Talk:Sequent

message: I propose merging this page into the sequent calculus page, and turning this page into a redirect to the latter. ---- Charles Stewart 17:33, 18 Aug 2004 (UTC)

message: Note to self and any others who embark on search & correct missions to correct succede to succeed (the non-math-literate ones like myself, anyway) - succedent is in fact a real word and correctly spelled as such.  Those naturally drawn to the subject matter of which this entry is a part no doubt already knew that - just a note for others.  Longshot14 05:45, 2 March 2006 (UTC)
thread_title:  Spelling Context 

message: I just got to this page by clicking on the word assertion in another page. This word was interpreted as Logical assertion which was then redirected here. I think we need to have a separate article for assertion, even if the content of the latter might be minimal. Thanks! dcresti 19:39, 31 May 2007 (UTC)
thread_title:  Careful with the redirects! 

message: Isn't the rule given in the "Rules" section upside down? It is certainly sound (tough not so useful, and atypic), but the other way round it would be the cut-rule, which is probably what was intended. Mattias Ulbrich 17:24, 15 February 2007 (UTC)
thread_title:  Wrong rule 

message: I removed the following:
:It may help to point out that we assign these meanings to these symbols. The :rules themselves behave in a mechanical nature and do not carry underlying :meaning. See Gödel's incompleteness theorems for more on that topic.
as it is not clear to me what this is trying to say, and how it is helpful.
Zero sharp 19:43, 11 May 2007 (UTC)
thread_title: Removed a phrase

message: What's meant with logical formulae? Maybe Well-formed formula? 
Also, this article would benefit from a real example. Thanks, --Abdull (talk) 18:46, 25 October 2009 (UTC)
thread_title:  logical formulae? 

message: I think I've just been wasting my time adding references to the "Sequent" page. The "Sequent calculus" page makes this page completely superfluous. Anything useful in the "Sequent" page should be moved to the "Sequent calculus" page and the "Sequent" page should redirect to it, as suggested in the comment made 10 years ago at the top of this Talk-page.
--Alan U. Kennington (talk) 14:48, 10 June 2014 (UTC)
replies: text: I like the approachable character of this page. You have made our life easier with your work. What if we were to transfer it as an introduction in the target page, which is less fun to read. --Ancheta Wis    (talk  | contribs) 15:21, 10 June 2014 (UTC)

text: There may be a way around this. To me, constructivist mathematics and Gentzen go hand in hand, and to me, constructivism is the base for computer programming, i.e., hacking. Based on this I found P. Wadler (1993), "Curry-Howard isomorphism for Sequent Calculus", a private communication which is citation #13, p.75 in Hugo Herbelin (1994). "A Lambda-calculus Structure Isomorphic to Gentzen-style Sequent Calculus Structure" Computer Science Logic 933 (1994) 61--75.  Herbelin is already cited in the Curry-Howard isomorphism article citations, along with a provisional history, from which to work backward to Suppes. Probably Suppes has no citations other than Gentzen? --Ancheta Wis    (talk  | contribs) 16:29, 11 June 2014 (UTC)
thread_title:  Merge sequent page into sequent calculus page 

message: Now I've spent 24 hours improving (I hope) the sequent calculus page. I added a couple of introductory paragraphs in plain English for non-specialists (like me), highlighted some of the text with clearer section headings, and added a few references. So far, so good. However, I think I have discovered a kind of historic discontinuity between the old clunky Gentzen sequent calculus and the modern practical kind of sequent calculus which I have read in the books of Suppes (1957), Lemmon (1965) and Margaris (1967). I cannot find a historic link between the early sequent calculus using tree diagrams and the more modern sequent calculus which looks more like a computer program in nice neat lines with dependency lists (sequent left hand sides) for each line. I think of the Suppes/Lemmon/Margaris approach as dinosaur logic. So to me, the old Gentzen sequent calculus is a kind of trilobite logic.
The Gentzen style of sequent calculus has great historical interest, but it is so far out of date, it's really an embarrassment to talk about it. So ideally I would like to add a section to the sequent calculus page to present the modern style of practical sequent calculus (which by the way, I have shown is semantically sound, but that's another story). However, I cannot really do that because there is a big historical discontinuity between the old and modern styles of sequent calculus. I cannot find any link between them in the 45 books on logic in my personal collection. So I can't link the modern and old sequent calculus styles on that wikipedia page. It is quite possible that Suppes rationalised and modernized sequent calculus some time before 1957. Unfortunately, he gives no references at all in his book. He must have published the development somewhere.
Question:
Can anyone tell me how the Gentzen style of sequent calculus is historically linked to the Suppes (1957) style?
I cannot put the semi-modern Suppes style up on wikipedia unless I can find the historical link. And by the way, the more modern style (which I have developed myself from the Margaris style) is very nice indeed for proving theorems. It's a real breeze and a delight compared to other styles of predicate calculus and FOL that I have used. But wikipedia forbids original research. So you won't be seeing my system here for another 5 years at least. But that's another story....
--Alan U. Kennington (talk) 14:22, 11 June 2014 (UTC)
replies: text: There may be a way around this. To me, constructivist mathematics and Gentzen go hand in hand, and to me, constructivism is the base for computer programming, i.e., hacking. Based on this I found P. Wadler (1993), "Curry-Howard isomorphism for Sequent Calculus", a private communication which is citation #13, p.75 in Hugo Herbelin (1994). "A Lambda-calculus Structure Isomorphic to Gentzen-style Sequent Calculus Structure" Computer Science Logic 933 (1994) 61--75.  Herbelin is already cited in the Curry-Howard isomorphism article citations, along with a provisional history, from which to work backward to Suppes. Probably Suppes has no citations other than Gentzen? --Ancheta Wis    (talk  | contribs) 16:29, 11 June 2014 (UTC)
thread_title: Trilobite logic

message: First, the material which I had thought of adding to the sequent page is actually already in existence (although only briefly) in the System L article. Therefore I personally have nothing to add to the sequent page.
Second, the word "sequent", according to my literature searches in the last couple of days, is used exclusively in the sense of sequent calculus with disjunctive lists of propositions on the right hand side of the assertion symbol. In other words, the word "sequent" is not applied to the natural deduction style of assertions which always have one and only one proposition on the right of the assertion symbol. (This is unfortunate, because there does not seem to be a standard terminology for such single-output assertions. The word "judgement" or "judgment" seems to be non-standard. It isn't in any of my 45 mathematical logic books.) Gentzen made a crystal clear sharp division between natural deduction with single-proposition outputs and "logistic" deduction with disjunctive output proposition-lists. This sharp division seems to have survived up to the current time.
There is a further division of natural deduction into tree-diagram layout in the old Gentzen style, and tabular layout in the Suppes/Lemmon style, which apparently is called System L, although I have not read any original sources on this system specifically using this name "System L".
Thus there seems to be the following division of the subject.
Sequent calculus. All sequents belong in this topic.
Natural deduction. This does not use sequents.
Tree-diagram layout. This is most often used in the metamathematical theory literature.
Tabular layout. This is used mostly for the purposes of teaching logic in textbooks. It is also very useful for practical applications by anyone who actually wants to use logic to do some mathematics.
Conclusion:
The set of topics which come under sequent but not under sequent calculus is the empty set. So the sequent page is redundant.  — Preceding unsigned comment added by Alan U. Kennington (talk • contribs) 11:58, 12 June 2014 (UTC)
replies: text: Per Martin-Löf discusses the use of the term 'judgement' in  Martin-Löf attributes your observation to a linguistic divide (the English-speaking world neglects Kant), which may be worth an article. --Ancheta Wis    (talk  | contribs) 13:49, 12 June 2014 (UTC)
thread_title:  System L and sequents. Should evacuate and redirect this page soon. 

message: One argument in favour of the retention of the "sequent" page is that fact that it is translated literally into three other languages, namely Spanish, Japanese and Chinese. The "sequent" page in Italian has more information than the English-language page, including many of the inference rules expressed in numerator/denominator form, while the Russian page has almost no information at all.
The argument against retaining the "sequent" page in English is that the true sequent has disjunctive semantics for multiple propositions on the right hand side, and therefore the "sequent" topic is a subset of the "sequent calculus" topic. It is not possible to explain a sequent without also explaining "sequent calculus". It's like trying to explain a screwdriver without also explaining what a screw is! The one makes no sense without the other.
However, if one did wish to retain the "sequent" page as a separate page, it could be a useful place to explain the difference between a sequent and a simple conditional assertion with only one proposition on the right. There is currently no wikipedia page for "conditional assertion". In fact, if you do a search for it with double-quotes, you get the text which I wrote myself on the "sequent" page. So I think this would be an excellent place to clarify the categories of assertions:
# Unconditional assertion.
# Simple conditional assertion.
# Sequent.
This would parallel the text which I added to the "sequent calculus" page to clarify the difference between:
# Hilbert-style logical calculus.
# Natural deduction.
# Sequent calculus.
The "sequent calculus" contains many confusing technicalities for non-specialists. This is an argument in favour of having a less technical "sequent" page.
Another thing that could be clarified on the "sequent" page is the curious notion of a "judgement" or "judgment" (which is spelled very inconsistently on wikipedia). This is not familiar to most people in the logic area, in my opinion. There is no point in explaining assertions in terms of "judgments" if the word "judgment" is even less familiar to readers than the word "assertion". Normally an encyclopedia should explain less well-known words in terms of more well-known words!
--Alan U. Kennington (talk) 04:00, 15 June 2014 (UTC)
thread_title:  Other-language versions of the sequent page 

message: After doing some editing of the sequent page for a few days (and doing some concomitant research of the literature), it seems to me now that there is a distinct role for a sequent page as opposed to the sequent calculus page. The distinction is more or less analogous to the distinction between the proposition page and the propositional calculus page. The sequent and proposition pages deal with just the logical concepts themselves, and their meanings, whereas the corresponding calculus pages are concerned more with the detailed rule-sets for the respective calculi and the theoretical properties and implications of such calculi.
Therefore it seems to me now that the sequent page should not be deleted. It does have a useful role. This is particularly so because the rule-sets and properties on the sequent calculus page are really incomprehensible to non-specialists, and in fact it is an esoteric subject. For the non-specialist, it is the natural deduction systems along the lines of Lemmon's System L which are more comprehensible and relevant outside the mathematical logic research context.
--Alan U. Kennington (talk) 07:56, 17 June 2014 (UTC)
thread_title:  The distinction between the sequent page and the sequent calculus page. 

message: Although Gentzen 1934/35 and Church 1944/58 are crystal clear in saying that "⊢" is the same as "⇒", which is part of the object language, not the metalanguage, there may have been a "shift of paradigm"! Curry 1963/76, page 184, and Lemmon 1965, page 12, seems to say fairly clearly that "⊢" signifies provability. (However, Lemmon defined sequents to be single-assertion natural deduction style conditional assertions only.) My suspicion is that the developments in model theory during later decades led to a reinterpretation of history. Gentzen's original paper was saying anyway that the system was complete, and therefore there was no difference between syntactic and semantic implication. So it did not make any difference at that time, and model theory had not really started as a serious mathematical subject as it is today.
So I think there may be two paradigms: the classical paradigm and the modern paradigm. Although Margaris (1967) did not talk about sequents, he did present a kind of natural deduction in his book, and definitely he defined "⊢" to signify provability.
I think it would be very interesting to trace where the paradigm shift happened exactly, and who did it!
--Alan U. Kennington (talk) 18:09, 19 June 2014 (UTC)
thread_title:  Apparent shift of paradigm?