Talk:Atomic sentence

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Contents 1 edits 2 Proposals for this article 3 Interpretation (logic) 4 Merge with Atomic formula 5 Definition by subformula 6 new definition in lead 7 Proposed New Introduction

[edit] edits

Edited all accept the examples: --Philogo 12:23, 3 September 2007 (UTC) Added more examples --Philogo 18:41, 3 September 2007 (UTC) Deleted some redundant para, tidied up lead definition - pretty sure its sound. Moved a para to bottom under "thoughts" pre tem. OK everybody? I propose we discuss future edits here before making them; saves time and effort and tempers--Philogo (talk) 14:38, 20 February 2008 (UTC)

[edit] Proposals for this article

Carl (CBM · talk) says I think that a more nontechnical definition (one that attempts to capture what an atomic sentence would be in natural language) would improve the article. Is this a good idea or should the article stay with dealing with the term Atomic Sentence as used in Symbolic Logic? Problem 1. It would be on what an atomic sentence would be in natural language, as used in Logic (see article title) not what is. If the term is not used in Logic in connection with natural language then this would be the proscribed original reseach and or not to do wuth Logic, more perhaps Philosophy of Logic of Philosophy of Language. Great topic, but is this the place fot it? If you look up Wittgenstein. TLP para 4.21 to 4.24 you will see that whatever W. used in Germn is transalted as elementary proposition not atomic sentence. He said:
4.24 The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs
but then says:
4.24 ..I write elementary propositions as functions of names, so that they have the form 'fx', ψ(x,y) etc. or I indicate them by the letters 'p', 'q', 'r'. so HE was referring to an artificial language, but beleives - I understand - that there are facts which are what is the case and the latter is the existence of a state of affairs (2) and with 4.24 he must believe that there are elementary facts which correspond to (are pictured by) elementary propositions. But W. said Russell and Moore would never understand TLP so what hope have I? And is this article the place to discuss these issues, or should they not better come under a Phil o Logic Article?--Philogo (talk) 20:47, 20 February 2008 (UTC)

[edit] Interpretation (logic)

I am thinking about moving the material under the interpretation section to this other article. It is a better treatment of the subject. Pontiff Greg Bard (talk) 16:52, 31 December 2007 (UTC)

Which material is a better treatment of what subject? Do you,htink that this article (Atomic Sentence) can do without the material under its interpretation section?

--Philogo (talk) 13:54, 31 January 2008 (UTC)

We don't have to move it, we could duplicate the material. Provide a small summary, here, and put a "Main article:Interpretation (logic)" in the section hatnote. Pontiff Greg Bard (talk) 17:53, 31 January 2008 (UTC) (Done)

You mean use the same examples? I rather favour 'mini-articles' like this one Atomic Sentence to explain technical terms - longer that a dictionary definition and with explains and links to related terms.

BTW I would favour this and similar aticles to be in the format 'Atomic sentence (Logic)' as in Argument (Logic). Similarly I think 'Disjunction (Logic)' is a better heading than 'Logical Disjunction' etc. If you agree would your rename this artice - I would not know how.--Philogo (talk) 02:06, 9 February 2008 (UTC)—Preceding unsigned comment added by Philogo (talk • contribs) 01:58, 9 February 2008 (UTC)

Please check out Wikipedia:WikiProject Integration. It has basically the opposite mission as yours. It is much better to have comprehensive articles with lesser supporting articles. Perhaps there is a place for a small summary of this article in some other more comprehensive article, and a link to this whole one from it (or some similar plan).

There are a bunch of articles that mathematicians have split up in to concept (math), and concept (everyone else). It seems that if there is anything non-mathematical to say about something that it needs to get kicked out resulting in, for example, lemma (mathematics) and lemma (logic). Even though we all know that these really are the exact same thing.

The result is that there are many of these stub and start class articles that could be worked into some B article making it an A. Instead they languish. I think the argument article, for instance, is a good christmas tree with ornaments type of arrangement.

Perhaps we could work content on molecular sentences in with this somehow too. Be well, Pontiff Greg Bard (talk) 23:41, 21 March 2008 (UTC)

[edit] Merge with Atomic formula

Kaustuv Chaudhuri said, 13:14, 26 September 2007 (UTC), in Talk:Atomic formula:- Atomic sentence as a Wikipedia article is a disgrace; ideally these two articles should be merged, but the turf wars that will result with that attempt are entirely too predictable. You might consider reading more than a single textbook on logic.

Is it that bad? —Preceding unsigned comment added by Philogo (talk • contribs) 13:57, 31 January 2008 (UTC)

No merge. They are very closely related, however, that justifies keeping them separate, so that the distinction is clear. Pontiff Greg Bard (talk) 19:40, 31 January 2008 (UTC)

At some point I have been planning to work on all these split- up articles on formulas, sentences, etc. "Atomic sentences" are a strange concept to emphasize, as they are simply the sentences of propositional logic. Eigenvector is a nice example of how intimately-related topics can be covered in a single article. — Carl (CBM · talk) 22:21, 31 January 2008 (UTC)

...as long as it is presented in such a way that one gets confused. People may be looking up one or the other so as to be able to distinguish them from each other, etc. I think having a separate article helps, but then it will always be a short one. I am very supportive of the goals of WP:INT. Pontiff Greg Bard (talk) 22:32, 31 January 2008 (UTC)

Are not 'Fa', 'Gab', 'Habc' etc. atomic sentences although not sentences of propositional logic?

--Philogo (talk) 14:17, 11 February 2008 (UTC) I am not sure that to have an article on 'Atomic Sentence' means that we believe it should be emphasised, but that it is the sort of term somebody might come across and want to know what it means, with a bit more discussion than you get in a dictionary. It is most true than a single article could explain a whole number of terms in a very readable way. An article on the internal compustion engine coud cover gaskets, big ends, plugs and all that it an interesting way. But if I wondered what a 'big end' and why I would not want it blown, I might not know to read an article about combustion engines, and I might not want to read all other stuff. I think that it rather in the aature of an encyclopedia that it has seperate short pieces called "articles" which could all be covered in a book. We have enclopedias so we don't have to read a whole goodamn book. That has to be our skill, write artilces that are not too long and not too short and directing the reader to other articles. --Philogo (talk) 02:19, 9 February 2008 (UTC)

One issue is that atomic sentence is a virtually unknown term in the literature. It makes sense as a term, but only a a corner case of atomic formula when there are some 0-ary relations in the language. Since ordinarily one doesn't work with 0-ary relations in the context of first-order logic (as a matter of practice), the term really isn't particularly important. It's right up there with putting an atomic formula in prenex form - it has one, but prenex form is only interesting for formulas with quantifiers. Similarly, the idea of sentence is only interesting for formulas with variables. — Carl (CBM · talk) 02:22, 9 February 2008 (UTC) — Carl (CBM · talk) 02:22, 9 February 2008 (UTC)

Are you sure that 'atomic sentence' is a virtually unknown term in the literature? I thought it a common term. Elliot Mendelson for exammple uses it in Introduction to Mathematical Logic and if you google it there are many hits. I would think that the notion of a sentence is extremely basic to Logic - the first think you learn about, followed by the distinction between Atomic and Compound sentences and thus truth functonal connectives. Much more basic than prenex form for example. For an encycopedia I would expect us to cover all the basic concepts and terms in Logic, just as I would expect there to be articles on basic terms like Mass, Momentum, Velocity written by the Physisists.--Philogo (talk) 02:42, 9 February 2008 (UTC)

In the context of propositional (sentential) logic, perhaps. But this article uses the definition of sentence from first-order logic, in terms of free variables. Perhaps the issue is that the article is simply confused. — Carl (CBM · talk) 03:08, 9 February 2008 (UTC)

The definition 'In (sentential logic and predicate logic), an atomic sentence is an atomic formula in which no variable occurs free.' would appear to define the term atomic sentence for both sentential and first order logic. Since there ARE no variables in the former, it would follow that in the former (but not the latter) an atomic sentence is the same as an atomic formula. If the definition does not hold for higher order logics then it should be amended to say 'In (sentential logic and first order predicate logic),...'--Philogo (talk) 14:24, 11 February 2008 (UTC)

If the following examples given are indeed NOT atomic sentences:

• x(F(x))
• z(G(a,z))

then parts of the article above need revision, e.g.
'An atomic sentence then is an atomic formula in which no occurrence of a variable is free. An occurrence of a variable in a wff is bound if it is within the scope of a variable-binding operator (e.g. a quantifier or description operator) and otherwise it is free.' since this definition would allow variables in atomic sentences provided they are bound. Either that is wrong or the examples are wrong. PS what is a 'description operator' as opposed to a quantifier? --Philogo (talk) 14:29, 11 February 2008 (UTC)

No, those aren't atomic sentences, since they are not even atomic formulas. — Carl (CBM · talk) 15:14, 11 February 2008 (UTC)

As I said if they are not then their examples should be removed from the articel, but... If they are not atomic formulae then Mendelson, Introduction to Mathermatical Logic, 1964, Chapter 1 page 46 is wrong when he writes:

The predicate letters applied to terms yield the atomic formulae, i.e. if A is a predicate letter and t1..tn are terms, then A(t1..tn) is an atomic forumula. [Terms are previously defined as just variables, individual constants and functions] Also see: Formal Logic/Predicate Logic/Formal Syntax i. A sentence letter (a zero-place predicate letter) is a well-formed formula. ii. If A is an n-place predicate letter (n greater than 0) and a1..an are terms, then A(a1..an)is a well-formed formula.

An atomic formula is one formed solely by formula formation clause {i} or {ii}. Put another way, an atomic formula is one in which no sentential connectives or quantifiers occur. —Preceding unsigned comment added by Philogo (talk • contribs) 14:18, 12 February 2008 (UTC)

Do you have alternative authority? We should get this exactly right if we are writing an encyclopedia article.

If Mr Mendelson is correct then the question remaining is whether such an atomic formula containing no free variables is an atomic sentence. Mendelson calls wfs with no free variables closed wfs (he himslef does not appear to call them atomic senences) In other words is it correct to apply to closed atomic wfs the term atomic sentence? —Preceding unsigned comment added by Philogo (talk • contribs) 14:07, 12 February 2008 (UTC)

Notice the explanation you quoted: "Put another way, an atomic formula is one in which no sentential connectives or quantifiers occur." That is exactly the normal definition of an atomic formula. So since there are no quantifiers, and no free variables, there must be no variables whatsoever in an "atomic sentence". I looked at three references quickly this morning, and none of them even included in the definition the possibility of 0-ary relation symbols in a first-order language. So while it would be possible to define an atomic sentence in first-order logic as a sentence consisting of a single 0-ary relation symbol and nothing else, this definition is essentially unheard of, because it has little practical interest in first-order logic. — Carl (CBM · talk) 14:40, 12 February 2008 (UTC)

I see the source of some confusion here. The article says:

" the following wfs are atomic sentences:

• p
• F(a)
• H(b,a,c)

The following wfs are atomic formulae but not atomic sentences because they include free variables:

• F(x)
• G(a,z)
• H(x,y,z)
  

" I MEANT to say above:
" If the following examples given are indeed NOT atomic sentences:
* x(F(x))

• z(G(a,z))
• F(a)
• H(b,a,c)

then parts of the article above need revision

" --Philogo (talk) 13:26, 13 February 2008 (UTC)

Things like F(a) are atomic sentences, yes, in the sense that they are atomic formulas and sentences. Now the question is the extent to which this topic is covered in logic texts. My experience is that it is not a topic of much interest, and rarely covered. — Carl (CBM · talk) 13:41, 13 February 2008 (UTC)

Well then the definition of atomic sentence and the examples given are OK afer all, but the topic you feel is not of much interest. Perhaps we should move on then Carl? --Philogo (talk) 14:18, 13 February 2008 (UTC) —Preceding unsigned comment added by Philogo (talk • contribs) 14:08, 13 February 2008 (UTC)

Please note that the topic of this section is about merging this with the article on atomic formulas. I'm glad that you've figured out the definitions, so that we can get back to talking about that. — Carl (CBM · talk) 14:42, 13 February 2008 (UTC)

I am glad you agree the defintions and examples are OK after all. Gregbard says no merge and I agree. The article needs a rewrite, as parts of it are sloppy. You do not think the topic is of much interest but I think it is basic. —Preceding unsigned comment added by Philogo (talk • contribs) 01:30, 14 February 2008 (UTC)

[edit] Definition by subformula

User CBM writes "It isn't true that an atomic sentence is same as a sentence with no subsentence - "\forall x (Rx)"", though 'subsentence' is often defined so that for any term t not containing x free, Rt is a subformula of (x)Rx (and likewise for the existential case). By this generally accepted definition of subformula, an atomic formula is the same thing as a formula with no proper subformula. Perhaps someone can give a reference which gives a different definition of subformula. Nortexoid (talk) 21:38, 19 February 2008 (UTC)

Even using that definition, which I agree with, you would need to know that there is some constant symbol in the signature in order to produce a subsentence of . So "no proper subsentence" can't be used as a general definition of atomic sentence. (In general, I suppose that if there are no constant symbols in the signature, and thus no closed terms, and moreover there are no 0-ary relation symbols, then there are no atomic sentences whatsoever. That is, of course, if an atomic sentence is meant to be an atomic formula that is also a sentence.) — Carl (CBM · talk) 23:05, 19 February 2008 (UTC)

As you say, if there are no closed terms then there are no atomic sentences besides propositional constants, which are never considered part of a first-order language anyway, except perhaps for the falsum or truth constant. So it is safe to assume the language has closed terms. Also, that there are no individual constants does not imply (in general) that there are no closed terms, as there may be e.g. term-forming operators in the language.

I find the definition in terms of there being no proper subsentence to be both clearer and simpler, which is why I used it for the introduction. It was then followed by a more involved definition relying essentially on particular features of the kind of language under discussion. This reliance has been removed from the current version. Also, what does a logic have to do with the definition of a sentence? We're talking about languages of logics. I find the current introduction slightly, though nonetheless, inaccurate. (Also, does 'the language of predicate logic' preclude higher order languages? If not then you need to state in general terms what an atomic sentence is for every order of language.) Nortexoid (talk) 23:51, 19 February 2008 (UTC)

I wish that the lede had a more nontechnical definition of atomic sentences, actually; I think I can find one once I get a chance to look at some references, but it will be a while before I can do that. I split some of the material into sections, lower down, for propositional and predicate logic. The details can always be spelled out there.

I am using predicate logic as a generic term, per that article. The definition of atomic sentence is, presumably, the same for all of them. The "standard" presentation of these admits only constant symbols, variables, and function symbols for forming terms; what other operators are you thinking of?

Description operators, e.g., are quite popular in the literature, especially in philosophical logic. They're variable-binding so they can form closed terms without there being any individual constants. Abstraction operators are another popular term-forming operator (for higher-order terms). And there are others.

I don't agree it's safe to assume there are closed terms, or that the definition should do so. I would normally ask whether there are references that do so, but I don't think there are many references that define atomic sentences for predicate logic at all. The main interest I have seen in atomic sentences is in theories of truth, not in technical contexts. — Carl (CBM · talk) 00:07, 20 February 2008 (UTC)

Do you mean atomic sentences specifically or atomic formulas in general? They play a vital role in both preservation theorems in modal logic and in computational logic, they form the basis of induction definitions of valuations for complex formulas, that any set of atomic formulas has a model is a central (for a number of reasons) theorem in model theory. Any text I've seen that defines formulas inductively gives a definition of atomic sentence as atomic formula with no free variable occurrences. Nortexoid (talk) 09:27, 20 February 2008 (UTC)

The texts I am familiar with define sentences and atomic formulas, but not specifically atomic sentences; this is probably just a difference in the collection of texts we are familiar with. If you have a few minutes, would you mind adding a few paragraphs about the theorems of modal logic and computational logic you alluded to? This article would be improved by a more thorough coverage of the actual use of atomic sentences, rather than just the definition. I am hoping to add more about the relationship between atomic sentences and theories of truth, but it will be a while before I can get to it. — Carl (CBM · talk) 13:07, 20 February 2008 (UTC)

[edit] new definition in lead

I removed a senetence Philogo added to the top of the lede. On one hand, it seemed to be redundant to the two sentences below it. But it also had a technical issue - there are no free variables at all in sentential logic (the concept doesn't make sense there), so we can't mention free variables when defining an atomic sentence in the context of propositional logic.

I support Philogo's attempt to add a better definition, though. I think that a more nontechnical definition (one that attempts to capture what an atomic sentence would be in natural language) would improve the article. — Carl (CBM · talk) 14:13, 20 February 2008 (UTC)

user before reverted mid my edit! Be calm. The lead defeinition "In Logic an atomic sentence is an atomic formula in which no variable occurs free." is sound I am sure. the following two sentences follow from it. User before last edited mid my edit. he defeintion is strictly about the term atomic snecne as used i the artifial languages of senetential logoc and predicate logic - not natural langauges. I was intending to add a discussion on the latter in another para. Will you join me there and leave the defintion as it is for now? (unless there are serious objections) --Philogo (talk) 14:30, 20 February 2008 (UTC)

Yes, let's take some time and get a good definition for the lede. I'll leave it alone for the rest of the day. There is still a technical issue that an atomic sentence in predicate logic may involve closed terms, not simply constant symbols as the current version of the lede says. — Carl (CBM · talk) 14:33, 20 February 2008 (UTC)

It also appears that the section on propositional logic is using things that belong only to predicate logic, such as terms and relation symbols; this will need to be moved to the correct section. The term propositional variable is well established for th e letters that are used in propositional logic that can be either true or false in an interpretation. — Carl (CBM · talk) 14:35, 20 February 2008 (UTC)

Can you give me an example of any atomic sentence in predicate logic involving closed terms, not simply constant symbols?

The article as I recal referred to propositional variables and propositional constants; I have used the term sentential letter to refer to the symbols in sentential logic. I used to talk about propositions and propositional logic but I have followed the lead of others that we should avoid the use of the word proposition since the term is controversial at least since the time of Quine. The terms sentence and wff etc as used by me in this article and in text books refers simply to the marks (as Frege termed them) on pieces of paper; such marks do not raise the ontological issues that proposition and statement do. It would be great to air these issues, but in another Philosphy of Logic aricel; lets give the poor reader a break and keep it simple here. We should aim for more endoresments like this: I've actually found the articles quite helpful and in many cases much clearer than the textbook. The fact that there are few other well organized online resources makes this a much more critical resource IMHO. left on my talk page --Philogo (talk) 20:16, 20 February 2008 (UTC)

re the section on propositional logic is using things that belong only to predicate logic. If there was such a section its not there now, so THAT issue is now moot. —Preceding unsigned comment added by Philogo (talk • contribs) 20:00, 20 February 2008 (UTC)

If the signature has a unary relation R, a unary function f, and a constant symbol a, then R(f(f(a))) is an atomic sentence, based on the closed term f(f(a)).

I can't agree to the current lead,

"In Logic an atomic sentence is an atomic formula in which no variable occurs free. It follows that in sentential logic, an atomic sentence is single sentential letter; in predicate logic, it is an n-ary preditate letter followed by n individual constants."

In the setting of sentential logic, it is meaningless to ask whether any variable occurs free; there is no definition of free or bound variables in sentential logic. So the claim that "it follows" isn't meaningful, since the definition in the first sentence doesn't make formal sense in sentential logic. The easiest way to avoid this problem is to give the definition in two pieces, one for sentential logic and one for predicate logic.

In the setting of predicate logic, as I pointed out, there is also the issue of closed terms that may appear in atomic sentences. — Carl (CBM · talk) 20:18, 20 February 2008 (UTC)

For what it's worth, I am looking at Philogo's latest work, and I think I like Carl's formulation better. Please include the relationship of an atomic sentence with well-formed formula, and sentential logic. I think in all cases where we can designate a particular major part of logic we should (rather than just saying "In logic, blah blah blah...") Thanks for both of your guys works. Be well, Pontiff Greg Bard (talk) 20:49, 20 February 2008 (UTC)

Re. closed terms discussion above.

If we define an atomic wff:

In Logic a term is a variable, an individual constant or a n-place function letter followed by n terms; an atomic formula is an expression consisting of either a sentential letter or an n-place predicate letter followed by n terms.

then the definition of an atomic sentence which I have proposed for this article, i.e. :

In Logic an atomic sentence is an atomic formula in which no variable occurs free.<br /> or
In Logic an atomic sentence is an atomic formula containing no variables.<br />

will allow atomic sentence to be (a) a sentential letter (b) an n-ary predicate followed by n individual symbols or n functions providing only that it contains no (free) variables.br /> (i.e its a atomic formula with no quantifiers or variables)

Thus the two definitions together allow all that should be allowed and prevent all that is disallowed. As in many subjects there is not just ONE way of defining a set of interrelated terms; you can choose which to start with (in the same way you can choose what axioms you want.) Thus in Geometry you can define a point in terms of lines (intersection of two lines) or instead define a point as that which has position but no length; a line as that which has length but no breadth (Euclid). Or a line is the intersection of two plains. You choose what you want to start with as undefined terms. See Frege e.g. on definitions. I was attempting to have a definition of an Atomic Sentence which was good for sentential as well as predicate logic. If we give different definitions for each, then up jumps Socrates and say "I don't want examples; I want to know what all atomic sentences have in common". You can use the subpart idea suggested above. E.g. we might say "A sentence is a string of characters which under an interpretation is either true or false. An atomic sentence is sentence no sub-string string of which is a sentence". Or as text books tend to do like "If X is a sentential letter then X is an atomic sentence. If X is an atomic sentence the ~X is a compound sentence. If X and Y are sentences then X & Y is a sentence [etc. for ach logical connective. Then if t is an individual constant then t is a term. If T is an nary function followed by n terms then X is a term. If Y is an nary predicate letter then Y followed by n terms is an atomic sentence. Finally, if X is a sentence but not an atomic sentence then it a compound sentence." That's the sort of thing you get in text books; but is it best for an article here? Anyway we are only defining what is called an atomic sentence is our artificial languages. If a definition is both concise and comprehensible we can move on. There are to my mind more interesting things to talk about. E.g. were it not for this prolonged debate I was going to draft a Para along thee lines, under heading of "considerations" or the like.
All atomics sentences are mutually compatible; no atomic sentence entails any other atomic sentence or its negation. No atomic sentence or its negation is logically true. (cf Wittgenstein, TLP para 4.211 It is a sign of a proposition's being elementary that there can be no elementary proposition contradicting it ) Therefore Ma is not implied by Ba under any interpretation. But if we assign "is a bachelor" to B and "is a man to" to M and John to a; then "If John a bachelor then John is a man" is represented by "BA -> Ma". The latter is a contingency but the former is necessarily true. Therefore either "Ba -> MA" does not represent the underlying logical structure of "If John a bachelor then John is a man" or it is a necessary truth not provable in sentential Logic. Obviously the former is the case. Therefore "If John a bachelor then John is a man" although grammatical a simple not a compound sentence nevertheless has a compound logical structure, i.e. "If (John is a man and John is unmarried) then John is a man". So If X a simple sentence in English how can we tell whether its logical structure is nevertheless compound? Coding it up in symbolic logic will not tell us; the correct coding PRESUPPOSES we have the correct logical analysis. So the question remains, how do we tell if the underlying structure is simple or compound? The answer lies in our earlier point: no atomic sentence entails any other atomic sentence or its negation . (I.e. W's 4.211 point). If John is a bachelor entails any other logically atomic sentence then "John a bachelor" is not logically simple. So what about ""This is an electron" - simple or compound? Does it entail any logically simple sentence? Suppose the answer so far as we know is no. But tomorrow some sub-atomic particle is discovered so that "This is an electron" entails "This has three stubby particles" then "This is an electron" turns out to be logically compound not simple. But discovered empirically!. I am attempting to illustrate that we could we talking about more interesting things about atomic sentences than the best way to define them. —Preceding unsigned comment added by 82.27.226.211 (talk) 01:35, 21 February 2008 (UTC) —Preceding unsigned comment added by 82.27.226.211 (talk) 01:35, 21 February 2008 (UTC)

I agree there are interesting things this article can discuss. Personally, I will need time to look up references before I can add them, but I encourage other people to do so. I want to point out there are a couple technical issues with what you said;

• "n-place predicate letter followed by n individual symbols or functions." - how does R(f(g(x),h(y))) fit into this scheme? It is a unary relation symbol R followed by a single term f(g(x),h(y)).

R(f(g(x),h(y))) is an atomic formula if R is an n-place predicate followed by n terms
R is a one place predicate So R(f(g(x),h(y))) is an atomic formula if f(g(x),h(y)) is one term
f(g(x),h(y)) is term if it is an m-place function letter followed by m terms
f is a two place function letter so f(g(x),h(y)) is a term if g(x),h(y) are two terms g(x),h(y)) are two one place function letters each followed by one individual constants or variable Thefefore R(f(g(x),h(y))) is an atomic formula
R(f(g(x),h(y))) is an atomic sentence if it is an atomic formula containing no variables.
Therefore R(f(g(x),h(y))) is an atomic sentence just in case 'x' and 'y' are not variables
'x' and 'y' and normally used to represent variables rather than individual constants. On this assumption:
R(f(g(x),h(y))) is an atomic formula but it is not an atomic sentence since it contains the (unbound) variables 'x' and 'y'

--82.27.226.211 (talk) 02:42, 21 February 2008 (UTC) --82.27.226.211 (talk) 02:39, 21 February 2008 (UTC) --82.27.226.211 (talk) 02:34, 21 February 2008 (UTC)

• " An atomic sentence is sentence no sub-string string of which is a sentence" is simply not correct, as I explained above. The counterexample is in a signature with no constant symbols whatsoever.
  

I am not sure what you are arguing here. contains a quantifier; ergo it is not an atomic formula a fortiori it is not an atomic sentence.
--82.27.226.211 (talk) 02:39, 21 February 2008 (UTC) --82.27.226.211 (talk) 02:34, 21 February 2008 (UTC) PS waht is a "signature"?

I would like as much as everyone else to get into the article definitions that are at least technically correct, to move on to other issues. — Carl (CBM · talk) 01:51, 21 February 2008 (UTC)

The defeintions of an atomic formula are based on those of Mendelson (p 16 & 46) and Mates, the texts I cited. There is no one correct way of defining these terms, but all the definitions amount to the same thing in the end. Be happy. Philogo--82.27.226.211 (talk) 02:42, 21 February 2008 (UTC) --82.27.226.211 (talk) 02:39, 21 February 2008 (UTC)

See signature (mathematical logic). The point above is that in general has no subsentences, but is not an atomic sentence, so the lack of subsentences is not a defining characteristic. Would you mind to fix the issue with the current wording that you reintroduced today, that I pointed out above? In particular, it doesn't make sense to talk about free variables (or bound variables) in the context of sentential logic. — Carl (CBM · talk) 02:59, 21 February 2008 (UTC)

Re: Now I see an agree: it is not an atomic sentence and yet has no part which is a sentence Ergo we cannot define atomic sentence as a sentence lacking a sub-sentence. Re: fix the errors you pointed out. See propose new introduction below. Stay happy.--Philogo (talk) 13:13, 21 February 2008 (UTC)

[edit] Proposed New Introduction

I propose the following REVISED new introduction (that is (a) independent of the article on atomic formula and (b) addresses the issue of the equivalents of atomic sentences in natural languages):--Philogo (talk) 13:16, 22 February 2008 (UTC)

In Logic, sentences (that is declarative sentences, also variously called propositions or statements) are those strings of words or symbols which are either true or false, i.e. are truthbearers. The truth of some such sentences is a function of (is determined by) simpler sentences. (E.g. the truth of ‘John is Greek and John is Happy’ is a function of the truths of ‘John is Greek’ and ‘John is Happy’). The simplest kind of sentence, an elementary sentence, would not be a function of any logically simpler sentence, it is reasoned; other sentences built up from the elementary sentences using logical connectives such as and and or would be logically compound sentences, see Logical atomism.

Logic has developed artificial languages, for example sentential calculus and predicate calculus partly with the purpose of revealing the underlying logic of natural languages statements, the surface grammar of which may conceal the underlying logical structure; see Analytic philosophy. In these artificial languages an Atomic Sentence is a string of symbols which can represent an elementary sentence in a natural language, and it can be defined as follows.

In a formal language, a well formed formula (wff) is a string of symbols consituted in accordance with the rules of syntax of the language. A term is a variable, an individual constant or a n-place function letter followed by n terms. An atomic formula is an wff consisting of either a sentential letter or an n-place predicate letter followed by n terms. A sentence is a wff in which any variables are bound. An atomic sentence is an atomic formula containing no variables. It follows that an atomic sentence contains no logical connectives, variables or quantifiers.
--Philogo (talk) 14:18, 21 February 2008 (UTC)

I like that, modulo some minor copyediting. The last paragraph is redundant with the next-to-last paragraph, though; they could be merged or the last paragraph removed. — Carl (CBM · talk) 14:47, 21 February 2008 (UTC)<br /

Yes redundant, a bit of rhetorical repetition I suppose. How about Del from 1. thru 2. So the preceding definition says what an atomic sentence is and the last para says, by implication, what it is not? i.e. The last para would then read:

It follows that an atomic sentence contains no logical connectives, variables or quantifiers.

Now implemented in revised version above. --Philogo (talk) 02:58, 23 February 2008 (UTC) Couple of points for consideration.

1. A while back it was asked if the definition were intended to cover second order predicate logic. In so far as second order predicate logic introduces symbols for (a) predicate variables (b) predicate ‘‘‘quantifiers’’’ then any atomic formula which included either of them would not be an atomic sentence (under our definition) because, as we concluded, an atomic sentence contains no logical connectives, variables or quantifiers Intuitively I think our definition would be right to do so. E.g. consider (everything is true of a). This is logically equivalent to A1(a) & A2(a) & ... &An(a) for all the n predicates in the domain of R. So our definition rightly excludes it.
2. What about identity/equality. Does and should our definition exclude any atomic formulae containing (i) an identity/equality symbol as a logical constant or (ii) a predicate letter representing identity? By our definition I(alb) is and a=b is not an atomic sentence (because we allow in effect just sentential letters, predicate letters, individual constants and function letters, and "=" is none of these). But if, in an interpretation we associate I(_,_) with all those couples <x,y> such that x is identical with y or x is equal to y, then in that interpretation I(a,b) and a=b would mean the same thing and yet the former is and the latter is not an atomic sentence. That is paraodoxical is it not? User:Philogo|Philogo]] (talk) 21:33, 21 February 2008 (UTC)

Amed revision to proposed new introduction above--Philogo (talk) 13:16, 22 February 2008 (UTC)