Talk:Tautology (logic)
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[edit] 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)
[edit] 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
[edit] But Tautological arguments are circular!
This topic seems to be missing something to me. 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)
[edit] 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)
[edit] 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)
[edit] 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, 
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)
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- 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)
- 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)
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- Comparing "2 + 3 = 5" to "" 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
and 
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)
- Comparing "2 + 3 = 5" to "
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- 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)
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- 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 symbol means. — Carl (CBM · talk) 03:20, 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
- 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)
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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)
[edit] 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)
