Talk:Tautology (logic)

message: This page was created by cut-and-paste from tautology, followed by editing to cut down to the relevant part. See that article for history, and talk:tautology for prior discussions. --Trovatore 02:35, 24 March 2006 (UTC)
thread_title: Origin of page

message: this page makes use of a notational system for precisely representing logical concepts, but this system is not named or explicitly referred to, either in the text or in the "see also" section.  can someone please add this?  it would help me understand the article if i could first learn the symbol system. Eupedia 14:52, 6 May 2006 (UTC)
replies: text: I found it finally (logic symbols), and linked it in.  other logic pages might benefit from this, too, if anyone wants to volunteer :) [and thanks, Trovatore, for moving my comment to where it belonged.  i live, i learn] Eupedia 19:58, 11 May 2006 (UTC)
replies: text: It is too hard to understand the actual definition of tautology by using the logical symbols.  Try to make this article easier for beginers. (btw, I got an A in college logic).--Nick Dillinger 00:00, 21 May 2006 (UTC)
thread_title: Logical notation

message: This topic seems to be missing something. I recall from logic 101 that a tautological argument is one in which the conclusion is the same as the premise. In other words, it is the most basic form of circularity, proving nothing. Tautologies are frequently buried in an argument, requiring deep analysis in order to be exposed. If, logically, one can reduce any essential part of an argument to tautology, one has proven the argument to rely on circularity, making it dismissable. 
One could easily read this article and assume just the opposite, and need I point out how badly that could poison every discussion that relies on logic?
Sevenwarlocks 13:50, 27 September 2006 (UTC)
replies: text: One cannot easily read this article, at least not in its current state. The page probably looks rather incomprehensible and certainly needs to be remade to be more accessible to the average reader. Anyway, similarly to you, I've always seen tautology as a type of logical fallacy that utilizes circular argument by following this "X=Y therefore Y=X" line of reasoning where the conclusion, however disguised, is basically a slight variation of its own premise. Again, the article doesn't even touch the fallacy angle, doesn't provide a succinct definition of the term, and seems to be entirely self-absorbed with it own formulas that are too abstract, technical, and unwieldy to be of any value to the uninitiated crowd.  Rankiri (talk)  —Preceding comment was added at 20:07, 17 February 2008 (UTC)
thread_title:  But Tautological arguments are circular! 

message: In the first section of this article, it is asserted that "not a tautology is an inconsistency and not an inconsistency is a tautology". But the next section provides an example statement from predicate logic that is asserted to be a "validity", which is "not a tautology", yet implying that it is also "not an inconsistency". Obviously this is inconsistent with "not a tautology is an inconsistency"
This article needs to be clarified. Presumably "not a tautology is an inconsistency" does not hold for the definition of a tautology in "Predicate logic"? This needs to be stated clearly in the section dealing with Validities to avoid any confusion. It should also be stated clearly that we are now dealing with a definition of tautology that is fundamentally different from its definition in the section preceding it, and that any conclusions reached in preceding section do not apply to this section. Also, it would be helpful to know just what exactly the relationship is between tautology, validity and inconsistency in Predicate logic. 
--Barfly42 15:49, 4 December 2006 (UTC)
thread_title:  Section "Tautologies vs Validities" is logically inconsistent with the first section 

message: I contend that this article as it is written is extremely difficult to understand. It lacks any real context to allow someone with limited knowledge of the topic to gain even a basic understanding of it. It needs to be more clearly defined, use English more effectively, and employ linking that provides greater understanding of how tautologies have been employed, who employed them and why, and related links that provide for both more specific and general understanding. To this end, I make the following proposals: 
A More General Definition of Logical Tautology
The general definition needs to be a whole lot more simple and clear. The root origin of the word is fine, but the description that follows seems far too tautological to be properly understood by a layman. I suggest something along the lines of: 
"Any statement that logically proves itself to be true because its premise and its conclusion are the same."
 
Examples in English first, please  
While there is a content heading for examples of logical tautologies, one finds when one goes there only to find examples of Logical_symbols instead. I suggest some examples that make use of the English language. Either blend the two by using sentences and their corresponding symbols, or use sentences first, and demonstrate the symbols in the next section. Additionally, I noticed that some of the logic symbols clearly demonstrate syllogistic reasoning. It seems logical that syllogisms might be used here to better illustrate what those logical symbols actually mean.
Linking is Fundamental
There are few links, or references to any philosophical ideas, people, or movements, that are closely related to this topic. An example of a philosophical movement involved in the use of tautologies would be logical_positivism. Also there were movements that were more-or-less opposed to the use of tautologies in a philosophical framework, namely pragmatism. 
Two-way reference linking 
It seems to me that, if a reader looks up "tautology" and finds it too difficult to grasp, they might want to try backing up to understand it from a more general perspective first. As this is the case, it would be more useful if, in addition to the links already present, there were also links to more general topics in this field, such as philosophy, logic, and syllogisms, to name a few. 
Tanstaafl28 10:37, 29 September 2007 (UTC)
thread_title:  This Article Really Needs to be Re-written so it's Easier to Understand

message: why not add something about the tautology meaning that the last column in a truth table will be all trues if it is a tautology?
--Royalasa (talk) 01:07, 25 July 2008 (UTC)
replies: text: This one is certainly more reasonable than your other proposal. The only real issue with it is that it might be a little too specific to one form of calculation. Still, I think some language could be worked out that would be OK. --Trovatore (talk) 01:12, 25 July 2008 (UTC)
replies: text: This is already in the article - it explains truth tables in detail and says "Because each row of the final column shows T, the sentence in question is verified to be a tautology" just after the truth table example. — Carl (CBM · talk) 13:32, 25 July 2008 (UTC)
thread_title:  truth tables (propositional calculus) 

message: The article states: "without the pejorative connotations it originally enjoyed"
Anyway, I just thought this was sort of funny since people and things wouldn't normally enjoy being considered pejorative.  I know the context isn't literally that of the emotion of joy, but doesn't this usage of the word "enjoy" usually accompany a positive attribute?--210.172.229.198 (talk) 01:22, 19 August 2008 (UTC)
replies: text: The term has a traditional use for things that are not actually positive. See OED definition 4, second paragraph. But the article has since been changed to use less flowerly language. — Carl (CBM · talk) 21:08, 2 December 2008 (UTC)
thread_title:  Does one enjoy being pejorative? 

message: As this is a logic page, why not provide some examples that use real-life imagery instead of letters and abstract symbols, which might be meaningless to non-specialized students?M. Frederick (talk) 20:48, 2 December 2008 (UTC)
replies: text: The motivation for using formulas is that this article is intended to be about tautologies in propositional logic, not about English sentences that have the form of tautologies. Indeed, the article has a hatnote that says "This article is about a technical notion in formal logic." A non-specialized student might need to read about propositional logic in order to follow it. Unfortunately our article on propositional logic isn't very good, though. — Carl (CBM · talk) 21:19, 2 December 2008 (UTC)
thread_title: Better examples

message: This article seems to suggest Tautologies do not occur in FOPL.  I would consider FA v ~FA to be a tautlogy.  Am I wrong?--Philogo (talk) 01:13, 1 February 2009 (UTC)
replies: text: I looked at the article but on first sight didn't see what you mean. Can you be a bit clearer? By FOPL, do you mean first-order predicate logic? What does FA mean? Which section suggests FA v ~FA is not a tautology? --Hans Adler (talk) 01:21, 1 February 2009 (UTC)

text: Please see Tautology_(logic)#Tautologies_versus_validities_in_first-order_logic. the fundamental definition of tautology is in the context of propositional logic, but as that section explains it is sometimes extended to first-order logic. — Carl (CBM · talk) 03:36, 1 February 2009 (UTC)
replies: text: Tautologous sentences are true under every interpretation because of the arrangements of their truth functional connectives. Nontautologous valid sentences depend on their quantifiers to make them true under every interpretation. This is why there are no valid sentences of propositional logic that are not tautologies --they have no quantifiers.Pontiff Greg Bard (talk) 06:42, 1 March 2009 (UTC)
thread_title: Tautologies in FOPL

message: I undid some changes to the lede (from this to this). Here are some specific comments:
# "Tautologies" are first and foremost a concept in propositional logic, where there are no "atomic sentences". However, the new text had this definition: "In the formal language of mathematical logic a tautology is a formula within some formal system that is true under any possible interpretation." This defines the set of logical valid formulas, not the set of tautologies. These only coincide for propositional logic. 
# The thing about Wittgenstein is covered in the history section already
# Most of the stuff about first-order logic was copied from lower in the article. There is no reason to repeat it in the lede, but I added a sentence to summarize it. The lede is not meant to be a long discussion, just a short summary of what is covered later. 
# There is an effective method for deciding whether a first-order formula is a tautology, but not a procedure for deciding if a first-order formula is logically valid. So the following claim in the new text was false: "There is an effective method of identifying tautologies in propositional logic, but not in first-order logic."
# This article used Harvard referencing, so references should not be put into footnotes. 
— Carl (CBM · talk) 13:59, 9 November 2009 (UTC)
replies: text: Concerning tautology ≠ valid formula for propositional logic: That's what I learned in Freiburg, but I am not sure if the distinction is universally observed. At least Rautenberg doesn't distinguish.  Hans Adler 14:37, 9 November 2009 (UTC)
replies: text: I have the impression that Rautenberg is idiosyncratic there. Enderton and Kleene make the distinction and are already cited in the article, and I just checked that Shoenfield, Hinman, and Smullyan  also maintain the distinction. Smullyan is particularly direct about it. — Carl (CBM · talk) 20:05, 9 November 2009 (UTC)

text: In logic, a tautology (from the Greek word ταυτολογία) is a logical truth whose truth is entirely due to the meanings of the logical connectives it contains, and not at all to the meanings of any atomic sentences it contains. In the formal language of mathematical logic a tautology is a formula within some formal system that is true under any possible interpretation. 
Now, a tautology is not a "logical truth", it is a formula. Also, the phrase "formal language of mathematical logic" has no meaning; there are many formal languages that are studied in mathematical logic, but mathematical logic itself is not a formal language. — Carl (CBM · talk) 20:13, 9 November 2009 (UTC)
thread_title:  Lede 2009-11-9 

message: I am going to go through and cut down the lede again:
 This article is not about propositions in natural language or about "logical truths" in general. It is about formulas in formal languages that are true in every interpretation.  SO "One is one" is not a tautology at all in the sense of this article, because it is a natural language sentence rather than a well formed formula.  It is certainly a tautology (rhetoric), which is a different concept that this article does not treat. 
 The lede is supposed to be a short summary of the rest of the article. Adding more and more material to the lede, while not expanding on that material lower down, isn't any good. In particular, the following seems extraneous for the lede:
Tautologies do not exclude any logical possibilities, and are therefore sometimes said to be "empty" or "uninformative." Being tautologous or contradictory is a formal property, whereas being a law of logic is a non-formal metalogical property (i.e. "One is one" is a tautology, whereas "No tautology is a contradiction" is not a tautology.). A tautology is in a logical form such that all substitution instances of any variables in it will cause for it to be true. Every proposition in the same logical form as a tautology is also a tautology. Epistemically, every proposition that can be known to be true by purely logical reasoning is a tautology.
Also, the following is expanded on later in the article already; it does not need to be repeated in the lede
 Therefore the task of determining whether or not the formula is a tautology is a finite, mechanical one. One need only evaluate the truth value of the formula under each of its  interpretations. A tautology is a formula that can be checked to be true for all interpretations by a finite truth table method, whereas some logically valid formulas of predicate logic cannot be so checked. The central thesis of logicism is that every proposition of mathematics is a tautology.
I merged the stuff about interpretations into the appropriate section. I don't think that the article on tautologies is the place to talk about logicism anyway, but the sentence quoted just above is literally false. Example: 2 + 2 \not = 1 + 1 + 1  is not a tautology, although it is a true proposition of mathematics. The logicists would not argue that 2 + 2  \not= 1 + 1 + 1  is a tautology in the sense of this article. 
 Things such as "Tautologies do not exclude any logical possibilities, and are therefore sometimes said to be "empty" or "uninformative." seem to be about rhetorical tautologies, not about well-formed formulas. 
— Carl (CBM · talk) 00:59, 19 December 2009 (UTC)
thread_title:  Edits 2009-12-18 

message: Twice in disparate literature I've run into this, and I found it useful -- to be "(logically) tautologuos"  means to possess a structural property that is inherited under modus ponens and substitution. The first time I ran into this was in Emil Post's 1921 PhD thesis (see van Heijenoort 1967:264ff). The second, more accessible place, is in Nagel and Newman's 1958 Goedel's Proof pp. 109ff. Here, as does Post, N & N remove any appeal to "truth" and "falsity", i.e. any "interpretation", and move an explanation of the notion of tautology toward a more fundamental "structural" basis. They proceed by dividing the outcome of "an evaluation" of a propositional formula into two mutually exclusive and exhaustive classes K1 and K2, with the formal definition that any formula is defined as "tautologous" if its output always falls into class K1 no matter what the classes (i.e. either K1 or K2) its constituents come from (or fall into, cf p. 111). The fundamental outcome of this is that "the property of being a tautology is hereditary under the Rule of Detachment. (The proof that it is hereditary under the Rule of Substitution) will be left to the reader)" (page 113); N & N proceed to then provide a simple proof of the first assertion.  
When I finally understood this notion it really cleared up some lingering confusion about "tautologous": the notion of "logical tautology" has nothing to do with "truth" or "falsity" but only with a mechanistic property that becomes important/useful in substitution and detachment (and therefore in proof theory). I'd like to add this, or see it added, to the article but am uncertain how/if to proceed. It's rather abstract and requires understanding of the purposes/import of "substitution" and "modus ponens". I think this appeal to the "mechanical" would remove lingering confusions between tautology (logic) and [[tautology {rhetoric)]].Comments? Bill Wvbailey (talk) 00:53, 18 June 2010 (UTC)
replies: text: The definition you suggest has to be qualified. While it is necessary that a set of formulae be closed under uniform substitution (of atoms for arbitrary formulae) and modus ponens to be the set of classical tautologies, it is clearly not also sufficient. Here are three cases that show insufficiency. (1) The empty set is closed under substitution and modus ponens. (2) The entire language is closed under substitution and modus ponens. (3) Sets of formulae in "extended" (e.g. modal) languages may be closed under substitution and modus ponens and not contain precisely the classical tautologies. If we restrict ourselves to a classical propositional language then it should be remarked that the definition is only good in the non-limit cases (i.e. cases excluding (1) and (2)) due to a result of Post. Nortexoid (talk) 18:56, 18 June 2010 (UTC)

text: The semantic aspect that a tautology is true in every interpretation is indeed important. It's also important that, in propositional logic, it is possible to effectively determine whether a given formula is a tautology.  But in first-order logic, it is not possible to effectively determine whether a given formula is true in every interpretation, which leads to the different definition of tautologies in that context as a proper subset of the logical validities. — Carl (CBM · talk) 19:19, 18 June 2010 (UTC)
thread_title:  Tautology: notion of "inheritence", removal of notions "true" and "false" 

message: I suggest this article would be greatly improved if the scope was expanded slightly to include examples of logical tautologies in English sentence form.  We should have them somewhere in Wikipedia, and they clearly do not belong in Tautology (rhetoric), so I suggest there should be a short section of examples here.  After all the name of this article is Tautology (logic), not Tautology (propositional logic). 
Examples:
* http://grammar.about.com/od/tz/g/tautolterm.htm
* http://www.buzzle.com/articles/tautology-examples.html
* http://www.wordiq.com/definition/Tautology#Logical_tautologies
--Born2cycle (talk) 03:53, 3 January 2011 (UTC)
replies: text: This article is intended to be about the notion of tautology in formal or mathematical logic; it's related to the ordinary-English sense of the word tautology only by analogy.  I don't think anything outside of formal logic belongs in the article.  I might be persuaded that a move to tautology (formal logic)  is plausible, but I'm totally against expanding the scope of the article outside of that. --Trovatore (talk) 23:09, 3 January 2011 (UTC)

text: Restricting the discussion to Tautology in logic (rather than in rhetoric) as apparently intended by [User:Born2cycle|Born2cycle]]:  A perfectly legitimate  question is whether the term tuatology includes not only  (a) formula which is true in every possible interpretation (b) any sentence formed by the interpretation of such a formlula (or in other words any sentence whose logical form is such a formula). (c) any statement made by or any proposition meant by such a sentence.   If we refer to Wittgenstein TL_P, 4.46 in particular, he applies the term tautology to what he calls (in translation) propositions, and this term does not appear to be synonymous with "formula" (see eg 4.2).  It may be that the term came to be used by later authors to be applicable to formulae and exclude sentences/statements/propositions of tautologically form.  Some citations might settle the matter. — Philogos (talk)  19:58, 13 June 2011 (UTC)
thread_title:  Tautology in English sentence form 

message: From the article body - "In 1884, Gottlob Frege proposed in his Grundlagen that a truth is analytic exactly if it can be derived using logic. But he maintained a distinction between analytic truths (those true based only on the meanings of their terms) and tautologies (statements devoid of content)."
I recall something similar. Also from others, e.g., devoid of "information", devoid of saying anything about "the world", etc. 
*1. Does anyone know what German word Frege used for "content"? 
*2. Does anyone have sources for the many other "nothing" aspect of tautologies? 
*3. Does anyone know what Frege's working definition of "content" was, or literature to read about it? PPdd (talk) 05:10, 10 June 2011 (UTC)
thread_title:  Its [[nothing]] - "Content" free 

message: The first sentence currently says:
:In logic, a tautology (from the Greek word ταυτολογία) is a formula that is true in every possible interpretation.
I thought that was a "validity" rather than a "tautology".  My understanding of tautologies is that they're formulas that can be seen to be true in every interpretation without considering the quantifiers.  That is, they're validities of the propositional calculus, not just of the predicate calculus.
So for example "if there is at least one man, and every man is mortal, then some man is mortal" is a validity, but not a tautology.  On the other hand, "either every man is mortal, or not every man is mortal" is a tautology.
I thought this was the common usage in logic, or at least in mathematical logic.  Is that not so?  In any case, shouldn't we have at least a short article on the propositional-calculus sense, which is very much simpler than the predicate-calculus sense, and of importance in its own right? --Trovatore (talk) 21:35, 9 October 2015 (UTC)
thread_title:  Tautology vs validity 

message: As of 9-26-2018, there are literally two citations, and they're both in the three-sentence 'in natural language' section. I believe that this page was copied from another page, perhaps someone could go back there and copy the citations as well.  — Preceding unsigned comment added by WillEaston (talk • contribs) 02:49, 27 September 2018 (UTC)
thread_title:  This really needs in-line citations 

message: What does this mean - 'Tautology is sometimes symbolized by "Vpq", and contradiction by "Opq".'? If no defence is offered, I'll delete it. Note that in Polish V is used for verum, and O is used for falsum, but the p and q have no role in connection with those. 31.50.156.4 (talk) 17:08, 9 June 2019 (UTC)
thread_title:  Vpq and Opq? 

message: Under References, [3] just loops back to the same page.  — Preceding unsigned comment added by 81.204.149.15 (talk) 17:46, 11 January 2024 (UTC)
replies: text: The editor left the following note alongside the reference. "This is an Easter Egg. By referencing itself, the article becomes itself tautological."— Ineuw talk 12:43, 13 March 2024 (UTC)
thread_title:  Third reference loops back to itself 

message: This article has improved considerably in the last 10 years, but I still see a problem with the definition. A basic issue here is that textbooks of logic are not consistent in their use of the term 'tautology'. Some use it in a broad sense in such a way that it is synonymous with 'validity' or 'valid formula'. Others use it in a narrow sense to mean a logical truth of the propositional calculus. There are plenty of books on both sides, which is unfortunate, but a Wiki article should not try to hide this, but point it out. 
In the narrow sense, \lnot (P \land \lnot P) is a tautology, as is \lnot (\exists x Fx \land \lnot \exists x Fx), but not (\forall x Fx) \to \lnot \exists x \lnot Fx. In the broad sense, all three are tautologies. 
The lede of the article currently states that a tautology is a formula that is true in every interpretation. The concept of interpretation is commonly used in quantifier logic, so this definition suggests the broad sense of tautology. But further down the article says that not all logical validities are tautologies of first-order logic. As a result, the reader is likely to be confused. 
I think it would be better if the article stated that the term tautology is ambiguous between these meanings. For example, Hedman, "A First Course in Logic" and Rautenberg, "A Concise Introduction to Mathematical Logic" both use the broad definition. But Enderton,"Mathematical Introduction to Logic" and Hinman, "Fundamentals of Mathematical Logic" both use the narrow definition. Dezaxa (talk) 20:18, 12 May 2024 (UTC)
thread_title:  Ambiguity