Talk:Tautology (logic)

Origin of page

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)

Logical notation

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)

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)

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)

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

But Tautological arguments are circular!

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)

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)

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)

Section "Tautologies vs Validities" is logically inconsistent with the first section

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)

This Article Really Needs to be Re-written so it's Easier to Understand

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)

Flagged as lacking context

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, $(A\land B)\lor (\lnot A)\lor (\lnot B)$ 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)

It isn't "fancy talk" to use the correct terminology when describing something. There are wikilinks for propositional logic and valuation, and then the lede gives a concrete example with two propositional variables. There's no reason to avoid precision in the lede - we expect the reader to use links for terms that aren't already known. The "context" is established by "In propositional logic", so a reader who isn't already familiar knows to start by learning about that. — Carl (CBM · talk) 21:21, 22 October 2007 (UTC)

Rather than changing the lede, which is precise, clear, and provides a good summary of the overall article, it would be reasonable to expand the first section somewhat. But remember this is not a textbook; we don't need to give large numbers of examples, be overly informal, or build things up as might be done in a classroom. Wikipedia's best articles just state what's going on, provide one or two short examples, and then move along to other things. — Carl (CBM · talk) 21:31, 22 October 2007 (UTC)

Or, convert the example in the lead from mathematical notation to English. I don't think that using English would be any less precise and it would make the article more approachable. — DIEGO ^talk 22:31, 22 October 2007 (UTC)

A tautology is a formal expression, rather than a natural language expression. So it wouldn't be ideal to replace the example of a tautology something that isn't a tautology. That would be like removing "2 + 3 = 5" from equation and attempting to replace it with English. — Carl (CBM · talk) 22:43, 22 October 2007 (UTC)

Comparing "2 + 3 = 5" to "$(A\land B)\lor (\lnot A)\lor (\lnot B)$" is not really an apt comparison. The average Wikipedia reader understands the value of integers and meaning behind the symbols "+" and "=", so that equation would not need additional context to be understood in the lead of an article. If the definition of tautology absolutely requires the use of the formal expression, then at least explain the symbols used. You must understand that symbols like $\lor$ and $\lnot$ mean nothing to the average reader without additional context. I am aware of that including an English translation after the expression would be introducing a tautology to the definition of tautology. However, I think it would nonetheless make the meaning more clear and the article more accessible — DIEGO ^talk 02:30, 23 October 2007 (UTC)

While it may well be that the accessibility of this article is subject to some possible improvements, I wonder if you understand what it's actually about? There's a different article, tautology (rhetoric), that may treat the concept you're interested in. --Trovatore 02:43, 23 October 2007 (UTC)

Yes, I understand what it is about. When I wrote "I am aware of that including an English translation after the expression would be introducing a tautology to the definition of tautology", I meant to link the first "tautology" to Tautology (rhetoric) (it was a poor attempt at humor). Sorry to confuse matters. — DIEGO ^talk 02:48, 23 October 2007 (UTC)

A completely naive reader couldn't read the first paragraph of distributor cap with no background knowledge, or Scale (music). Similarly, there is no expectation they can read the first paragraph here if they know nothing about propositional logic. That's why the article starts with "In propositional logic" - so the reader who doesn't have the background to read this article can find the right place to start. The lede is not meant to be completely self-sufficient; it's meant to be a concise summary of the article for a reader who is somewhat familiar with the background ideas in the area. Not every article about formal logic needs to explain what the $\lor$ symbol means. — Carl (CBM · talk) 03:20, 23 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:

1 "tautology: in logic, a statement so framed that it cannot be denied without inconsistency."

2 "In the propositional calculus, a logic in which whole propositions are related by such connectives as ⊃ (“if . . . then”), • (“and”), ∼ (“not”), and ∨ (“or”), even complicated expressions such as [(A ⊃ B)•(C ⊃ ∼B)] ⊃ (C ⊃ ∼A) can be shown to be tautologies by displaying in a truth table every possible combination of T (true) and F (false) of its arguments A, B, C and after reckoning out by a mechanical process the truth-value of the entire formula, noting that, for every such combination, the formula is T."

3 Wittgenstein apparently "argued in the Tractatus Logico-Philosophicus (1921) that all necessary propositions are tautologies" and extended this via predicate calculus [?] to the notion of validity. Carnap amended this: "The Logical Positivists held that, in general, every necessary truth (and, thus, every tautology) is derivable from some rule of language; its only necessity is its being prescribed by a rule in a certain system."

Bill Wvbailey 16:36, 23 October 2007 (UTC)

A note at the top to point to tautology (rhetoric) would be helpful. That article should point out the relationship with analytic truth. In mathematics, we don't worry about such things very often, but here's an overview of the situation as I understand it.

The philosophers like Kant who worried about such things distinguished analytical truths ("all bachelors are unmarried") from synthetic truths ("all bachelors are lonely"). The former are (kind of) what the Greeks had called tautologies - they are true based only on the meanings of the words involved, without any need for additional facts. Synthetic truths, on the other hand, require some knowledge or experience to verify.

Wittgenstein argued that the truths of mathematics are tautologous, based on some prior meaning he had of the word tautologous. But others began to take that as the definition of mathematical truth, and thus lose the former meaning of tautology.

The contemporary use of the term tautology in logic is entirely about truth, in the form of validity. Thus a statement of propositional logic is a tautology exactly when it is "logically valid". These are now synonymous concepts, and either one can be used to define the other. The fact that tautologies are verifiable (provable, if you want) is important, but not part of the definition of a tautology.

There is an interesting article that has more info on the development of the word tautology: Dreben, Burton, and Floyd, Juliet, “Tautology: How Not To Use A Word,” Synthese 87, 23-49, 1991. — Carl (CBM · talk) 17:14, 23 October 2007 (UTC)

Improvements made?

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)

Fixing the disambiguation problem was easy, so I have done that. As a vague explanation, a tautology is a statement (i.e., in the context of this article, a statement in propositional logic) that is true "in every possible world", for some exact definition of "possible worlds". The negation of a tautology is another statement, and of course it is false "in every possible world". It seems that some people call such a statement a contradiction.

I agree that this article isn't very good, neither for experts nor for laypeople. --Hans Adler (talk) 17:27, 18 June 2008 (UTC)

Thanks for that, it's a bit clearer now. I think another problem is putting equations in the lead. When Stephen Hawking was writing A Brief History of Time his editor told him that every equation would halve his readership. so the book ended up with only one - E = MC². I think would be better keep them out of the lead and only introduce them once the terms have been explained. Those who are trying to make this article simpler have my admiration, but I'm afraid it's got some way to go before the layman has any chance of understanding it - if that's ever going to be possible. Richerman (talk) 22:37, 18 June 2008 (UTC)

There are no equations in the lede, just examples of tautologies. If you don't want to be confronted with formulas of propositional logic, then you are probably reading the wrong article because you won't be interested in any tautologies in the sense of this article. Were you looking for tautology (rhetoric)? I just noticed that there is a disambiguation notice missing; I will add it now. --Hans Adler (talk) 23:33, 18 June 2008 (UTC)

Actually that notice shouldn't be there, because the title is already disambiguated. --Trovatore (talk) 23:45, 18 June 2008 (UTC)

Oops, they're not equations are they - just rather unfamiliar symbols. I got to this page via a link from another article and I think it should have gone to the rhetoric page but someone probably hadn't checked the link. Problem is I can't remember which article I was reading so I can fix the link. I'll have to check "what links here." Still, I've learned a bit about something I knew nothing about - although probably just enough to know I'll never understand it! Still, maybe with some more work on the article by you guys I will one day. Thanks for your patience. Richerman (talk) 00:10, 19 June 2008 (UTC)

Oh, and as there are only two articles on rhetoric I think a reference to the other article at the top of each page is very useful, even if it's not strictly necessary. Anything that makes things clearer is good in my book. Richerman (talk) 00:19, 19 June 2008 (UTC)

Yes, if disambiguation notes on disambiguated articles are strictly forbidden, I would consider that a rule that needs changing, or at least ignoring per WP:IAR. – I have just gone through "what links here" and corrected some of the most obvious (and easy to fix) inappropriate links. --Hans Adler (talk) 00:42, 19 June 2008 (UTC)

Oh, I don't know that they're strictly forbidden -- personally, though, I routinely remove them, on the grounds that they're useless. Why would a person wind up at a page with a disambiguated title, when looking for a different page? If you think that this particular page is especially subject to that, then maybe the parenthetical (logic) is not specific enough, and we should look for a better disambiguating word or phrase. --Trovatore (talk) 03:43, 19 June 2008 (UTC)

Well in this case I ended up there from a bad link from British English (I've fixed it now) and then assumed that the logic in the title referred to logical thought rather than mathematical logic - especially as some philosophers were mentioned. However you can also inadvertantly bypass the DAB page by getting the title wrong slightly, doing a search and making the wrong choice from it. I can't see that a line of blue text is particularly ugly, and they are quite often helpful. It can also just say "this article is about X, for other uses see disambiguation page" Richerman (talk) 11:42, 19 June 2008 (UTC)

list of major tautologies

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 do not think this is a good idea. This is an encyclopedia article, not a table from a handbook. We discuss the concept and give a few examples, not a huge list of them. --Trovatore (talk) 01:09, 25 July 2008 (UTC)

How would you feel of another article, "list of propositional tautologies"? It would need to have a clearly limited scope, of course, but I think that if we limit it to tautologies that are given a specific name by a reliable source, that will limit it enough. I think the table may be of interest as a "lookup table", like the CRC manuals are. — Carl (CBM · talk) 13:34, 25 July 2008 (UTC)

While there might be some value in cataloging the common formulas, this seems a futile effort, like counting angels dancing on heads of pins. Such a cataloging appears in Reichenbach 1947 p. 38-39 (there are about 50 of them, with names). Wouldn't it make more sense to show how (or at least point out that) one can create all the tautologies starting with the assertion of "TRUTH". For example, A V ~A is TRUE for all evaluations of A. Ditto for (A & B) V ~(A & B), (~A & ~B) V (A V B), etc. There should be 2^(2^N) of these (reduced) formulas given the number of variables N. We can start with TRUTH and the various logical equivalences, for example A V TRUE = TRUE and A & A = A and A V FALSE = A, and use substitution and distribution and association etc to create formulas galore.

For us "visual" folks the way to proceed is with a Karnaugh map filled with T's. How many formulas can be devised to fill the map with ones? You can use disjunctive normal form to see that for e.g. a 3-variable map you need to OR together the 8 minterms to fill the map. But you can OR together various sub-groups of the minterms and use the well-known reduction methods to form simpler formulas ... there are (2^(2^3)) different formulas (that are in properly-reduced form). Bill Wvbailey (talk) 15:21, 25 July 2008 (UTC)

I would expect to find 50 to 100 named tautologies; that seems like a reasonable length for an independent article. This article does describe the substitution lemma, in section 6, which can be used to generate an infinite collection of tautologies starting with any one tautology. — Carl (CBM · talk) 17:11, 25 July 2008 (UTC)

I'm generally not a fan of articles of that sort; they seem more like something you'd find in a handbook than in an encyclopedia. I think I was the one who nominated table of divisors for deletion. But anyway I lost, so it appears that others don't agree with me on that. --Trovatore (talk) 17:23, 25 July 2008 (UTC)

I wouldn't object either. If someone starts this I could certainly check, footnote with alternative names, etc. Reichenbach has catagorised his list into 7 perhaps-useful groupings, too. Bill Wvbailey (talk) 19:02, 25 July 2008 (UTC) Actually 8, I missed one: TAUTOLOGIES IN THE CALCULUS OF PROPOSITIONS: (1) Concerning one proposition, (2) Sum, (3) Product, (4) Sum and Product, (5) Negaiton, product, sum, (6) Implication,m negation, product, sum, (7) equivalence, implication, negation, product, sum, (8) One-sided implications. Bill Wvbailey (talk) 20:13, 25 July 2008 (UTC)

Is this incorrect?

• $\lnot (A\land B)\Leftrightarrow (\lnot A\lor \lnot B)$ ("if not both A and B, then not-A or not-B", and vice versa), which is known as de Morgan's law.

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)

truth tables (propositional calculus)

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)

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)

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)

Does one enjoy being pejorative?

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)

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)

Better examples

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)

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)

Tautologies in FOPL

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 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)

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)

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)

Lede 2009-11-9

I undid some changes to the lede (from this to this). Here are some specific comments:

1. "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.
2. The thing about Wittgenstein is covered in the history section already
3. 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.
4. 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."
5. This article used Harvard referencing, so references should not be put into footnotes.

— Carl (CBM · talk) 13:59, 9 November 2009 (UTC)

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. [1] Hans Adler 14:37, 9 November 2009 (UTC)

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 [2] also maintain the distinction. Smullyan is particularly direct about it. — Carl (CBM · talk) 20:05, 9 November 2009 (UTC)

There are other problems with the previous text as well. For example, that text starts

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)

Moved from lede 2009-11-27

I moved this from the lede:

"Tautology is also a general term used for any logical equivalence. It is in this sense that rules of inference such as "P $\leftrightarrow$ (P $\lor$ P)" and "P $\leftrightarrow$ (P $\wedge$ P)" may be called "tautology." The truth of a tautology 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."

This text is problematic for several reasons:

1. I don't believe that "tautology" is a general term for logical equivalence
2. I don't believe that texts in logic actually confuse inference rules with tautologies
3. Tautologies are primarily important in propositional logic, where the term "atomic sentence" doesn't even make sense. But in the context of first order logic, I don't see how "atomic sentences" are directly relevant to the fact that $(\forall x\,P(x))\lor (\lnot \forall x\,P(x))$ is a tautology.

— Carl (CBM · talk) 19:19, 27 November 2009 (UTC)

RE #1: I agree with what you wrote. Just today, while suffering through editing on the Euler diagram page I re-read pp. 38-39 in Reichenbach 1947 Elements of Symbolic Logic Dover, NY ISBN: 0-486-24004-5. These pages include approx. 46 "Tautologies in the calculus of Propositions", and the last 13 of which are categorized as "One-sided implications", meaning that the major connective is implication, not equivalence. Here is the example from the Euler page: Here we see a tautology at the major connective, shown in the yellow column. Clearly this tautology has nothing whatever to do with logical equivalence as the minor connectives (& on the left side, ~ on the right side) yield different patterns of 1's and 0's:

The Truth Table demonstrates that the formula ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) is a tautology as shown by all 1's in yellow column..

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

RE #2: the use of tautologies in a deduction (i.e. "making an inference"). This is murky; my guess is most books are lousy at explaining the following in detail (derived from Reichenbach et. al.) In my understanding of this, really the only tool (besides substitution) available in an inference is modus ponens, and for this to be useful, the premises A and A-->B, both must be known to be true; if so then we can detach B as true. Now if the A-->B is a tautology, then we can easily detach B -- we only need to know that A is true, and because it is a tautology we know that A-->B is always true. But if A-->B is true for only some cases of A and B assignments then only in some of the cases (lines in the truth table corresponding to variable assignments) can the modus ponens be applied:

Rule of inference. (Modus ponens.) If 'a ⊃ b' is true, and 'a' is true, 'b' can be asserted. ¶ This rule is usually indicated by the schema:

a ⊃ b

a

_______

b

"The first two lines of [the schema] are called premises; the third line is called conclusion. ... This consideration ["separation" or "detaching"] makes clear the distinction between inference and implication. An implication is a statement; it is used for inferences, but it cannot take the place of an inference because an inference is a procedure not a statement. This procedure can be described only in a rule, formulated in the metalanguage, and symbolically expressed by a schema. The inferential implication ... on the onther hand, belongs to the object language and therefore cannot replace this schema. ¶ It is important to relize that the rule of inference holds for every sort of implication. In [example 3], [very complicated formulas in a modus ponens] the implication is a tautological implication; in example (2) we have used a connective implication which is not tautological. The rule holds, however even if we use a merely adjunctive implciation, as in the example:

If snow is white, then sugar is sweet.

Snow is white

_______

Sugar is sweet.

The conclusion is true, although there is no connection in the implication of the first premise, which holds only in the adjunctive sense [etc]." (pp.64-65)

RE 3: Did you mean "propositional" logic as opposed to "positional logic?" Certainly tautologies exist in relations between atomic sentences. When applied to "the higher calculus of functions" Reichenbach says has a discussion of this at pp. 226ff, including a "General definition of tautologies: A tautology is a true formula which contains no empirical constants, or a formula resultiong from such a formula by specialization. [etc]" I don't feel qualified to opine further about the use of tautologies in "the higher calculus".

Bill Wvbailey (talk) 20:22, 27 November 2009 (UTC)

Yes, I meant propositional logic, fixed. — Carl (CBM · talk) 20:39, 27 November 2009 (UTC)

I think Carl's recent edits are just fine. I sure appreciate any content that can be preserved rather than panned. I have seen "tautology" used as the name of a rule of inference in several places, specifically Irving Copi. "Tautology" is a general term for logical equivalences (but only ones that always turn out to be true.) This isn't the greatest use of the term, however it is prevalent. Stay cool Carl. Pontiff Greg Bard (talk) 01:01, 30 November 2009 (UTC)

Edits 2009-12-18

I am going to go through and cut down the lede again:

1. 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.
2. 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 2 n  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.

3. 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)

Tautology: notion of "inheritence", removal of notions "true" and "false"

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)

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)

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)

Tautology in English sentence form

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:

• 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)

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)

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)

Its nothing - "Content" free

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)

Tautology vs validity

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)

This really needs in-line citations

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)

Vpq and Opq?

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)

Third reference loops back to itself

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)

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)

Ambiguity

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 $\lnot (\exists xFx\land \lnot \exists xFx)$, but not $(\forall xFx)\to \lnot \exists x\lnot Fx$. 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)

Lead sentence is not supported by any sources

The lead sentence says: a tautology is a formula or assertion that is true in every possible interpretation.

By the revision on 5 September 2024‎ by @Dezaxa, "An interpretation is not an assignment of values to a variable."

However, going through every (english) source listed:

• Britanica: "In the propositional calculus, a logic in which whole propositions are related by such connectives as ⊃ (“if…then”), · (“and”), ∼ (“not”), and ∨ (“or”), even complicated expressions such as [(A ⊃ B) · (C ⊃ ∼B)] ⊃ (C ⊃ ∼A) can be shown to be tautologies by displaying in a truth table every possible combination of truth-values—T (true) and F (false)—of its arguments A, B, C and after reckoning out by a mechanical process the truth-value of the entire formula, noting that, for every such combination, the formula is T. "

• Mathworld: "A sentence whose truth table contains only 'T' is called a tautology."

• Kleene, S.C.: Mathematical Logic: "Without knowing the truth values of the prime components, we can nevertheless say that the composite formula is true. Such formulas are said to be valid, or to be identically true , or (after Wittgenstein 1921) to be tautologies (in, or of, the propositional calculus)."

• A Mathematical Introduction to Logic: "[...] Hence we are left with: ∅ |= τ iff every truth assignment (for the sentence symbols in τ ) satisfies τ . In this case we say that τ is a tautology (written |= τ )."

• Elements of Symbolic Logic: "Definition of tautologies. A tautology is a formula that is true whatever be the truth-values of the elementary propositions of which it is composed."

For all of the given definitions, the only requirement has to do with the truth values of a formula's (predicate) variables. In fact, many of these sources do not contain the word "interpretation" at all. It is for this reason that I will be removing the definition involving "interpretation" in the lead sentence. Farkle Griffen (talk) 01:50, 5 September 2024 (UTC)

@Farkle Griffen Thanks for trying to improve the article and for your comment here. I reverted the earlier change you made because it provided an incorrect statement of what an interpretation means. As to the definition of tautology, it is problematic, as I noted in the section called Ambiguity immediately above this one. The word is quite simply ambiguous. It has a narrow meaning, under which it is a logical truth of propositional logic, and a broad meaning under which it is a synonym for a logical truth or valid sentence and hence means true under all interpretations. Both are in common use. Many older logicians such as Russell, Tarski and Gödel preferred the broader use. The broad definition is also given in textbooks such as Shawn Hedman, A First Course in Logic, Oxford University Press (2004) page 63; and Wolfgang Rautenberg, A Concise Introduction to Mathematical Logic, Third edition, Springer (2010) page 64.

This twofold use of the term tautology is noted by John Corcoran, in the entry "Tautology" in The Cambridge Dictionary of Philosophy, Cambridge University Press (1995). He says: "In the broad sense considered here, a tautology is a proposition whose negation is a contradiction. Equivalently, a tautology is a proposition that is logically equivalent to the negation of a contradiction. There are many other formulations." Then later he says, "There is a special subclass of tautologies called truth-functional tautologies that are true in virtue of a special subclass of logical terms called truth-functional connectives (‘and’, ‘or’, ‘not’, ‘if’, etc.)." So, Corcoran is using 'tautology' in the broad sense, and 'truth-functional tautology' for the narrow sense. But as you show in the texts you quote, many authors restrict 'tautology' to the narrow sense and use 'valid sentence' or 'validity' or 'logical truth' for the broad sense.

For myself, I think it would be helpful if the article stated in the lede that this ambiguity exists. I suggest the following:

In mathematical logic, a tautology is a proposition that is true regardless of the interpretation of its component terms, with only the logical constants having a fixed meaning.

Logicians use the word tautology in two different but related senses. In a narrow sense, a tautology is a logical truth of propositional logic. Logical truths of first order and higher order logics are distinguished by being called valid formulas or validities. In a broad sense, a tautology is a proposition that is true under all interpretations, or that is logically equivalent to the negation of any contradiction. In this broad sense, all valid formulas are tautologies, and the tautologies of propositional logic are distinguished by being called truth-functional tautologies. Although this ambiguity is potentially confusing, both senses are in common use within textbooks.

Let me know what you think and I'll update it. Dezaxa (talk) 18:15, 5 September 2024 (UTC)

Okay, I understand a little better now. I think I have two main issues that should be dealt with first.

(1) The definitions shouldn't be combined. Trying to create one "umbrella definition" that encompases both terms will only make the meaning more ambiguous, and the whole article more convoluted. The lead paragraphs should be something more along the lines of:

"In mathematical logic, a tautology has two related, but distinct definitions. In propositional logic, a tautology is.... " (This should take about a paragraph, going over the meaning and history)

(New paragraph) "In first-order and higher-order logics, a tautology is..."

(2) Keep in mind that the majority of readers of this article will be highschoolers or undergraduates learning basic propositional logic for the first time. The term "interpretation" is confusing and more abstract than most of these readers will be able to handle. See WP:Technical. It either shouldn't be in the lead, or it needs a basic explanaion immediately after.

Apart from these I have a few weaker suggestions, that should be considered but are mostly just my personal opinion

• In the first instance where you use "proposition," it really should be "formula" or "propositional formula". A simple explanation could be included here, "...a propositional formula, (propositions connected by logical connectives like and, or, implies, etc)..."
• Sections of the article should be clear which definition they are using, saying "In the propositional meaning" or "In the first order meaning...", or some other indication.
• I would keep the first section(s) to the very basic concepts of propositional tautologies, like explanations of truth tables and basic logical connectives.

I would recommend keeping the more broad definition to a minimum in the lead, and dedicate a section later in the article re-introducing the term. Most of the sections near the beginning of the article should be focused on the propositional logic meaning for the reasons meantioned above.

I think this would help clarify the article Farkle Griffen (talk) 20:56, 5 September 2024 (UTC)

What you propose doesn't really work. The problem with the term tautology is not that it means one thing in propositional logic and another thing in quantifier logic. The problem is that logicians use it in two different ways. I can illustrate by reference to the following two formulas: 1. $\lnot (P\land \lnot P)$ and 2. $(\forall xFx)\rightarrow \lnot (\exists x\lnot Fx)$. Both are logical truths or validities of classical logic and so are true in all interpretations. The first is a formula of propositional logic and the second is a formula of first-order logic. Now some logicians use 'tautology' in a narrow sense and will say that 1 is a tautology and 2 is not. Both are validities, but 1 is a tautological validity, 2 is not. Other logicians use 'tautology' in a wide sense and will say that both are tautologies. 1 is a truth-functional tautology, 2 is a tautology but not a truth-functional one.

This is the ambiguity that needs to be communicated and I don't think there is a simple way to do it. The reader should be told that the term is ambiguous otherwise they will get confused when they read books and papers on the subject. Dezaxa (talk) 00:03, 7 September 2024 (UTC)

I've made a modest update to the lede and put the stuff about tautology being ambiguous between a broad and narrow use in the historical section. Also, for good measure, I added a small section on tautologies in non-classical logics. Dezaxa (talk) 01:50, 8 September 2024 (UTC)

Thank you. This is much more descriptive than the previous version. With that said, I do still have one small issue. I don't think the word "interpretation" should be used in the lead paragraph; at least, not without an explanation. It's a technical term, but has a very common-use definition too, which is far more broad than would be intended, and this is going ti confuse most readers who aren't familiar with logic.

And similarly with the term "logical constant". The article linked to is way too abstract to be helpful to any readers.

The first paragraph should be written as if you're writing to a 10-year-old.

But, again, I would say this is already an improvement. Farkle Griffen (talk) 05:10, 8 September 2024 (UTC)