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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)
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)
There appears to be an error in the first para: the logical notation reads 'x and not x' when it should in fact read 'x or not x'. Unfortunately, I don't know how to edit it! Nick Jones
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)
Incorrect. You are talking about a tautology in Rhetoric, which means a 'circular argument' in a derogatory way. A tautology in Logic is entirely different, and means an argument that is true from all avenues of approach. — Preceding
unsigned comment added by
174.63.125.98 (
talk)
01:59, 11 October 2011 (UTC)
Unfortunately, this is just an instance of a general problem across all specific scientific / formal disciplines in Wikipedia. When the subject matter is advanced or specific enough, it reads like gibberish to the layman. Heck, I took Symbolic Logic in college, got an A, and I still find certain passages of this article to be kind of difficult to read. There are ways of explaining things where you can use more mainstream vocabulary in such a way that the reader's task is easier. And yes, Tautology means something different in Logic vs. Rhetoric, which is why when you wiki "Tautology" you are presented with the various flavors. However, it's pretty obvious why those two concepts are related. In both cases, the "sentence" or "statement" that is a tautology that does not provide the listener with any additional information. Perhaps the main difference is the perceived tone of the 2 terms. In Rhetoric it is a negative thing because presumably the speaker is arguing for something and if the argument does not provide the listener with any additional information then his statements appear to have no value. In Logic, it is useful as a template that allows terms to be simplified or facilitates intermediate steps in a proof. For example, DeMorgan's lets you convert an "AND" statement into its logically equivalent "OR" statement. 2001:4898:80E8:0:0:0:0:7C7 ( talk) 20:58, 9 October 2015 (UTC)
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)
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)
Examples table
Would it be possible to add vertical lines to the example table to separate the columns? The column headings all seem to run together and it's difficult to tell where one statement ends and the next begins.
dcraig
17:48, 6 October 2007 (UTC)
re suggestion to define tautology as : "Any statement that logically proves itself to be true because its premise and its conclusion are the same." Tautologies (in Logic) do not have premises and conclusions, although the word tautology is used in a different way outside of Logic. You are thinking of arguments of the form P therefore P.
It is ARGUMENTS that have premises and conclusions, not statments be they tautologies or no. The premises and conclusion of an argument are statements. Arguments are not statements. However there is a statement corresponding to any argument known as the correspoding conditional(q.v.). The correspsondong conditional for an argument of form P thefore P wold be P->P and this indeed is a tautology.--
Philogo
14:02, 7 November 2007 (UTC)
I've reprinted this here so we can look at it:
In propositional logic, a tautology (from the Greek word ταυτολογία) is a sentence that is true in every valuation (also called interpretation) of its propositional variables, independent of the truth values assigned to these variables. For example, is a tautology, because any valuation either makes A and B both true, or makes one or the other false. According to Kleene (1967, p. 12), the term was introduced by Ludwig Wittgenstein (1921).
The negation of a tautology is a contradiction, a sentence that is false regardless of the truth values of its propositional variables, and the negation of a contradiction is a tautology. A sentence that is neither a tautology nor a contradiction is logically contingent. Such a sentence can be made either true or false by choosing an appropriate interpretation of its propositional variables.
The reason someone usually cares about a tautology is because of its use in deductive reasoning via e.g. modus ponens, i.e. the fundamental of the rules of inference. Perhaps move the fancy-talk about "valuation" and "interpretation" and "propositional variables" to later. Just start with the notion that, IF "statements" ("propositions", "formulas", etc) are connected in certain ways, for example modus ponens, THEN some constructions (e.g. modus ponens) can be shown to always yield "true" no matter the truth or falsity of the statements used in the construction -- these "well-constructed logical strings" are called "tautologies". Thus the notion of a tautology has to do with "immediate consequence", and not the truths of the "sentences" used in the construction. This has to do with the notion of "provability" as opposed to "truth".
Thus it's entirely possible to start with one falsehood ("pigs fly") or two falsehoods ("pigs fly", "pigs bring babies") and construct a "correct" argument.
The modus ponens argument is correctly-formed by virtue of its form and the notions of AND and IMPLY no matter whether or not we agree that "pigs fly" is FALSE, "pigs bring babies" is FALSE, "pigs don't fly" is TRUE, and "pigs are mammals" is TRUE. This is why arguers should always check first to see if the argument is "well-formed" (e.g. reducible to a tautology). Only then should they tackle the truth of the premises. Logical strings can be checked for tautology by use of truth tables -- the highlighted row on the left that corresponds to the "THEN" column is all T (true).
( A | & | ( A | → | B ) ) | => | B | ( A | & | ( A | → | B ) ) | => | B | ||
F | F | F | T | F | T | F | IF | ( pigs fly | & | (pigs fly | implies | pigs bring babies) ) | THEN | pigs bring babies | |
F | F | F | T | T | T | T | IF | ( pigs fly | & | (pigs fly | implies | pigs are mammals) ) | THEN | pigs are mammals | |
T | F | T | F | F | T | F | IF | ( pigs don't fly | & | ( pigs don't fly | implies | pigs bring babies) ) | THEN | pigs bring babies | |
T | F | T | T | T | T | T | IF | ( pigs don't fly | & | ( pigs don't fly | implies | pigs are mammals) ) | THEN | pigs are mammals |
Bill Wvbailey 21:08, 22 October 2007 (UTC)
Diego's confusion was my confusion. After a little definitional research in my trusty dictionary and Encyclopedia Britannica I realized that there are at least two different definitions of “tautology”. There is a third "issue" around "validity" (which I know little about) in the sense of "truth" and "falsity" as opposed to "provable". I recommend the page begin with a "disambiguation note" -- for tautology(rhetoric) and perhaps something for validity.
I also realized that this article is discussing the "Formalist" notion of a tautology. For instance, all 11 of Hilbert 1927’s logical axioms are tautologies. But a demonstration of this implies an “interpretation” of “0” and “1” or “T” and “F” as (the only) values to be substituted for the “propositions”, plus a description of the behavior of each logical sign (i.e. the relations indicated by the signs). I have to go away and mull this over some more.
Merriam-Webster's 9th Collegiate dictionary) defines tautology as "2: a tautologous statement", and then tautologous as "2: true by virtue of its logical form alone".
Encylopedia Britannica 2006 offers three distinctions, the last of which reintroduces "truth" and "falsity" via validity:
Bill Wvbailey 16:36, 23 October 2007 (UTC)
I have tried to add some things to make the article more accessible. I would appreciate any comments about whether this was successful. — Carl ( CBM · talk) 13:49, 24 October 2007 (UTC)
I'm afraid I can't even understand the lead although it may be that this will never be comprehensible to the layman because of all the technical terms needed to describe it. My fist problem is that "valuation" goes to a disambiguation page with three possibilities, and I've no idea which is the right one, and secondly most of the links are to pages that are equally incomprehensible. As an example of what I don't understand what does "A tautology's negation is a contradiction" mean? Richerman ( talk) 16:21, 18 June 2008 (UTC)
I am going to add this list of major tautologies.
Law of the Excluded Middle
[S ∨ [~S]] Law of Noncontradiction [~[S ∧ [~S]]]
Law of Identity
[S ⇔ S] Law of Double Negation [S ⇔ [~[~S]]] De Morgan's Law for Conjunction [[~[S ∧ T]] ⇔ [[~S] ∨ [~T]]] De Morgan's Law for Disjunction [[~[S ∨ T]] ⇔ [[~S] ∧ [~T]]] Law of Negation of the Conditional [[~[S ⇒ T]] ⇔ [S ∧ [~T]]] Commutative Law for Conjunction [[S ∧ T] ⇔ [T ∧ S]] Commutative Law for Disjunction [[S ∨ T] ⇔ [T ∨ S]] Commutative Law for the Biconditional [[S ⇔ T] ⇔ [T ⇔ S]] Associative Law for Conjunction [[[S ∧ T] ∧ U] ⇔ [S ∧ [T ∧ U]]] Associative Law for Disjunction [[[S ∨ T] ∨ U] ⇔ [S ∨ [T ∨ U]]] Distributive Law of Conjunction over Disjunction [[S ∧ [T ∨ U]] ⇔ [[S ∧ T] ∨ [S ∧ U]]] Distributive Law of Disjunction over Conjunction [[S ∨ [T ∧ U]] ⇔ [[S ∨ T] ∧ [S ∨ U]]] Distributive Law of Implication over Conjunction [[S ⇒ [T ∧ U]] ⇔ [[S ⇒ T] ∧ [S ⇒ U]]] Law of Contraposition [[S ⇒ T] ⇔ [[~T] ⇒ [~S]]] Law of Equivalence for the Conditional [[S ⇒ T] ⇔ [[~S] ∨ T]] Self-distributive Law for Conjunction [[S ∧ [T ∧ U]] ⇔ [[S ∧ T] ∧ [S ∧ U]]] Self-distributive Law for Disjunction [[S ∨ [T ∨ U]] ⇔ [[S ∨ T] ∨ [S ∨ U]]]
Self-distributive Law for the Conditional
[[S ⇒ [T ⇒ U]] ⇔ [[S ⇒ T] ⇒ [S ⇒ U]]] Law of Conjunctive Selection [[S ∧ T] ⇒ S] Law of Disjunctive Alternative [[[~S] ∧ [S ∨ T]] ⇒ T] Law of Disjunctive Implication [S ⇒ [S ∨ T]] Cyclical Law of Implication [S ⇒ [T ⇒ S]] Law of Joint Implication [[[S ∧ T] ⇒ U] ⇔ [S ⇒ [T ⇒ U]]]
Law of Denied Implication
[[[~T] ∧ [S ⇒ T]] ⇒ [~S]]
Law of Contradictory Implication
[[~S] ⇒ [S ⇒ T]]
Law of Conjunctive Implication
[S ⇒ [T ⇒ [S ∧ T]]]
Law of Biconditional Implication
[S ⇒ [T ⇒ [S ⇔ T]]]
Law of Contraposition for the Biconditional
[[S ⇔ T] ⇔ [[~S] ⇔ [~T]]] Law of Alternative Implication [[[S ⇒ T] ∧ [U ⇒ T]] ⇒ [[S ∨ U] ⇒ T]]
Disjunctive Law for Conditionals
[[S ⇒ T] ∨ [T ⇒ S]]
Transitive Law of Implication
[[[S ⇒ T] ∧ [T ⇒ U]] ⇒ [S ⇒ U]]
Transitive Law of the Biconditional
[[[S ⇔ T] ∧ [T ⇔ U]] ⇒ [S ⇔ U]]
-- Royalasa ( talk) 01:03, 25 July 2008 (UTC)
I am saying this because it only makes sense if you write the two forms differently, its not enough to say vice versa, or saying it this way can be misleading. See http://en.wikipedia.org/wiki/De_Morgan%27s_laws -James — Preceding unsigned comment added by 65.60.245.62 ( talk) 21:41, 21 March 2015 (UTC)
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)
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)
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)
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)
I undid some changes to the lede (from this to this). Here are some specific comments:
— Carl ( CBM · talk) 13:59, 9 November 2009 (UTC)
There are other problems with the previous text as well. For example, that text starts
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)
I moved this from the lede:
This text is problematic for several reasons:
— Carl ( CBM · talk) 19:19, 27 November 2009 (UTC)
Square # | Venn, Karnaugh region | x | y | z | (~ | (y | & | z) | & | (x | → | y)) | → | (~ | (x | & | z)) | ||
0 | x'y'z' | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | ||
1 | x'y'z | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | ||
2 | x'yz' | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | ||
3 | x'yz | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | ||
4 | xy'z' | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | ||
5 | xy'z | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | ||
6 | xyz' | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | ||
7 | xyz | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 |
Bill Wvbailey ( talk) 20:22, 27 November 2009 (UTC)
I am going to go through and cut down the lede again:
Also, the following is expanded on later in the article already; it does not need to be repeated in the lede
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: is not a tautology, although it is a true proposition of mathematics. The logicists would not argue that is a tautology in the sense of this article.
— Carl ( CBM · talk) 00:59, 19 December 2009 (UTC)
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)
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).
(2) In logic, a statement that is unconditionally true by virtue of its form alone; for example, "Socrates is either mortal or he's not." Adjective: tautologous or tautological.
Examples:
-- Born2cycle ( talk) 03:53, 3 January 2011 (UTC)
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.
The first sentence currently says:
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)
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)
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)
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)
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, is a tautology, as is , but not . 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)
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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)
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)
There appears to be an error in the first para: the logical notation reads 'x and not x' when it should in fact read 'x or not x'. Unfortunately, I don't know how to edit it! Nick Jones
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)
Incorrect. You are talking about a tautology in Rhetoric, which means a 'circular argument' in a derogatory way. A tautology in Logic is entirely different, and means an argument that is true from all avenues of approach. — Preceding
unsigned comment added by
174.63.125.98 (
talk)
01:59, 11 October 2011 (UTC)
Unfortunately, this is just an instance of a general problem across all specific scientific / formal disciplines in Wikipedia. When the subject matter is advanced or specific enough, it reads like gibberish to the layman. Heck, I took Symbolic Logic in college, got an A, and I still find certain passages of this article to be kind of difficult to read. There are ways of explaining things where you can use more mainstream vocabulary in such a way that the reader's task is easier. And yes, Tautology means something different in Logic vs. Rhetoric, which is why when you wiki "Tautology" you are presented with the various flavors. However, it's pretty obvious why those two concepts are related. In both cases, the "sentence" or "statement" that is a tautology that does not provide the listener with any additional information. Perhaps the main difference is the perceived tone of the 2 terms. In Rhetoric it is a negative thing because presumably the speaker is arguing for something and if the argument does not provide the listener with any additional information then his statements appear to have no value. In Logic, it is useful as a template that allows terms to be simplified or facilitates intermediate steps in a proof. For example, DeMorgan's lets you convert an "AND" statement into its logically equivalent "OR" statement. 2001:4898:80E8:0:0:0:0:7C7 ( talk) 20:58, 9 October 2015 (UTC)
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)
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)
Examples table
Would it be possible to add vertical lines to the example table to separate the columns? The column headings all seem to run together and it's difficult to tell where one statement ends and the next begins.
dcraig
17:48, 6 October 2007 (UTC)
re suggestion to define tautology as : "Any statement that logically proves itself to be true because its premise and its conclusion are the same." Tautologies (in Logic) do not have premises and conclusions, although the word tautology is used in a different way outside of Logic. You are thinking of arguments of the form P therefore P.
It is ARGUMENTS that have premises and conclusions, not statments be they tautologies or no. The premises and conclusion of an argument are statements. Arguments are not statements. However there is a statement corresponding to any argument known as the correspoding conditional(q.v.). The correspsondong conditional for an argument of form P thefore P wold be P->P and this indeed is a tautology.--
Philogo
14:02, 7 November 2007 (UTC)
I've reprinted this here so we can look at it:
In propositional logic, a tautology (from the Greek word ταυτολογία) is a sentence that is true in every valuation (also called interpretation) of its propositional variables, independent of the truth values assigned to these variables. For example, is a tautology, because any valuation either makes A and B both true, or makes one or the other false. According to Kleene (1967, p. 12), the term was introduced by Ludwig Wittgenstein (1921).
The negation of a tautology is a contradiction, a sentence that is false regardless of the truth values of its propositional variables, and the negation of a contradiction is a tautology. A sentence that is neither a tautology nor a contradiction is logically contingent. Such a sentence can be made either true or false by choosing an appropriate interpretation of its propositional variables.
The reason someone usually cares about a tautology is because of its use in deductive reasoning via e.g. modus ponens, i.e. the fundamental of the rules of inference. Perhaps move the fancy-talk about "valuation" and "interpretation" and "propositional variables" to later. Just start with the notion that, IF "statements" ("propositions", "formulas", etc) are connected in certain ways, for example modus ponens, THEN some constructions (e.g. modus ponens) can be shown to always yield "true" no matter the truth or falsity of the statements used in the construction -- these "well-constructed logical strings" are called "tautologies". Thus the notion of a tautology has to do with "immediate consequence", and not the truths of the "sentences" used in the construction. This has to do with the notion of "provability" as opposed to "truth".
Thus it's entirely possible to start with one falsehood ("pigs fly") or two falsehoods ("pigs fly", "pigs bring babies") and construct a "correct" argument.
The modus ponens argument is correctly-formed by virtue of its form and the notions of AND and IMPLY no matter whether or not we agree that "pigs fly" is FALSE, "pigs bring babies" is FALSE, "pigs don't fly" is TRUE, and "pigs are mammals" is TRUE. This is why arguers should always check first to see if the argument is "well-formed" (e.g. reducible to a tautology). Only then should they tackle the truth of the premises. Logical strings can be checked for tautology by use of truth tables -- the highlighted row on the left that corresponds to the "THEN" column is all T (true).
( A | & | ( A | → | B ) ) | => | B | ( A | & | ( A | → | B ) ) | => | B | ||
F | F | F | T | F | T | F | IF | ( pigs fly | & | (pigs fly | implies | pigs bring babies) ) | THEN | pigs bring babies | |
F | F | F | T | T | T | T | IF | ( pigs fly | & | (pigs fly | implies | pigs are mammals) ) | THEN | pigs are mammals | |
T | F | T | F | F | T | F | IF | ( pigs don't fly | & | ( pigs don't fly | implies | pigs bring babies) ) | THEN | pigs bring babies | |
T | F | T | T | T | T | T | IF | ( pigs don't fly | & | ( pigs don't fly | implies | pigs are mammals) ) | THEN | pigs are mammals |
Bill Wvbailey 21:08, 22 October 2007 (UTC)
Diego's confusion was my confusion. After a little definitional research in my trusty dictionary and Encyclopedia Britannica I realized that there are at least two different definitions of “tautology”. There is a third "issue" around "validity" (which I know little about) in the sense of "truth" and "falsity" as opposed to "provable". I recommend the page begin with a "disambiguation note" -- for tautology(rhetoric) and perhaps something for validity.
I also realized that this article is discussing the "Formalist" notion of a tautology. For instance, all 11 of Hilbert 1927’s logical axioms are tautologies. But a demonstration of this implies an “interpretation” of “0” and “1” or “T” and “F” as (the only) values to be substituted for the “propositions”, plus a description of the behavior of each logical sign (i.e. the relations indicated by the signs). I have to go away and mull this over some more.
Merriam-Webster's 9th Collegiate dictionary) defines tautology as "2: a tautologous statement", and then tautologous as "2: true by virtue of its logical form alone".
Encylopedia Britannica 2006 offers three distinctions, the last of which reintroduces "truth" and "falsity" via validity:
Bill Wvbailey 16:36, 23 October 2007 (UTC)
I have tried to add some things to make the article more accessible. I would appreciate any comments about whether this was successful. — Carl ( CBM · talk) 13:49, 24 October 2007 (UTC)
I'm afraid I can't even understand the lead although it may be that this will never be comprehensible to the layman because of all the technical terms needed to describe it. My fist problem is that "valuation" goes to a disambiguation page with three possibilities, and I've no idea which is the right one, and secondly most of the links are to pages that are equally incomprehensible. As an example of what I don't understand what does "A tautology's negation is a contradiction" mean? Richerman ( talk) 16:21, 18 June 2008 (UTC)
I am going to add this list of major tautologies.
Law of the Excluded Middle
[S ∨ [~S]] Law of Noncontradiction [~[S ∧ [~S]]]
Law of Identity
[S ⇔ S] Law of Double Negation [S ⇔ [~[~S]]] De Morgan's Law for Conjunction [[~[S ∧ T]] ⇔ [[~S] ∨ [~T]]] De Morgan's Law for Disjunction [[~[S ∨ T]] ⇔ [[~S] ∧ [~T]]] Law of Negation of the Conditional [[~[S ⇒ T]] ⇔ [S ∧ [~T]]] Commutative Law for Conjunction [[S ∧ T] ⇔ [T ∧ S]] Commutative Law for Disjunction [[S ∨ T] ⇔ [T ∨ S]] Commutative Law for the Biconditional [[S ⇔ T] ⇔ [T ⇔ S]] Associative Law for Conjunction [[[S ∧ T] ∧ U] ⇔ [S ∧ [T ∧ U]]] Associative Law for Disjunction [[[S ∨ T] ∨ U] ⇔ [S ∨ [T ∨ U]]] Distributive Law of Conjunction over Disjunction [[S ∧ [T ∨ U]] ⇔ [[S ∧ T] ∨ [S ∧ U]]] Distributive Law of Disjunction over Conjunction [[S ∨ [T ∧ U]] ⇔ [[S ∨ T] ∧ [S ∨ U]]] Distributive Law of Implication over Conjunction [[S ⇒ [T ∧ U]] ⇔ [[S ⇒ T] ∧ [S ⇒ U]]] Law of Contraposition [[S ⇒ T] ⇔ [[~T] ⇒ [~S]]] Law of Equivalence for the Conditional [[S ⇒ T] ⇔ [[~S] ∨ T]] Self-distributive Law for Conjunction [[S ∧ [T ∧ U]] ⇔ [[S ∧ T] ∧ [S ∧ U]]] Self-distributive Law for Disjunction [[S ∨ [T ∨ U]] ⇔ [[S ∨ T] ∨ [S ∨ U]]]
Self-distributive Law for the Conditional
[[S ⇒ [T ⇒ U]] ⇔ [[S ⇒ T] ⇒ [S ⇒ U]]] Law of Conjunctive Selection [[S ∧ T] ⇒ S] Law of Disjunctive Alternative [[[~S] ∧ [S ∨ T]] ⇒ T] Law of Disjunctive Implication [S ⇒ [S ∨ T]] Cyclical Law of Implication [S ⇒ [T ⇒ S]] Law of Joint Implication [[[S ∧ T] ⇒ U] ⇔ [S ⇒ [T ⇒ U]]]
Law of Denied Implication
[[[~T] ∧ [S ⇒ T]] ⇒ [~S]]
Law of Contradictory Implication
[[~S] ⇒ [S ⇒ T]]
Law of Conjunctive Implication
[S ⇒ [T ⇒ [S ∧ T]]]
Law of Biconditional Implication
[S ⇒ [T ⇒ [S ⇔ T]]]
Law of Contraposition for the Biconditional
[[S ⇔ T] ⇔ [[~S] ⇔ [~T]]] Law of Alternative Implication [[[S ⇒ T] ∧ [U ⇒ T]] ⇒ [[S ∨ U] ⇒ T]]
Disjunctive Law for Conditionals
[[S ⇒ T] ∨ [T ⇒ S]]
Transitive Law of Implication
[[[S ⇒ T] ∧ [T ⇒ U]] ⇒ [S ⇒ U]]
Transitive Law of the Biconditional
[[[S ⇔ T] ∧ [T ⇔ U]] ⇒ [S ⇔ U]]
-- Royalasa ( talk) 01:03, 25 July 2008 (UTC)
I am saying this because it only makes sense if you write the two forms differently, its not enough to say vice versa, or saying it this way can be misleading. See http://en.wikipedia.org/wiki/De_Morgan%27s_laws -James — Preceding unsigned comment added by 65.60.245.62 ( talk) 21:41, 21 March 2015 (UTC)
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)
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)
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)
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)
I undid some changes to the lede (from this to this). Here are some specific comments:
— Carl ( CBM · talk) 13:59, 9 November 2009 (UTC)
There are other problems with the previous text as well. For example, that text starts
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)
I moved this from the lede:
This text is problematic for several reasons:
— Carl ( CBM · talk) 19:19, 27 November 2009 (UTC)
Square # | Venn, Karnaugh region | x | y | z | (~ | (y | & | z) | & | (x | → | y)) | → | (~ | (x | & | z)) | ||
0 | x'y'z' | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | ||
1 | x'y'z | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | ||
2 | x'yz' | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | ||
3 | x'yz | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | ||
4 | xy'z' | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | ||
5 | xy'z | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | ||
6 | xyz' | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | ||
7 | xyz | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 |
Bill Wvbailey ( talk) 20:22, 27 November 2009 (UTC)
I am going to go through and cut down the lede again:
Also, the following is expanded on later in the article already; it does not need to be repeated in the lede
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: is not a tautology, although it is a true proposition of mathematics. The logicists would not argue that is a tautology in the sense of this article.
— Carl ( CBM · talk) 00:59, 19 December 2009 (UTC)
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)
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).
(2) In logic, a statement that is unconditionally true by virtue of its form alone; for example, "Socrates is either mortal or he's not." Adjective: tautologous or tautological.
Examples:
-- Born2cycle ( talk) 03:53, 3 January 2011 (UTC)
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.
The first sentence currently says:
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)
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)
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)
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)
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, is a tautology, as is , but not . 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)