Talk:Sentence (mathematical logic)

Needs references (Enderton? Smullyan?) Could mention sentences of infinitary or other non-first-order logic. Should terms such as "formula" be defined? --Trovatore 28 June 2005 18:39 (UTC)
 * I do need references. but it's a type of grammar, not a definition of a sentance in logic. so I have to oppose merge. I have a pretty good book by robert L. causey called "Logic, Sets, and Recursion"(2nd ed) that discusses this breifly.  I'm just really bad with doing references.
 * I am doing to make some tentative changes. Do change them back if they are not good. I will try to defend my changes. The issue here is that we cannot give an exact definition of sentence without first fixing a predicate logic. However, in linguistics, there are different kinds of langauges, yet there can be a satisfactory definition of a sentence. That is what I'm trying to do in my changes.DesolateReality 05:14, 9 June 2007 (UTC)
 * Looks good so far. However I think the following sentence needs some thought:
 * A sentence is distinguished from a formula as it assumes a fixed truth value given a structure of the predicate logic.
 * One problem is the word "assumes", which some readers may take to mean "makes an assumption" rather than "takes on". Another is "distinguished from a formula", when a sentence is a formula. But I think your general thrust is on-target. --Trovatore 07:27, 9 June 2007 (UTC)
 * Thank you Trovatore for the careful reading. Do you have suggestions on how I might improve the sentence (no pun intended!) in concern?DesolateReality 15:40, 10 June 2007 (UTC)
 * I've made the edits recently. it involved quite some rephrasing. do edit it at will!--DesolateReality 00:50, 11 June 2007 (UTC)
 * Looks good to me. --Trovatore 01:08, 11 June 2007 (UTC)

Plain English
In plain English, this sentence is interpreted to mean that every member of the structure concerned is the square of a member of that particular structure.

Em, this is plain English? --Abdull 19:03, 2 December 2007 (UTC)

intro para
In the follwing the first sentnce is clear, but the remainder not at all. I propose to delete all but first sentence. Any objections?

In mathematical logic, a sentence of a predicate logic is a well formed formula with no free variables. A sentence is viewed by some as expressing a proposition. It makes an assertion, potentially concerning any structure of L. This assertion has a fixed truth value with respect to the structure. In contrast, the truth value of a formula (with free variables) may be indeterminate with respect to any structure. As the free variables of a formula can range over several values (which could be members of a universe, relations or functions), its truth value may vary. --Philogo (talk) 22:54, 18 March 2009 (UTC)


 * I just rewrote that, perhaps its clearer now? linas (talk) 21:27, 13 June 2011 (UTC)
 * Well that response took a long time (two years)! Yes it seems to be clearer.  However I note it says A sentence can be viewed as expressing a proposition, something that may be true or false. and in the foreword it refers to the article Statement (logic).  The existence of propositions is disputed, and the usefullness of the concept of a statement as proposed by Strawson is not unverslly accepted.  The existence of meaningful declarative sentences is less often disputed, but whether they them selves are true or false (as opposed to utterances thereof) and whether they only in so far as thet make statments or express propositions  is debated.  The article on truthbearers explores these issues.  Technical terms like proposition and statement should be used with great care, or not at all.  &mdash; Philogos (talk)  21:43, 13 June 2011 (UTC)
 * The lede now says A sentence can be viewed as expressing a proposition, something that may be true or false. If there are such things as propositions surely a sentence in a formal language would only express one of them under an interpretation: an unintepreted sentence in a FL surely expresses nothing and is neither true nor false - it's just a string sf symbols.&mdash; Philogos (talk)  21:59, 13 June 2011 (UTC)


 * Err, it seems you know much more about the historical/philosophical side of this than I do; please strike or reword the part about propositions as desired. Myself, I mostly just understand things from the formal perspective. So, yes, a sentence is an 'uninterpreted sentence'; its just a string of symbols.  Perhaps the article could have two sections: an informal/historical discussion/introduction, and a formal-math definition? Myself, I just can't write the former. linas (talk) 17:48, 14 June 2011 (UTC)
 * Rather than A sentence can be viewed as expressing a proposition, something that may be true or false. would it not be more accurate to say "A sentence under an interpretation is either true or false." (no need to mention propositions or statements, or use the wooly can be viewed ) The information specified in the interpretation provides enough information to give a truth value to any atomic formula, after each of its free variables, if any, has been replaced by an element of the domain. (from Interpretation (logic))  19:50, 14 June 2011 (UTC) &mdash;  Philogos (talk)

Revision
Hello, This is a good article. I propose the following revisions of this section because Wikipedia mathematics guidelines state 1) "identify the nature of the entity being defined" for formulas, 2) "avoid contentless clichés" e.g. "In plain English" and "on the other hand." 3) avoid redundant and unnecessary symbols and words (Halmos, P.R. (1970), "How to Write Mathematics", Enseignements Mathématiques, 16: 123–152, doi:10.5169/seals-43857. Reprinted in ISBN 0821800558). Manual of Style/Mathematics

Example
For the interpretation of formulas, consider these structures: the positive real numbers, the real numbers, and complex numbers. The following example in first-order logic
 * $$\forall y \ \exists x \ (y=x^2)$$

a sentence. This sentence means that for every y, there is an x such that $y=x^{2}.$  This sentence is true for positive real numbers, false for real numbers, and true for complex numbers.

However, the formula


 * $$\exists x \ (y=x^2)$$

is a sentence because of the presence of the free variable y. For real numbers, this formula is true if we substitute (arbitrarily) $y=2,$ but is false if $y=-2.$

It is the presence of a free variable, rather than the inconstant truth value, that is important; for example, even for complex numbers, where the formula is always true, it is still not considered a sentence. Such a formula may be called a predicate instead..

TMM53 (talk) 07:08, 3 April 2023 (UTC)