contains x free.
Is there perhaps a mistake here? 
The page also says: "US. Let \phi(p) be a formula with one or more instances of the propositional variable p, and let \psi be another formula.  Then from \phi(p), infer \phi(\psi)."
This also seems wrong to me. Let \phi(p) be  p \rightarrow q  and let \psi be \neg q.
Then we have from p \rightarrow q, infer \neg q \rightarrow q, which is clearly wrong. 
I think what it's meant to be is:
"US. Let \phi(p) be a theorem with one or more instances of the propositional variable p, and let \psi be a formula.  Then \phi(\psi) is also a theorem."
In other words, if  \vdash\!\phi(p)  then  \vdash\!\phi(\psi) 
AndreRD (talk) 11:50, 16 June 2016 (UTC)
replies: text: I agree. These seem like valid complaints. They need fixing. Let me add dubious tags 67.198.37.16 (talk) 02:13, 15 December 2018 (UTC)
thread_title:  Problems with Universal generalisation and Uniform substution 

message: Am I reading this right? 
A -> B -> A ^ B
I'm not seeing any mention of the branching rule used. This should be explicitly mentioned somehow. I'm assuming left branching and reading it this way:
"(A implies B) implies (A and B)" 
This would mean that if we accept that A implies B we have to accept B. What? And it means that if we accept that A implies B we have to accept A. What? Something seems to be wrong here.
If we assume some sort of right branching scheme it could be:
"A implies (B implies (A and B))"
Which seems fine. But right branching schemes require reading all the way to the end of a potentially very long expression before you can even figure out how to group terms, and then going back to the beginning of the expression and employing the previously discovered grouping while reading the expression. Could this really be how a standard notation for a Hilbert system works? It seems to me there must be something wrong here. Perhaps the expression in question was formulated incorrectly. But either way, to avoid confusion, some convention should be employed to allow a reader to figure out notation conventions. Perhaps a section on notation should be included in each article that covers a logical system. If Wikipedia employs certain agreed upon conventions about logical notations, then one or more articles about those conventions should be crafted and those articles could be linked to in a sufficiently prominent way from articles about logical systems. Readers should not have to guess, and notation systems used should be made explicit somehow. Comiscuous (talk) 19:48, 19 December 2021 (UTC)
thread_title:  Conjunction introduction and elimination