Talk:First-order logic

message: all t P(x,t) => P(x,y) (quantifier axiom)
P(x,y)=> exist z P(z,y) (same quantifier axiom)
all t P(x,t) => exist x P(z,y) (Using prepositional logic tautology for the above two, ((p->z)^(q->r))->(p->r)) 
all y (all t P(x,t)=> exist z P(z,y)) (Generalizing wrt y)
all t P(x,t) => all y exist z P(z,y) (Using q axiom 3 w= allt P(x,t) z(y)= exist z P(z,y))
all x ( all t P(x,t)=> ally exist z P(z,y)) (Generalizing wrt x)
exist x all t P(x,t) => all y exist z P(z,y) ( Q axiom 4)
exist x all y P(x,y) => all y exist x P(x,y) (changing dummy variables)
Please edit if there are errors and I want to know what is wrong and why.   
Thanks
Supems (talk) 17:37, 16 April 2008 (UTC)
thread_title: Proof for  Exist x all y P(x,y) => all y exist x P(x,y)

message: Have I missed it or is there no mention at all of the importance and very little of the applications in many fields of first-order logic (as compared with other logics)? Ht686rg90 (talk) 11:30, 9 October 2008 (UTC)
thread_title:  Importance and applications 

message: The argument for why FOL is needed over propositional logic is flawed.  The flaw is in the translation from English to propositional logic.  The proper translation would be:
* Socrates \rightarrow Man
* Man \rightarrow Mortal
* \therefore Socrates \rightarrow Mortal 
A better example is needed for why FOL is more powerful than propositional logic.
Drunkpotato (talk) 20:40, 5 December 2008 (UTC)
thread_title:  FOL vs Propositional Logic 

message: Some time ago, I added Generalized modus ponens to the inference rules (take a look at the link), but it got removed with the explanation "hardly a readable addition and not one of basic inference rules." Admittedly, it isn't very well written. But still, the more information, the better? What do others think? Offliner (talk) 17:13, 15 January 2009 (UTC)
replies: text: I think "The more information, the better" is a provably false motto, and thus a terrible one to live by (such as in the case of "generalized modus ponens"). Nortexoid (talk) 18:13, 17 January 2009 (UTC)
replies: text: Why exactly don't you support including it? Offliner (talk) 18:27, 17 January 2009 (UTC)
thread_title:  Generalized modus ponens 

message: The current article often deploys the acronym FOL, instead of just printing "First Order Logic".  That kind of shorthand doesn't seem useful in a general encyclopedia.  Most Wikipedia articles manage to avoid such acronyms, for example the United States article doesn't substitute "USA" every few paragraphs. --AC (talk) 07:47, 22 April 2009 (UTC)
replies: text: Sounds great to me. Make the edits.--Heyitspeter (talk) 00:55, 6 June 2009 (UTC)
thread_title:  "FOL" acronym is worthwhile because? 

message: If anybody in interested in pursuing that issue (not in this article, of course) there's a volume edited by Dov M. Gabbay called "What is a logical system?". The first essay there by Ian Hacking is "what is logic?" Pcap ping 12:01, 18 September 2009 (UTC)
thread_title:  OT: What is logic? 

message: Could someone explain why it redirects here.  I removed a statement about the term, but I reverted myself, as I'm not sure it isn't correct.  — Arthur Rubin  (talk) 01:59, 19 November 2009 (UTC)
replies: text: It's a less common term for first-order logic but I have seen it before. — Carl (CBM · talk) 03:47, 19 November 2009 (UTC)
replies: text: Could it not be confused with the study of more general types of quantifiers beyond for all and there exist. Such as there are aleph-1 or more of whatever. Or infinite conjunctions. See infinitary logic. Or Henkin quantifiers. JRSpriggs (talk) 04:00, 19 November 2009 (UTC)
replies: text: I agree that the term sounds more broad than just first-order logic. But Mendelson's Introduction to mathematical logic, for example, uses "quantification theory" as the title of the chapter on first-order logic. — Carl (CBM · talk) 04:28, 19 November 2009 (UTC)
replies: text: This justifies the redirect, but not this prominent mention in the article. The claim that quantification theory is the "theoretical study of first-order logic" also seems to be misleading to me; if it makes any sense at all then surely only in a philosophy context? Hans Adler 09:09, 19 November 2009 (UTC)

text: The sentence was worded a little strongly, so I just merged the term into the list of synonyms. — Carl (CBM · talk) 12:10, 19 November 2009 (UTC)
thread_title:  quantification theory 

message: I think the following sentence should be removed from the introduction paragraph.
"A history of first-order logic and an account of its emergence over other formal logics is provided by Ferreirós (2001)."
There's nothing wrong with it wrt its /content/, but it's a statement about the article rather than about FOL, I would say, and the introduction should be kept as concise and relevant as possible.
I'm putting it up for discussion here, but unless someone strongly object, I would like to delete it (and fit the sentence into another section, later in the article).
-- Minvogt (talk) 13:54, 13 January 2010 (UTC)
replies: text: It is usual for articles to have history sections. This article does not. The sentence you mentioned is the last sentence in the lead and may be regarded as a substitute for having a history section. There is no better place for it in this article. So leave it where it is. JRSpriggs (talk) 10:39, 15 January 2010 (UTC)
replies: text: What about moving it to the "references" or "further reading" section? along with a label?--Heyitspeter (talk) 20:42, 16 January 2010 (UTC)
replies: text: The reference is already there — "*.". The sentence just informs the reader that it is there. JRSpriggs (talk) 00:33, 17 January 2010 (UTC)

text: I don't think there is a much of another section where the sentence would fit. Really, the article could use a history section. But I don't like the idea of not mentioning the Ferrerios reference in the article at all. It's a high-quality reference, and readers looking at an encyclopedia article may well want to know about the history of the topic, so pointing out that reference seems like a good thing for the article to do. — Carl (CBM · talk) 03:19, 18 January 2010 (UTC)
replies: text: I see your point. I still think it's looks sort of clumsy, but I guess 'form follows function' applies here. -- Minvogt (talk) 02:01, 2 March 2010 (UTC)

text: This sentence is somehow unusual but acceptable, provided that Ferreirós 2001 is really a/the fundamental book on the matter. --Nemo 15:34, 31 March 2017 (UTC)
thread_title:  sentence doesn't belong to introduction IMO 

message: Third point in the definition says "Binary connectives. x is free in (φ --> ψ) if and only if x is free in either φ or ψ. x is bound in (φ --> ψ) if and only if x is bound in either φ or ψ. The same rule applies to any other binary connective in place of -->."
But this seems contradictory. If x is free in φ and bound in ψ, both conditions are satisfied. Surely x must be bound in φ and ψ to be bound in (φ --> ψ). Kronocide (talk) 16:33, 30 December 2010 (UTC)
replies: text: A variable can have both free and bound occurrences in the same formula; like "open" and "closed" in topology these are not exclusive. However, in practice people usually avoid formulas such as (\exists x)(P(x,y) \land (\forall y) Q(x,y)) by renaming one of the ys to something else. In that formula, y has both a free occurrence and a bound occurrence. The definition in the article here is identical to the one in Hinman, Fundamentals of mathematical logic, 2005, p. 100. — Carl (CBM · talk) 17:04, 30 December 2010 (UTC)
replies: text: I see. I was (erroneously) thinking of the variables rather than their symbols (such that variables would be individuated by distinct scopes in the formula). Thanks for putting me straight. Kronocide (talk) 14:28, 2 January 2011 (UTC)
thread_title:  Mistake in definition of free variable? 

message: I am starting a new thread because the previous one is getting unmanageable.  Yes, collective versus distributive is the issue.  The grammar page you referred me to is understandably not sensitive to the problem of infinite sets.  At plural quantification there is obviously more awareness of it.  Now this page cannot be solely a page about grammar, as first order logic is commonly used in math as you might imagine.  Numerous mathematicians think of quantification over sets of individuals as second-order quantification.  This is what was causing the misunderstandings over the lede during the past few months.  I think the mathematicians' viewpoint should be reflected here as well.  Tkuvho (talk) :32, 10 January 2011 (UTC)
:Sorry to refer you to a grammar page: to my suprise there was no wiki logic/philosophy articles on distributive and non-distributive predication. The following gives a better account:  http://plato.stanford.edu/entries/plural-quant/index.html It seems to me that (a) if you have only distrib preds then you can use FOL preds in the normal way (b) if you have only non-distrib preds you can represent then by second order preds or you could allow plural quantification in FOL. (c) if you have both distrib preds and non-distrib preds (as in the examples discussed above and, I think, in the famous Some critics only admire on another then you  would have to either us SOL with SO predicates and FO predicates or do some fancy footwork with plural quantification.  The situation represents a dilemma for philosophy (of logic). (a) If we allow SOL (and higher) without the caveat that it is a mere convenience and can be reduced to FOL then we pay the price of ontological commitment to the existence of predictates, sets and propositions in addition to objects.  The fear of Realism and what it entails is over two thousand years old. (b) If we do not allow SOL (and higher) then we seem unable to handle very simple problems, e.g. the logic of adjectives and adverbs. Surely "all green men are men" and "all who are running quickly are running" are logical truths but tricky to cast into FOL, as is "Roses are red and red is a colour".  My impression is that mathematicians are not too bothered by Realism. I do not know whether there are mathematical equivalents of the adjectives and adverbs problem described, but essentially the same problem arises with any predicates applicable non-distributively, an eg being (if I understnad it) yours regarding the boundaries of bounded sets. Philogo (talk) 02:25, 10 January 2011 (UTC)
replies: text: What do you see as the problem with Realism in your comment on item (a)?  Tkuvho (talk) 06:20, 10 January 2011 (UTC)
replies: text: Oh dear: I have two "a"s above. I do not think item (a1) creates any problems. (you have only distrib preds and use FOL preds in the normal way)  Ontological commitment is restricted to the elements of the domain.  (If the domain includes abstract objects, the problem is not created by the use of FOL.) With (a2) (use SOL and higher (without the caveat that it is a mere convenience and can be reduced to FOL) we pay the price of ontological commitment to the existence of predictates, sets and propositions in addition to objects  In other words we are ontologically committed  to entities in addition to the objects (initially) cited as elements of the domain, i.e a "torrent of universals".  
"By treating predicate letters as variables of quantification we precipitated a torrent of universals against which intuition is powerless" - Quine, Reification of Universals, para 6, in From a logical point of view, Harper, 1953/1961.Philogo (talk) 21:30, 10 January 2011 (UTC)
replies: text: Right, that's what I thought you would say.  I hasten to mention that I am sympathetic to Quine's criticism.  The ontological assumptions made in current mathematical practice could be questioned from a philosophical perspective.  However, it is not the purpose of an encyclopedia to be prescriptive.  The current mathematical practice being what it is, the "torrent of universals" is ironically a day-to-day reality one can't ignore.  Mathematicians do assume the existence of sets.  That's what ZFC is all about.  Editing this page so as to fight such a reality is not really what we should be doing.  Tkuvho (talk) 06:05, 11 January 2011 (UTC)
replies: text: I have no argument with that. Philogo (talk) 20:22, 11 January 2011 (UTC)

text: Hi JR, Thanks for your contribution.  What would you recommend as far as the lede is concerned?  Tkuvho (talk) 10:43, 10 January 2011 (UTC)
thread_title:  plural quantification 

message: The paragraph Restrictions, extensions and variations does not mention polish notation under Restricted languages.  Philogo (talk) 20:41, 11 January 2011 (UTC)
replies: text: Of course not, it isn't a restriction, just a different way of writing the same thing. The article does mention Polish notation in the 'Notational conventions' section, which is appropriate. — Carl (CBM · talk) 20:53, 11 January 2011 (UTC)

text: Mot sure about "of course". Some of the other "restictions" mentioned in the paragraph are also surely equaly "ways of writing the same thing." Perhaps they should be under variations.Philogo (talk) 21:10, 11 January 2011 (UTC)
replies: text: Polish notation has all the same quantifiers, connectives, and types of symbols that the usual syntax has. It is written in a different order, and has a different convention for parentheses. The restrictions listed in the restrictions section actually restrict things (e.g. no existential quantifiers, no binary functions) which are not restricted in Polish notation. — Carl (CBM · talk) 21:17, 11 January 2011 (UTC)
replies: text: Precisely; and "different convention for parentheses" would be a 'variation', would it not?Philogo (talk) 22:55, 11 January 2011 (UTC)
replies: text: It seems to me that the "notational conventions" section, where Polish notation is already mentioned, is a fine place for it.  I don't see why it needs to be mentioned a second time just because some section title has the word "variations"? Really that word could be removed, I think that "restrictions and extensions" would match the content there. — Carl (CBM · talk) 23:04, 11 January 2011 (UTC)

text: It make sense that the lable sense what is in the tin.  Philogo (talk) 23:08, 11 January 2011 (UTC)
thread_title:  Restrictions, extensions and variations 

message: Redirects here, so this paper, also as (pre)print, specifically, the giant theorem 1.1, which pieces together several (famous) results should probably be mentioned. I'm adding it to further reading for now. (I discovered it while I was looking for an accessible formulation of Kamp's theorem to add at Temporal logic). Tijfo098 (talk) 23:17, 20 April 2011 (UTC)
replies: text: I don't think there is a section now where that would fit. The idea of first-order definable languages on words is a pretty specialized application of first-order logic (similarly to the way that results on definability in o-minimal structures are a specialized case of general first-order definability).  I think that characterizations of languages in computability probably  belong in more specialized articles, along with results on o-minimality. Even saying what it means for a language to be first-order definable takes a little work. — Carl (CBM · talk) 00:07, 21 April 2011 (UTC)
replies: text:  Ok, I'll create a stub on that at some point. For important classes, like regular language, we have separate articles that cover the many ways to define/characterize them. Tijfo098 (talk) 09:01, 21 April 2011 (UTC)
replies: text: That seems reasonable. We do have an article on Büchi automaton that should also link to it. — Carl (CBM · talk) 10:42, 21 April 2011 (UTC)
thread_title:  First-order language 

message: First order logic is used in (or is) the foundation of mathematics, i.e. set theory in the form of ZFC and similar axiom systems.
But set theory seems to be ubiquitous in mathematical logic too. Much of mathematical logic seems to be formulated using set theory (naive version),
Most proofs of the basic things like compactness I have bumped into use ZFC in a blatant way. Sets, natural numbers, infinity,
the notions of cardinality, uncountability, functions, relations, the axiom of choice, etc. Basically the whole package is used.
What is the relation between mathematical logic and Set Theory on a basic level? Can one be formulated at all without some naive version of the other?
How about a little section discussing this? YohanN7 (talk) 20:25, 2 May 2012 (UTC)
replies: text: I doubt that it is possible to fully separate them. However in practice, one uses vague informal notions (in natural languages like English) to develop the rudimentary formal theories, and then builds on them to get the more elaborate formal theories. Notice that one can do simple logic without knowing anything about: soundness, consistency, completeness, compactness, the Löwenheim–Skolem theorem, etc.. These results which use advanced set theory are not part of first-order logic itself, rather they are part of its meta-theory; that is, they are about first-order logic rather than within it. JRSpriggs (talk) 06:48, 3 May 2012 (UTC)
thread_title:  Set Theory and Mathematical Logic 

message: Typo: In section "Loving Relations", the formula for 'no rows are empty' should read: ∀y∃xLxy. Simon.
:I don’t think so, rows in the tables correspond to the first argument of the L predicate.—Emil J. 14:36, 6 March 2013 (UTC)
thread_title:  Typo 

message: The organization of and terminology used in material in many related pages is shoddy at best. It's been almost a decade since I've (formally) studied formal logic, but even with that background I found the distinctions both pages made confusing. I figure both pages are important, especially the former, and likely to be visited by relative newbies. I've put this note here so that someone might have the opportunity to amend these pages - unfortunately I won't have time to edit them for a few months. Patrick R (talk) 19:04, 30 March 2013 (UTC)
replies: text: Vague general criticism is not helpful. Please be specific about what you think should be changed. JRSpriggs (talk) 12:30, 31 March 2013 (UTC)

text: The article predicate logic is just meant to direct readers to articles about specific logics, but someone had added quit a bit of idiosyncratic stuff there. I pruned it back down. — Carl (CBM · talk) 15:00, 31 March 2013 (UTC)
thread_title: Material in [[predicate calculus]] / [[monadic predicate calculus]] etc. 

message: I think this ambiguity example can be made easier if we also show it the following way. Show that the stronger version is "There is someone (who can be fooled every time)" and the weaker version is "(There is someone who can be fooled) every time". Next, the two versions can be transcribed into different logical statements as in the section. This is exactly how I first understood those paragraphs. FrenzY (talk) 04:03, 29 April 2013 (UTC)
thread_title:  "There is a fool" example 

message: The article says:
"The most common convention is that a formula with free variables is said to be satisfied by an interpretation if the formula remains true regardless which individuals from the domain of discourse are assigned to its free variables. This has the same effect as saying that a formula is satisfied if and only if its universal closure is satisfied."
However, there is an alternative view, and this concerns the interpretation of quantifier-free formula (QFF), an important class of formulas as many quantifier-free first order theories are decidable. In this view, we say that satisfiability of a QFF is equivalent to satisfiability of the formula in which the variables are all existentially quantified. OTOH, validity of a QFF is equivalent to validity of the formula in which the variables are all universally quantified (see for example Calculus of Computation by Bradley and Manna). Can the main page be changed to reflect this?  — Preceding unsigned comment added by Houseofwealth (talk • contribs) 21:10, 13 May 2014 (UTC)
replies: text: "Satisfiability" is defined as it is to ensure that the rules of inference at List of rules of inference#Rules of classical predicate calculus work. In particular, universal introduction would fail, if your definition were used. JRSpriggs (talk) 10:02, 14 May 2014 (UTC)
thread_title:  Question about interpreting formula with free variables 

message: Could anybody please explain to me what is meant by "they are examples" in section First-order logic#Loving relation? I have no idea who is "they" (The individuals a,...,e? The sentences/formulas 1,...,10? The matrices?) and of what they are examples. Thanks in advance. - Jochen Burghardt (talk) 19:00, 18 October 2014 (UTC)
replies: text: Apparently that sentence meant that all but some diagrams (they are the relations expressed by the diagrams) represent relations. I removed that sentence since it's ambiguous and can be easily be interpreted in a way that it's erroneous. The article needs a major cleanup anyway. Regards. Mario Castelán Castro (talk) 17:44, 19 October 2014 (UTC)
replies: text: In the removed sentence "Except for the sentences 9 and 10, they are examples.", "they" refers to the diagrams (that is, relations). For sentences 9 & 10, there is only one possible diagram which satisfies those sentences, namely the one shown. For any of the other sentences, there are other diagrams besides the one shown which also satisfy the sentence. Thus the diagram is merely an example in those cases, not a definitive representation of the sentence. JRSpriggs (talk) 10:52, 20 October 2014 (UTC)

text: To Jochen Burghardt: I like your suggested changes. JRSpriggs (talk) 06:38, 21 October 2014 (UTC)
thread_title:  Section "Loving relation" 

message: The sentence "First order logic is distinguished from propositional logic by its use of quantified variables" is ambiguous.  — Preceding unsigned comment added by 109.154.238.115 (talk) 09:18, 21 October 2014 (UTC)
thread_title:  Ambiguity 

message: The first line of that section contains the symbol for phi in text ("...M satisfies φ...") and then a formula in LaTeX. The phi in LaTeX renders differently for me (only for me or for everyone?).
So one time it's variant a of small phi and then it's variant b (see Phi). That might be confusing, because it almost looks like a different symbol.
One solution would be to use \varphi instead of \phi in LaTeX. But I don't know about the Wikipedia guidlines here and whether you even consider this a problem. (It did confuse me until I realized) Cptwunderlich (talk) 11:23, 29 October 2014 (UTC)
replies: text: The symbols look different in my browser, too, and they certainly should be made looking similar. I'm not aware of a policy preferring one or the other version; anyway, minimizing confusion should be more important than adhering to (other) style policies. - Jochen Burghardt (talk) 12:12, 29 October 2014 (UTC)
thread_title:  Different symbol in "Validity, satisfiability, and logical consequence" 

message: In the section of Expressiveness, subsection Formalizing natural languages, there it is stated that single-sorted first order logic cannot express many natural language features. Here single-sorted is used for the first time and left unexplained.  — Preceding unsigned comment added by 37.4.86.141 (talk) 11:41, 2 June 2015 (UTC)
replies: text: Being brought to the section =Limitations= by your pointing to an obvious deficit, I allow myself to articulate strong reservations in connecting formal languages with natural languages in the way it is handled in this section. I cannot see how the term natural languages is made available to a formal treatment. I feel the need of someone appropriately knowledgeable to improve this section. Run of the mill linguists may lack intimate experience with formal languages and their possibilities of being incomplete. Purgy (talk) 09:58, 3 June 2015 (UTC)
thread_title:  Single-Sorted First Order Logic 

message: ... have been reverted by  calling them style changes. I'm not a native speaker so I have no verdict on this, but I experienced more than a style change in the original edit, and find the reversion not only appropriate but absolutely necessary. I perceived the edits as if FOL were the ultima ratio in logic. Please, 72.251.70.97, discuss your intentions. Purgy (talk) 09:32, 8 June 2015 (UTC)
thread_title:  Recent edits by 72.251.70.97 ... 

message: You are right, the lead is messy. The reason is: First-order logic was merged with Predicate logic recently, and I tried to adapt the lead to discuss the relations between both. However, mentioning all subtleties that came to my mind made the text too long and complicated. Maybe, major parts of it should eventually be moved to an own section, called e.g. "Naming conventions and relations to other logics". On the other hand, at least the article and redirect names, i.e. "First-order logic" and "Predicate logic" should be introduced in the lead, and the former needs to be contrasted with "higher-order logic", and the latter with "propositional logic", so all the mess needs to be in the lead, doesn't it? Currently, I have no idea how to cope with this. Maybe, you have? - Jochen Burghardt (talk) 14:03, 6 December 2016 (UTC)
replies: text: I don't think that the lede is the place to try to talk about higher-order logic; this is already in a section in the article. Similarly, the article section "introduction" mentions propositional logic. In general, the lede should just be a short summary of the article, rather than developing significant points on its own. — Carl (CBM · talk) 15:12, 6 December 2016 (UTC)
thread_title:  Lead is messy 

message: I undid an edit by an IP editor but it may have been a little hasty. I was partly reacting to the edit summary's misspelling of grammar and on second look I'm not sure it wasn't a legitimate edit. The edit changed "there exists a man such as X is Socrates and X is a man" to "there exists X such that X is Socrates and X is a man" which actually seems like it's correct to me. I'll leave this up to someone familiar with the topic of the article. —DIY Editor (talk) 16:14, 11 December 2016 (UTC)
replies: text: It should be "such that" - I'll fix it. Thanks for the note here, I would not have noticed the edit otherwise. — Carl (CBM · talk) 16:35, 11 December 2016 (UTC)
replies: text: As a user of English only in non-native ways I do not dare to touch such delicate equipment, but may I remark that the formulation as of now is not very easy to grasp for me (and never was in any of the previous three versions). As far as I understand, it is about the difference between propositions formulated with and without variables (X). I imagine that it might be possible to express this still more amenable to non-precognizant readers. Thank you. -Purgy (talk) 17:43, 11 December 2016 (UTC)
thread_title:  Question regarding an edit 

message: An example of formulas speaks about a function symbol, althought function symbols are not defined in Formulas section.
Something is wrong! --VictorPorton (talk) 01:54, 8 November 2019 (UTC)
replies: text: Formulas may contain terms and those terms may contain function symbols. JRSpriggs (talk) 09:44, 8 November 2019 (UTC)
thread_title:  An error in the definition of formulas 

message: The section Notational conventions gives an order of precedence, but provides no source for this. I have been looking around, and I can find several sources (presentations from classes mainly), that seem to disagree. This answer from stackexchange quotes Herbert Enderton, A Mathematical Introduction to Logic, page 78, which also has quantifiers with higher precedence than ∧ and ∨. --Lasse Hillerøe Petersen (talk) 13:25, 3 May 2020 (UTC)
replies: text: If omitting a pair of parentheses would result in two or more reasonable interpretations of the string, then they should be retained. JRSpriggs (talk) 19:42, 3 May 2020 (UTC)
thread_title:  Precedence order (section 2.2.3 Notational conventions) 

message: There is an inconsistency of terminology between some different articles and I'm not sure how it should be fixed. This article currently has a section called Deductive systems, and it gives as examples of deductive systems, Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. The article on Formal system gives first order predicate logic as an example of a deductive system, i.e. it is a broader definition that potentially encompasses an entire logic. The article on proof calculus, or proof system lists things like Hilbert systems, natural deduction systems, sequent calculus, etc., as examples of proof calculi. So, is a deductive system a logic, or is it a proof system? Or is it not synonymous with either? Or is the terminology just inconsistent among different authors?     Dezaxa (talk) 21:31, 29 September 2021 (UTC)
replies: text: I believe first-order logic is not a deductive system, since the objects of a proof calculus are what are called deductions and intended to formalize deductions. A deduction is like a parse tree of the proven formula, it shows explicitly how the formula is constructed from the axioms via rules intended to preserve truth. But the parse tree of a formula in first-order logic doesn't deduce anything (about truth). C7XWiki (talk) 10:25, 25 September 2022 (UTC)
thread_title:  Inconsistency of terminology 

message: I would like to add function application to the abelian group example, as I think it would make the ability of FOL to apply functions more clear. For example we can construct a formula φ(x,y) that's true iff y = x + x + x, so we can said to have defined f(x) = x + x + x. Some formula such as \forall x(x\leq f(x)) makes sense in this context even though f isn't a function symbol in the language, since we read this as \forall x\exists y(\phi(x,y)\land x\leq y). But I don't have proper sourcing to write this section, this book by K. Kunen gives a similar example of how we read function expressions within a formula for "S(y)\in x" but not the general case, nor "f(x)=y". C7XWiki (talk)
Edit: Maybe Quine 1940 is another sufficient source. C7XWiki (talk) 18:48, 28 July 2023 (UTC)
thread_title:  Example of definable functions 

message: The introduction to this article is incorrect. It states: "First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable."
First, "Socrates is a man" is not existentially quantified, it is a simple predicate: "x is a man", which is symbolized as "Mx" where M[φ] =df "φ is a man", and x =df a variable (note: Socrates is not a variable, it is a single term "s"). So, "Socrates is a man" would be symbolized as "Ms," not ∃x(Mx & Sx), or "there exists an x such that x is a man and x is Socrates".
Second, the source for this sentence even confirms this: 
"First order logic allows the use of sentences that contain variables. So that rather than the proposition
Socrates is a man
one can have sentences of the form
X is a man
where X is a variable."
This explanation doesn't contain any quantifications and merely confirms the use of predicates in first-order logic.
If you wanted to display quantification and predication as an example, the ole "All men are mortal; Socrates is a man; therefore, Socrates is mortal" might make more sense:
1. ∀x(Mx → Ox)
2. Ms
3. /Os
Where: O[φ] =df φ is mortal. M[φ] =df φ is a man. s = Socrates.
Translated out, it would read:
1. For all x, if x is a man, then x is mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal. 173.13.91.9 (talk) 15:04, 6 July 2023 (UTC)
replies: text: I completely agree. Do you have a particular text suggestion how the (2nd sentence of the) introductory paragraph should look like? - Jochen Burghardt (talk) 16:13, 6 July 2023 (UTC)
replies: text: Perhaps something like: "First-order logic uses predicates and quantifiers (among other systems) to extend atomic propositional statements with variables. Whereas "all men are mortal" would be translated as "M → O" or "if you're a man, then you're mortal" in simple propositional logic, first-order logic can express the statement as "∀x(Mx → Ox)" or "for all x, if x is a man, then x is mortal." 173.13.91.9 (talk) 15:06, 7 July 2023 (UTC)
thread_title:  First Example is incorrect 

message: Would it help to include some information on the history of the syntax-semantics distinction? (E.g. according to Ebbinghaus's Ernst Zermelo (2007), pp.182--183, acceptance of it was what set "classic" set theory researchers such as Zermelo and Fraenkel apart from the younger generation of set theory researchers such as Goedel, Skolem, and von Neumann). In this way the syntax-semantics distinction appears much more recent than the advent of first-order logic. C7XWiki (talk) 21:23, 20 April 2024 (UTC)
thread_title:  Syntax-semantics distinction and history